positive%20definite
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1: 26.12 Plane Partitions
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►A plane partition, , of a positive integer , is a partition of in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns.
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26.12.3
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26.12.9
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26.12.13
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26.12.14
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2: 27.2 Functions
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►Functions in this section derive their properties from the fundamental
theorem of arithmetic, which states that every integer can be represented uniquely as a product of prime powers,
…Euclid’s Elements (Euclid (1908, Book IX, Proposition 20)) gives an elegant proof that there are infinitely many primes.
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►the sum of the th powers of the positive integers that are relatively prime to .
…This is the number of positive integers that are relatively prime to ; is Euler’s totient.
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►and if is the smallest positive integer such that , then is a primitive root mod .
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3: 35.3 Multivariate Gamma and Beta Functions
4: 13.30 Tables
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Slater (1960) tabulates for , , and , 7–9S; for and , 7D; the smallest positive -zero of for and , 7D.
Abramowitz and Stegun (1964, Chapter 13) tabulates for , , and , 8S. Also the smallest positive -zero of for and , 7D.
Zhang and Jin (1996, pp. 411–423) tabulates and for , , and , 8S (for ) and 7S (for ).
5: 35.5 Bessel Functions of Matrix Argument
6: 35.1 Special Notation
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complex variables. | |
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positive integer. | |
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space of positive-definite real symmetric matrices. | |
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is positive definite. Similarly, is equivalent. | |
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7: 21.1 Special Notation
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positive integers. | |
( times). | |
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complex, symmetric matrix with strictly positive definite, i.e., a Riemann matrix. | |
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intersection index of and , two cycles lying on a closed surface. if and do not intersect. Otherwise gets an additive contribution from every intersection point. This contribution is if the basis of the tangent vectors of the and cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is . | |
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8: Bibliography G
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Positive sums of the classical orthogonal polynomials.
SIAM J. Math. Anal. 8 (3), pp. 423–447.
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Algorithm 831: Modified Bessel functions of imaginary order and positive argument.
ACM Trans. Math. Software 30 (2), pp. 159–164.
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Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments.
ACM Trans. Math. Software 30 (2), pp. 145–158.
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Algorithm 939: computation of the Marcum Q-function.
ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
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Definite integrals of the complete elliptic integral
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J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.
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9: Bibliography
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Exact linearization of a Painlevé transcendent.
Phys. Rev. Lett. 38 (20), pp. 1103–1106.
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On the degrees of irreducible factors of higher order Bernoulli polynomials.
Acta Arith. 62 (4), pp. 329–342.
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Repeated integrals and derivatives of Bessel functions.
SIAM J. Math. Anal. 20 (1), pp. 169–175.
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Table of Definite and Infinite Integrals.
Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.
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Positive Jacobi polynomial sums. II.
Amer. J. Math. 98 (3), pp. 709–737.
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