About the Project

positive%20definite

AdvancedHelp

(0.001 seconds)

1—10 of 349 matching pages

1: 26.12 Plane Partitions
A plane partition, π , of a positive integer n , is a partition of n in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. …
26.12.3 B ( r , s , t ) = { ( h , j , k ) |  1 h r , 1 j s , 1 k t } .
26.12.9 ( h = 1 r j = 1 s h + j + t 1 h + j 1 ) 2 ;
26.12.13 h = 1 r j = 1 r h + j + t 1 h + j 1 ;
26.12.14 h = 1 r j = 1 r + 1 h + j + t 1 h + j 1 .
2: 27.2 Functions
Functions in this section derive their properties from the fundamental theorem of arithmetic, which states that every integer n > 1 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. … the sum of the k th powers of the positive integers m n that are relatively prime to n . …This is the number of positive integers n that are relatively prime to n ; ϕ ( n ) is Euler’s totient. … and if ϕ ( n ) is the smallest positive integer f such that a f 1 ( mod n ) , then a is a primitive root mod n . …
3: 35.3 Multivariate Gamma and Beta Functions
35.3.2 Γ m ( s 1 , , s m ) = 𝛀 etr ( 𝐗 ) | 𝐗 | s m 1 2 ( m + 1 ) j = 1 m 1 | ( 𝐗 ) j | s j s j + 1 d 𝐗 , s j , ( s j ) > 1 2 ( j 1 ) , j = 1 , , m .
35.3.6 Γ m ( a , , a ) = Γ m ( a ) .
35.3.7 B m ( a , b ) = Γ m ( a ) Γ m ( b ) Γ m ( a + b ) .
4: 13.30 Tables
  • Slater (1960) tabulates M ( a , b , x ) for a = 1 ( .1 ) 1 , b = 0.1 ( .1 ) 1 , and x = 0.1 ( .1 ) 10 , 7–9S; M ( a , b , 1 ) for a = 11 ( .2 ) 2 and b = 4 ( .2 ) 1 , 7D; the smallest positive x -zero of M ( a , b , x ) for a = 4 ( .1 ) 0.1 and b = 0.1 ( .1 ) 2.5 , 7D.

  • Abramowitz and Stegun (1964, Chapter 13) tabulates M ( a , b , x ) for a = 1 ( .1 ) 1 , b = 0.1 ( .1 ) 1 , and x = 0.1 ( .1 ) 1 ( 1 ) 10 , 8S. Also the smallest positive x -zero of M ( a , b , x ) for a = 1 ( .1 ) 0.1 and b = 0.1 ( .1 ) 1 , 7D.

  • Zhang and Jin (1996, pp. 411–423) tabulates M ( a , b , x ) and U ( a , b , x ) for a = 5 ( .5 ) 5 , b = 0.5 ( .5 ) 5 , and x = 0.1 , 1 , 5 , 10 , 20 , 30 , 8S (for M ( a , b , x ) ) and 7S (for U ( a , b , x ) ).

  • 5: 35.5 Bessel Functions of Matrix Argument
    35.5.3 B ν ( 𝐓 ) = 𝛀 etr ( ( 𝐓 𝐗 + 𝐗 1 ) ) | 𝐗 | ν 1 2 ( m + 1 ) d 𝐗 , ν , 𝐓 𝛀 .
    35.5.5 𝟎 < 𝐗 < 𝐓 A ν 1 ( 𝐒 1 𝐗 ) | 𝐗 | ν 1 A ν 2 ( 𝐒 2 ( 𝐓 𝐗 ) ) | 𝐓 𝐗 | ν 2 d 𝐗 = | 𝐓 | ν 1 + ν 2 + 1 2 ( m + 1 ) A ν 1 + ν 2 + 1 2 ( m + 1 ) ( ( 𝐒 1 + 𝐒 2 ) 𝐓 ) , ν j , ( ν j ) > 1 , j = 1 , 2 ; 𝐒 1 , 𝐒 2 𝓢 ; 𝐓 𝛀 .
    35.5.7 𝛀 A ν 1 ( 𝐓 𝐗 ) B ν 2 ( 𝐒 𝐗 ) | 𝐗 | ν 1 d 𝐗 = 1 A ν 1 + ν 2 ( 𝟎 ) | 𝐒 | ν 2 | 𝐓 + 𝐒 | ( ν 1 + ν 2 + 1 2 ( m + 1 ) ) , ( ν 1 + ν 2 ) > 1 ; 𝐒 , 𝐓 𝛀 .
    6: 35.1 Special Notation
    a , b complex variables.
    m positive integer.
    𝛀 space of positive-definite real symmetric matrices.
    𝐗 > 𝐓 𝐗 𝐓 is positive definite. Similarly, 𝐓 < 𝐗 is equivalent.
    7: 21.1 Special Notation
    g , h positive integers.
    g × × × ( g times).
    𝛀 g × g complex, symmetric matrix with 𝛀 strictly positive definite, i.e., a Riemann matrix.
    a b intersection index of a and b , two cycles lying on a closed surface. a b = 0 if a and b do not intersect. Otherwise a b gets an additive contribution from every intersection point. This contribution is 1 if the basis of the tangent vectors of the a and b cycles (§21.7(i)) at the point of intersection is positively oriented; otherwise it is 1 .
    8: Bibliography G
  • G. Gasper (1977) Positive sums of the classical orthogonal polynomials. SIAM J. Math. Anal. 8 (3), pp. 423–447.
  • A. Gil, J. Segura, and N. M. Temme (2004a) Algorithm 831: Modified Bessel functions of imaginary order and positive argument. ACM Trans. Math. Software 30 (2), pp. 159–164.
  • A. Gil, J. Segura, and N. M. Temme (2004b) Computing solutions of the modified Bessel differential equation for imaginary orders and positive arguments. ACM Trans. Math. Software 30 (2), pp. 145–158.
  • A. Gil, J. Segura, and N. M. Temme (2014) Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.
  • M. L. Glasser (1976) Definite integrals of the complete elliptic integral K . J. Res. Nat. Bur. Standards Sect. B 80B (2), pp. 313–323.
  • 9: Bibliography
  • M. J. Ablowitz and H. Segur (1977) Exact linearization of a Painlevé transcendent. Phys. Rev. Lett. 38 (20), pp. 1103–1106.
  • A. Adelberg (1992) On the degrees of irreducible factors of higher order Bernoulli polynomials. Acta Arith. 62 (4), pp. 329–342.
  • D. E. Amos (1989) Repeated integrals and derivatives of K Bessel functions. SIAM J. Math. Anal. 20 (1), pp. 169–175.
  • A. Apelblat (1983) Table of Definite and Infinite Integrals. Physical Sciences Data, Vol. 13, Elsevier Scientific Publishing Co., Amsterdam.
  • R. Askey and G. Gasper (1976) Positive Jacobi polynomial sums. II. Amer. J. Math. 98 (3), pp. 709–737.
  • 10: 25.7 Integrals
    For definite integrals of the Riemann zeta function see Prudnikov et al. (1986b, §2.4), Prudnikov et al. (1992a, §3.2), and Prudnikov et al. (1992b, §3.2).