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21: 32.14 Combinatorics

… ►The distribution function $F(s)$ given by (32.14.2) arises in random matrix theory where it gives the limiting distribution for the normalized largest eigenvalue in the Gaussian Unitary Ensemble of $n\times n$ Hermitian matrices; see Tracy and Widom (1994). … ►See Forrester and Witte (2001, 2002) for other instances of Painlevé equations in random matrix theory.

22: David M. Bressoud

… ► 227, in 1980, Factorization and Primality Testing, published by Springer-Verlag in 1989, Second Year Calculus from Celestial Mechanics to Special Relativity, published by Springer-Verlag in 1992, A Radical Approach to Real Analysis, published by the Mathematical Association of America in 1994, with a second edition in 2007, Proofs and Confirmations: The Story of the Alternating Sign Matrix Conjecture, published by the Mathematical Association of America and Cambridge University Press in 1999, A Course in Computational Number Theory (with S. …

23: 21.4 Graphics

… ►

21.4.1

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Referenced by:

Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1, Figure 21.4.1

Permalink:

http://dlmf.nist.gov/21.4.E1

Encodings:

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See also:

Annotations for §21.4 and Ch.21

►This Riemann matrix originates from the Riemann surface represented by the algebraic curve ${\mu }^3-{\lambda }^7+2{\lambda }^3\mu =0$; compare §21.7(i). … ►

21.4.2

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

… ►

21.4.3

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Symbols:

$\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.4.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.4 and Ch.21

… ►

►

►

Figure 21.4.5: The real part of a genus 3 scaled Riemann theta function: $\mathrm{\Re }\widehat{\theta }\left(x+\mathrm{i}y,0,0\right|{𝛀}_2)$, $0\le x\le 1$, $0\le y\le 3$. This Riemann matrix originates from the genus 3 Riemann surface represented by the algebraic curve ${\mu }^3+2\mu -{\lambda }^4=0$; compare §21.7(i). Magnify 3D Help

24: 26.15 Permutations: Matrix Notation

§26.15 Permutations: Matrix Notation

… ►The permutation $\sigma$ corresponds to the matrix in which there is a 1 at the intersection of row $j$ with column $\sigma (j)$, and 0’s in all other positions. The permutation $35247816$ corresponds to the matrix … ►The sign of the permutation $\sigma$ is the sign of the determinant of its matrix representation. The inversion number of $\sigma$ is a sum of products of pairs of entries in the matrix representation of $\sigma$: …

25: 21.6 Products

… ►Let $𝐓=[T_{jk}]$ be an arbitrary $h\times h$ orthogonal matrix (that is, $𝐓{𝐓}^{\mathrm{T}}=𝐈$) with rational elements. Also, let $𝐙$ be an arbitrary $g\times h$ matrix. … ►

21.6.1 $$𝒦={\mathbb{Z}}^{g\times h}𝐓/({\mathbb{Z}}^{g\times h}𝐓\cap {\mathbb{Z}}^{g\times h}),$$

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Defines:

$𝒦$: set of matrices (locally)

Symbols:

$\mathbb{Z}$: set of all integers, $\cap$: intersection, $g$: positive integer and $h$: positive integer

Permalink:

http://dlmf.nist.gov/21.6.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.6(i), §21.6 and Ch.21

►that is, $𝒦$ is the set of all $g\times h$ matrices that are obtained by premultiplying $𝐓$ by any $g\times h$ matrix with integer elements; two such matrices in $𝒦$ are considered equivalent if their difference is a matrix with integer elements. … ►

21.6.3 $$\prod _{j=1}^h\theta \left(\sum _{k=1}^h{T}_{jk}{𝐳}_k\right|𝛀)=\frac{1}{{𝒟}^g}\sum _{𝐀\in 𝒦}\sum _{𝐁\in 𝒦}{\mathrm{e}}^{2\pi \mathrm{i}\mathrm{tr}\left[\frac{1}{2}{𝐀}^{\mathrm{T}}𝛀𝐀+{𝐀}^{\mathrm{T}}[𝐙+𝐁]\right]}\prod _{j=1}^h\theta \left({𝐳}_j+𝛀{𝐚}_j+{𝐛}_j\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathrm{tr}𝐀$: trace of matrix, ${𝐀}^{\mathrm{T}}$: transpose of matrix, $g$: positive integer, $h$: positive integer, $𝛀$: a Riemann matrix, $T_{jk}$: element, $𝒦$: set of matrices and $𝒟$: number of elements

