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11: 26.13 Permutations: Cycle Notation

§26.13 Permutations: Cycle Notation

… ►An explicit representation of $\sigma$ can be given by the $2\times n$ matrix: … ►In cycle notation, the elements in each cycle are put inside parentheses, ordered so that $\sigma (j)$ immediately follows $j$ or, if $j$ is the last listed element of the cycle, then $\sigma (j)$ is the first element of the cycle. … ►is $\left(1,3,2,5,7\right)\left(4\right)\left(6,8\right)$ in cycle notation. …They are often dropped from the cycle notation. …

12: 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

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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

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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). …

13: 35.4 Partitions and Zonal Polynomials

… ►For any partition $\kappa$, the zonal polynomial $Z_{\kappa }:𝓢\to ℝ$ is defined by the properties … ►Alternative notations for the zonal polynomials are $C_{\kappa }(𝐓)$ (Muirhead (1982, pp. 227–239)), ${𝒴}_{\kappa }(𝐓)$ (Takemura (1984, p. 22)), and ${\mathrm{\Phi }}_{\kappa }(𝐓)$ (Faraut and Korányi (1994, pp. 228–236)). … ►

35.4.5 $$Z_{\kappa }\left(𝐇𝐓{𝐇}^{-1}\right)=Z_{\kappa }\left(𝐓\right),$$ $𝐇\in 𝐎(m)$.

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Symbols:

$\in$: element of, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $m$: positive integer and $\kappa$: vector of nonnegative integers

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http://dlmf.nist.gov/35.4.E5

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Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.6 $$\sum _{|\kappa |=k}{Z}_{\kappa }\left(𝐓\right)={(\mathrm{tr}𝐓)}^k.$$

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Symbols:

$\mathrm{tr}𝐀$: trace of matrix, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐓$: real symmetric $m\times m$ matrix, $k$: nonnegative integer and $\kappa$: vector of nonnegative integers

Permalink:

http://dlmf.nist.gov/35.4.E6

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Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.7 $${\int }_{𝐎(m)}{Z}_{\kappa }\left(𝐒𝐇𝐓{𝐇}^{-1}\right)\mathrm{d}𝐇=\frac{Z_{\kappa }\left(𝐒\right)Z_{\kappa }\left(𝐓\right)}{Z_{\kappa }\left(𝐈\right)}.$$

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Symbols:

$\int$: integral, $Z_{\kappa }\left(𝐓\right)$: zonal polynomial, $𝐈$: $m\times m$ identity matrix, $𝐒$: real symmetric $m\times m$ matrix, $𝐓$: real symmetric $m\times m$ matrix, $𝐎(m)$: space of $m\times m$ orthogonal matrices, $𝐇$: orthogonal $m\times m$ matrix, $\mathrm{d}𝐇$: normalized Haar measure on $𝐎(m)$ , $m$: positive integer and $\kappa$: vector of nonnegative integers

Referenced by:

§35.9

Permalink:

http://dlmf.nist.gov/35.4.E7

Encodings:

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See also:

Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

…

14: Bibliography H

… ►

Bessel functions of matrix argument. Ann. of Math. (2) 61 (3), pp. 474–523.

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External Links:

FullText, MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§35.1, §35.2, §35.3(ii), §35.5(i), §35.5(ii), §35.5(iii), §35.6(i), §35.6(ii), §35.6(iii), §35.6(iv), §35.7(i), §35.7(ii), §35.7(iv), §35.8(i), §35.8(ii), §35.8(iv), Ch.35, Notations P.

Encodings:

BibTeX

…

15: 18.2 General Orthogonal Polynomials

… ►The matrix on the left-hand side is an (infinite tridiagonal) Jacobi matrix. This matrix is symmetric iff $c_n=a_{n-1}$ ($n\ge 1$). … ►With notation (18.2.4_5), (18.2.5), (18.2.7) … ►When the Jacobi matrix in (18.2.11_9) is truncated to an $n\times n$ matrix …

16: 1.1 Special Notation

§1.1 Special Notation

►(For other notation see Notation for the Special Functions.) ► ►►►►

$x,y$	real variables.
…
${𝐀}^{-1}$	inverse of the square matrix $𝐀$
$𝐈$	identity matrix
…

►In the physics, applied maths, and engineering literature a common alternative to $\overline{a}$ is $a^{\ast }$, $a$ being a complex number or a matrix; the Hermitian conjugate of $𝐀$ is usually being denoted ${𝐀}^{†}$.

