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Numerals
((1, 0, 1)-Matrix
The number of distinct ((1; 0; 1)/-/n )n matrices (counting row and column permutations, the transpose, and multiplication by (1 as equivalent) having 2n different row and column sums for n 0 2, 4, 6, ... are 1, 4, 39, 2260, 1338614, ... (Kleber). For example, the 2 )2 matrix is given by ! (1 (1 ; 0 1 To get the total number from these counts (assuming that 0 is not the missing sum, which is true for n 5 10); multiply by (2n!)2 : In general, if an -matrix which has different column and row sums (collectively called line sums), then 1. n is even, 2. The number in f(n; 1 (n; 2 (n; . . . ; ng that does not appear as a line sum is either (n or , and 3. Of the largest line sums, half are column sums and half are row sums (Bodendiek and Burosch 1995, F. Galvin). See also ALTERNATING SIGN MATRIX, C -MATRIX, INTEGER MATRIX References
Bodendiek, R. and Burosch, G. "Solution to the Antimagic 0; 1; (1 Matrix Problem." Aufgabe 5.30 in Streifzuge ¨ durch die Kombinatorik: Aufgaben und Losungen aus ¨ dem Schatz der Mathematik-Olympiaden. Heidelberg, Germany: Spektrum Akademischer Verlag, pp. 250 Á/253, 1995....
((1, 1)-Matrix
See also HADAMARD MATRIX, INTEGER MATRIX References
´ Kahn, J.; Komlos, J.; and Szemeredi, E. "On the Probability that a Random 91 Matrix is Singular." J. Amer. Math. Soc. 8, 223 Á/240, 1995....
2
The number two (2) is the second POSITIVE INTEGER and the first PRIME NUMBER. It is EVEN, and is the only EVEN PRIME (the PRIMES other than 2 are called the ODD PRIMES). The number 2 is also equal to its FACTORIAL since 2! 02: A quantity taken to the POWER 2 is said to be SQUARED. The number of times k a given BINARY number bn Á Á Á b2 b1 b0 is divisible by 2 is given by the position of the first bk 01; counting from the right. For example, 12 0 1100 is divisible by 2 twice, and 13 0 1101 is divisible by 2 zero times. The only known solutions to the
n...
See also DOZEN, GROSS References
Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, 1986....
Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, p. 210, 1991....
See also 16-CELL, 24-CELL, 600-CELL, CELL, HYPERCUBE, PENTATOPE, POLYCHORON, POLYTOPE, SIMPLEX References
Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, p. 404, 1969. Coxeter, H. S. M. "Stellating ." §14.2 in Regular Polytopes, 3rd ed. New York: Dover, pp. 136 Á/137, 157, 264 Á/267, and 292, 1973....
Bold, B. Famous Problems of Geometry and How to Solve Them. New York: Dover, p. 70, 1982. Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, 1969. De Temple, D. W. "Carlyle Circles and the Lemoine Simplicity of Polygonal Constructions." Amer. Math. Monthly 98, 97 Á/108, 1991. Dickson, L. E. "Constructions with Ruler and Compasses; Regular Polygons." Ch. 8 in Monographs on Topics of Modern Mathematics Relevant to the Elementary Field (Ed. J. W. A. Young). New York: Dover, pp. 352 Á/386, 1955. Dixon, R. Mathographics. New York: Dover, p. 53, 1991. Hermes, J. "Ueber die Teilung des Kreises in 65537 gleiche Teile." Nachr. Konigl. Gesellsch. Wissensch. Gottingen, ¨ ¨ Math.-Phys. Klasse , pp. 170 Á/186, 1894....
where W0 is a constant of integration. See also ORDINARY DIFFERENTIAL EQUATION–SECONDORDER References
Boyce, W. E. and DiPrima, R. C. Elementary Differential Equations and Boundary Value Problems, 4th ed. New York: Wiley, pp. 118, 262, 277, and 355, 1986....
