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(0, 1)-Matrix
A (0; 1)/-INTEGER MATRIX, i.e., a matrix each of whose elements is 0 or 1, also called a binary matrix. The numbers of binary matrices with no adjacent 1s (in either columns or rows) for n 0 1, 2, ..., are given by 2, 7, 63, 1234, ... (Sloane’s A006506). For example, the binary matrices with no adjacent 1s are ! ! ! ! 01 00 00 00 ; ; ; 00 10 01 00 ! ! ! 10 10 01 ; ; ; 01 00 10 These numbers are closely related to the HARD The numbers of binary matrices with no three adjacent 1s for , 2, ..., are given by 2, 16, 265, 16561, ... (Sloane’s A050974).
SQUARE ENTROPY CONSTANT....
References
Daiev, V. "Problem 636: Greatest Divisors of Even Integers." Math. Mag. 40, 164 Á/165, 1967. Guy, R. K. "Residues of Powers of Two." §F10 in Unsolved Problems in Number Theory, 2nd ed. New York: SpringerVerlag, p. 250, 1994. Montgomery, P.-L. "New solution to 2^n 0 0 3 (mod n)." NMBRTHRY@listserv.nodak.edu posting, 24 Jun 1999. Sloane, N. J. A. Sequences A036236 and A050259 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/ eisonline.html. Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, pp. 41 Á/ 44, 1986....
(P.-L. Montgomery 1999). In general, the least satisfying 2n
k (mod n) for k 0 2, 3, ... are n 0 3, 4700063497, 6, 19147, 10669, 25, 9, 2228071, ... (Sloane’s A036236). See also 1, BINARY, 3, RULER FUNCTION, SQUARED, TWO-EARS THEOREM, TWO-FORM, TWO-GRAPH, TWOSC A L E EXPANSION, TW O- S HEETED HYPERBOLOID, ZERO...
Hurd, S. and Trautman, D. "The Knight’s Tour on the 15Puzzle." Math. Mag. 66, 159 Á/166, 1993. Johnson, W. W. "Notes on the ‘15 Puzzle. I."’ Amer. J. Math. 2, 397 Á/399, 1879. Kasner, E. and Newman, J. R. Mathematics and the Imagination. Redmond, WA: Tempus Books, pp. 177 Á/180, 1989. Kraitchik, M. "The 15 Puzzle." §12.2.1 in Mathematical Recreations. New York: W. W. Norton, pp. 302 Á/308, 1942. Liebeck, H. "Some Generalizations of the 14 Á/15 Puzzle." Math. Mag. 44, 185 Á/189, 1971. Loyd, S. Mathematical Puzzles of Sam Loyd, Vol. 1. New York: Dover, pp. 19 Á/20, 1959. Loyd, S. Jr. Sam Loyd’s Cyclopedia of 5,000 Puzzles, Tricks, and Conundrums. Lamb Pub., 1993. Mallison, H. V. "An Array of Squares." Math. Gaz. 24, 119 Á/ 121, 1940. Sloane, N. J. A. Sequences A046164 in "An On-Line Version of the Encyclopedia of Integer Sequences." http://www.research.att.com/~njas/sequences/eisonline.html. Spitznagel, E. L. Jr. Selected Topics in Mathematics. New York: Holt, Rinehart and Winston, pp. 143 Á/148, 1971. Spitznagel, E. L. Jr. "A New Look at the Fifteen Puzzle." Math. Mag. 40, 171 Á/174, 1967. Steinhaus, H. Mathematical Snapshots, 3rd ed. New York: Dover, pp. 14 Á/16, 1999. Story, W. E. "Notes on the ‘15 Puzzle. II."’ Amer. J. Math. 2, 399 Á/404, 1879. Whipple, F. J. W. "The Sign of a Term in the Expansion of a Determinant." Math. Gaz. 13, 126, 1926. Wilson, R. M. "Graph Puzzles, Homotopy, and the Alternating Group." J. Combin. Th. Ser. B 16, 86 Á/96, 1974....