Permalink:

http://dlmf.nist.gov/21.6.E3

Encodings:

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See also:

Annotations for §21.6(i), §21.6 and Ch.21

…

26: 21.3 Symmetry and Quasi-Periodicity

… ►

21.3.1 $$\theta \left(-𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.3.E1

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(i), §21.3 and Ch.21

►

21.3.2 $$\theta \left(𝐳+{𝐦}_1\right|𝛀)=\theta \left(𝐳\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.3.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(i), §21.3 and Ch.21

… ►

21.3.3 $$\theta \left(𝐳+{𝐦}_1+𝛀{𝐦}_2\right|𝛀)={\mathrm{e}}^{-2\pi \mathrm{i}\left(\frac{1}{2}{𝐦}_2\cdot 𝛀\cdot {𝐦}_2+{𝐦}_2\cdot 𝐳\right)}\theta \left(𝐳\right|𝛀),$$

ⓘ

Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.3.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(i), §21.3 and Ch.21

… ►

21.3.4 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶+{𝐦}_1}{𝜷+{𝐦}_2}\right]\left(𝐳\right|𝛀)={\mathrm{e}}^{2\pi \mathrm{i}𝜶\cdot {𝐦}_2}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀).$$

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Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Referenced by:

Erratum (V1.0.5) for Equation (21.3.4)

Permalink:

http://dlmf.nist.gov/21.3.E4

Encodings:

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Errata (effective with 1.0.5):

Originally the vector ${𝐦}_2$ on the right-hand-side was given incorrectly as ${𝐦}_1$.

Reported 2012-08-27 by Klaas Vantournhout

See also:

Annotations for §21.3(ii), §21.3 and Ch.21

… ►

21.3.6 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(-𝐳\right|𝛀)={(-1)}^{4𝜶\cdot 𝜷}\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀).$$

ⓘ

Symbols:

$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: Riemann theta function with characteristics, $𝛀$: a Riemann matrix, $𝜶$: $g$-dimensional vector and $𝜷$: $g$-dimensional vector

Permalink:

http://dlmf.nist.gov/21.3.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §21.3(ii), §21.3 and Ch.21

…

27: Morris Newman

… ►Newman wrote the book Matrix Representations of Groups, published by the National Bureau of Standards in 1968, and the book Integral Matrices, published by Academic Press in 1972, which became a classic. …

28: Ingram Olkin

… ►Olkin’s research covered a broad range of areas, including multivariate analysis, reliability theory, matrix theory, statistical models in the social and behavioral sciences, life distributions, and meta-analysis. …

29: 3.7 Ordinary Differential Equations

… ►where $𝐀(\tau ,z)$ is the matrix ►

3.7.6

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Defines:

$𝐀(\tau ,z)$: matrix (locally) and $A_{jk}(\tau ,z)$: matrix entry (locally)

Permalink:

http://dlmf.nist.gov/3.7.E6

Encodings:

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See also:

Annotations for §3.7(ii), §3.7 and Ch.3

… ►Let ${𝐀}_P$ be the $(2P)\times (2P+2)$ band matrix … ►

3.7.13 $${𝐀}_P𝐰=𝐛.$$

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Symbols:

$𝐀(\tau ,z)$: matrix, $𝐛(\tau ,z)$: vector and $P$: partitioning point

Referenced by:

§3.7(iii), §3.7(iv)

Permalink:

http://dlmf.nist.gov/3.7.E13

Encodings:

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See also:

Annotations for §3.7(iii), §3.7 and Ch.3

… ►This converts the problem into a tridiagonal matrix problem in which the elements of the matrix are polynomials in $\lambda$; compare §3.2(vi). …

30: 30.16 Methods of Computation

… ►For $d$ sufficiently large, construct the $d\times d$ tridiagonal matrix $𝐀=[A_{j,k}]$ with nonzero elements … ►Let $𝐀$ be the $d\times d$ matrix given by (30.16.1) if $n-m$ is even, or by (30.16.6) if $n-m$ is odd. …