Abhyankar’s Conjecture
For a FINITE GROUP G , let p(G) be the SUBGROUP generated by all the SYLOW P -SUBGROUPS of G . If X is a projective curve in characteristic p ! 0, and if x0 ; ..., xt are points of X (for t ! 0), then a NECESSARY and SUFFICIENT condition that G occur as the GALOIS GROUP of a finite covering Y of X , branched only at the points x0 ; ..., xt ; is that the QUOTIENT GROUP G=p(G) has 2g ' t generators. Raynaud (1994) solved the Abhyankar problem in the crucial case of the affine line (i.e., the projective line with a point deleted), and Harbater (1994) proved the full Abhyankar conjecture by building upon this special solution....
Absolutely Fair
A sequence of random variates X0 ; X1 ; ... is called absolutely fair if for n 0 1, 2, ..., (X1 ) 00 and (Xn'1 ½X1 ; . . . ; Xn ) 00 (Feller 1971, p. 210). See also MARTINGALE References
Feller, W. An Introduction to Probability Theory and Its Applications, Vol. 2, 3rd ed. New York: Wiley, 1971....
An abundant number is an INTEGER n which is not a PERFECT NUMBER and for which s(n) s(n) (n > n; (1) where s(n) is the DIVISOR FUNCTION. The quantity s(n) (2n is sometimes called the ABUNDANCE. The first few abundant numbers are 12, 18, 20, 24, 30, 36, ... (Sloane’s A005101). Abundant numbers are sometimes called EXCESSIVE NUMBERS. There are only 21 abundant numbers less than 100, and they are all EVEN. The first ODD abundant number is 945 0 3 3 × 7 × 5: That 945 is abundant can be seen by computing (2)...
where fn (x) is an EXPONENTIAL POLYNOMIAL. The actuarial polynomials are given in terms of the EXPONENTIAL POLYNOMIALS fn (x) by a (b) (x) 0 (1 ( t)b fn ((x) n
n Xb 0 f (k) ((x): n k k00...
Addition Chain
An addition chain for a number n is a SEQUENCE 1 0 a0 Ba1 B. . . B ar 0n; such that each member after a0 is the SUM of two earlier (not necessarily distinct) ones. The number r is called the length of the addition chain. For example, 1; 1 '1 02; 2 '2 04; 4 '2 06; 6 '2 08; 8 '6 014 is an addition chain for 14 of length r 0 5 (Guy 1994). See also BRAUER CHAIN, HANSEN CHAIN, SCHOLZ CONJECTURE References
Guy, R. K. "Addition Chains. Brauer Chains. Hansen Chains." §C6 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 111 Á/13, 1994....
References
Hinden, H. J. "The Additive Persistence of a Number." J. Recr. Math. 7, 134 Á/35, 1974. Sloane, N. J. A. Sequences A006050/M4683 and A031286 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/ sequences/eisonline.html. Sloane, N. J. A. "The Persistence of a Number." J. Recr. Math. 6, 97 Á/8, 1973. Weisstein, E. W. "Integer Sequences." MATHEMATICA NOTEBOOK INTEGERSEQUENCES.M....
Adjacent Value
The value nearest to but still inside an inner References
Tukey, J. W. Explanatory Data Analysis. Reading, MA: Addison-Wesley, p. 667, 1977.
FENCE....
See also ADJOINT, COMPLEX CONJUGATE, DAGGER, HERMITIAN MATRIX, SCHUR DECOMPOSITION, TRANSPOSE...
Affine Group
The set of all nonsingular AFFINE TRANSFORMATIONS of a TRANSLATION in SPACE constitutes a GROUP known as the affine group. The affine group contains the full linear group and the group of TRANSLATIONS as SUBGROUPS. See also AFFINE COMPLEX PLANE, AFFINE CONNECTION, AFFINE EQUATION, AFFINE GEOMETRY, AFFINE HULL, AFFINE PLANE, AFFINE SPACE, AFFINE TRANSFORMATION, AFFINITY References
Birkhoff, G. and Mac Lane, S. A Survey of Modern Algebra, 5th ed. New York: Macmillan, p. 237, 1996....
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