¨ A finite regular 4-D POLYTOPE with SCHLAFLI SYMBOL f3; 4; 3g: Coxeter (1969) gives a list of the VERTEX positions. The EVEN coefficients of the/D4/ lattice are 1, 24, 24, 96, ... (Sloane’s A004011), and the 24 shortest vectors in this lattice form the 24-cell (Coxeter 1973, Conway and Sloane 1993, Sloane and Plouffe 1995). The 24-cell is self-dual, and is the unique regular convex POLYCHORON which has no direct 3-D analog....
36 Officer Problem
How can a delegation of six regiments, each of which sends a colonel, a lieutenant-colonel, and major, a captain, a lieutenant, and a sub-lieutenant be ar-...
References
Riordan, J. Combinatorial Identities. New York: Wiley, p. 18, 1979. Roman, S. "The Abel Polynomials." §4.1.5 in The Umbral Calculus. New York: Academic Press, pp. 29 Á/0 and 72 Á/5, 1984....
Now, take y1) (3) minus y2) (2), y1 [yƒ 'P(x)y? 'Q(x)y2 ] (y2 [yƒ 'P(x)y? 'Q(x)y1 ] 00 2 2 1 1 (4) (y1 yƒ (y2 yƒ ) 'P(y1 y? (y? y2 ) 'Q(y1 y2 (y1 y2 ) 00 2 1 2 1 (y1 yƒ (y2 yƒ) 'P(y1 y? (y? y2 ) 00: 2 1 2 1 (5) (6)...
Abel’s Lemma
The pure equation x p 0C of PRIME degree p is irreducible over a FIELD when C is a number of the FIELD but not the p th POWER of an element of the FIELD. Jeffreys and Jeffreys (1988) use the term "Abel’s lemma" for another LEMMA related to ABEL’S UNIFORM CONVERGENCE TEST. See also ABEL’S IRREDUCIBILITY THEOREM, GAUSS’S POLYNOMIAL THEOREM, KRONECKER’S POLYNOMIAL ¨ THEOREM, SCHONEMANN’S THEOREM References
Dorrie, H. 100 Great Problems of Elementary Mathematics: ¨ Their History and Solutions. New York: Dover, p. 118, 1965....
(2) j zj20 z 2 : If the COMPLEX NUMBER is written z 0x 'iy; then the absolute square can be written (3) j x 'iyj20 x 2 'y 2 : An absolute square can be computed in terms of x and y using the Mathematica command ComplexExpand[Abs[z ]2, TargetFunctions- ! {Conjugate}]. An important identity involving the absolute square is given by...
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....
Now, we can identify the expression as consisting of three terms abody d2r ; dt 2 (12) (13) (14)...
where qn f (xn ; yn ): The method can then be extended to arbitrary order using the finite difference integration formula from Beyer (1987) fp dp 0 0 1 ' 1 9 ' 152 9 2 ' 3 9 3 ' 251 9 4 ' 29858 9 5 ' 19087 9 6 ' . . . fp 2 8 720 60480 (9) to obtain...
Adele ´
´ An element of an ADELE GROUP, sometimes called a REPARTITION in older literature (e.g., Chevalley 1951, ´ p. 25). Adeles arise in both NUMBER FIELDS and ´ FUNCTION FIELDS. The adeles of a NUMBER FIELD are Q the additive SUBGROUPS of all elements in kv ; where v is the PLACE, whose ABSOLUTE VALUE is B1 at all but finitely many v/s....
Adem Relations
Relations in the definition of a STEENROD which state that, for i B2j; Sq i ( Sq j (x) 0
ALGEBRA...
Affine Geometry
in which properties are preserved by from one PLANE to another. In an affine geometry, the third and fourth of EUCLID’S POSTULATES become meaningless. This type of GEOMETRY was first studied by Euler.
GEOMETRY PARALLEL PROJECTION...
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