17: 21.2 Definitions

… ►

21.2.1 $$\theta \left(𝐳\right|𝛀)=\sum _{𝐧\in {\mathbb{Z}}^g}{\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝐧\cdot 𝛀\cdot 𝐧+𝐧\cdot 𝐳\right)}.$$

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Defines:

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

Symbols:

$\pi$: the ratio of the circumference of a circle to its diameter, $\in$: element of, $\mathrm{e}$: base of natural logarithm, $\mathrm{i}$: imaginary unit, $\mathbb{Z}$: set of all integers, $g$: positive integer and $𝛀$: a Riemann matrix

Referenced by:

§21.2(ii)

Permalink:

http://dlmf.nist.gov/21.2.E1

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Annotations for §21.2(i), §21.2 and Ch.21

… ►

21.2.2 $$\widehat{\theta }\left(𝐳\right|𝛀)={\mathrm{e}}^{-\pi [\mathrm{\Im }𝐳]\cdot {[\mathrm{\Im }𝛀]}^{-1}\cdot [\mathrm{\Im }𝐳]}\theta \left(𝐳\right|𝛀).$$

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Defines:

$\widehat{\theta }\left(𝐳\right|𝛀)$: scaled Riemann theta function

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, $\mathrm{\Im }$: imaginary part and $𝛀$: a Riemann matrix

Permalink:

http://dlmf.nist.gov/21.2.E2

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Annotations for §21.2(i), §21.2 and Ch.21

… ►

21.2.6 $$\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)={\mathrm{e}}^{2\pi \mathrm{i}\left(\frac{1}{2}𝜶\cdot 𝛀\cdot 𝜶+𝜶\cdot [𝐳+𝜷]\right)}\theta \left(𝐳+𝛀𝜶+𝜷\right|𝛀),$$

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Symbols:

$\theta \left(𝐳\right|𝛀)$: Riemann theta function, $\theta \left[\genfrac{}{}{0.0pt}{}{𝜶}{𝜷}\right]\left(𝐳\right|𝛀)$: 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

Permalink:

http://dlmf.nist.gov/21.2.E6

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Annotations for §21.2(ii), §21.2 and Ch.21

… ►

21.2.7 $$\theta \left[\genfrac{}{}{0.0pt}{}{\mathrm{𝟎}}{\mathrm{𝟎}}\right]\left(𝐳\right|𝛀)=\theta \left(𝐳\right|𝛀).$$

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Symbols:

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

Permalink:

http://dlmf.nist.gov/21.2.E7

Encodings:

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Annotations for §21.2(ii), §21.2 and Ch.21

… ►For $g=1$, and with the notation of §20.2(i), …

18: Bibliography C

… ►

A History of Mathematical Notations, Volume II. Open Court Publishing Company, Chicago.

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Notes:

Reprinted by Cosimo in 2007.

External Links:

ZentralBlatt

Referenced by:

§24.1.

Encodings:

BibTeX

… ►

Théorie Générale de L’Équation de Mathieu et de quelques autres Équations différentielles de la mécanique. Masson et Cie, Paris (French).

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External Links:

MathReview (A. Erdélyi), ZentralBlatt

Referenced by:

§28.1, Notations C, Notations I, Notations I, Notations J, Notations J, Notations S.

Encodings:

BibTeX

… ►

Note on Nörlund’s polynomial $B_n^{(z)}$ . Proc. Amer. Math. Soc. 11 (3), pp. 452–455.

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External Links:

FullText, MathReview (M. P. Drazin), ZentralBlatt

Referenced by:

§26.1, §26.1, Notations S, Notations S.

Encodings:

BibTeX

… ►

$S$-matrix and $R$-matrix determination of the low-energy ${}_{}{}^5\mathrm{He}$ and ${}_{}{}^5\mathrm{Li}$ resonance parameters. Phys. Rev. C 55 (1), pp. 536–539.

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External Links:

Document

Referenced by:

2nd item.

Encodings:

BibTeX

… ►

Coulomb Wave Functions. Roy. Soc. Math. Tables, Vol. 11, Cambridge University Press, Cambridge.

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External Links:

ISBN 0-521-06156-3, MathReview (John Todd), ZentralBlatt

Referenced by:

item Curtis (1964a):, §33.14(i), §33.20(iv), §33.21(ii), §33.23(vi), 2nd item, §33.9(ii), Notations P, Notations Q.

Encodings:

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…

19: Mathematical Introduction

… ►

Notation for the Special Functions

… ►This section may also include important alternative notations that have appeared in the literature. … ►The exceptions are ones for which the existing notations have drawbacks. … ►The Notations section includes all the notations for the special functions adopted in this Handbook. … ►

Common Notations and Definitions

…

20: 18.39 Applications in the Physical Sciences

… ►This is also the normalization and notation of Chapter 33 for $Z=1$ , and the notation of Weinberg (2013, Chapter 2). … ►, the J-matrix elements) as in Gautschi (1968), Golub and Welsch (1969), Gordon (1968). … ►

§18.39(iv) Coulomb–Pollaczek Polynomials and J-Matrix Methods

… ►

Discretized and Continuum Expansions of Scattering Eigenfunctions in terms of Pollaczek Polynomials: J-matrix Theory

… ►As this follows from the three term recursion of (18.39.46) it is referred to as the J-Matrix approach, see (3.5.31), to single and multi-channel scattering numerics. …