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Lanczos vectors

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11: 35.4 Partitions and Zonal Polynomials

… ►A partition

\kappa=(k_{1},\dots,k_{m})

is a vector of nonnegative integers, listed in nonincreasing order. Also,

|\kappa|

denotes

k_{1}+\dots+k_{m}

, the weight of

\kappa

;

\ell(\kappa)

denotes the number of nonzero

k_{j}

;

a+\kappa

denotes the vector

(a+k_{1},\dots,a+k_{m})

. … ►

35.4.4

Z_{\kappa}\left(\boldsymbol{{0}}\right)=\begin{cases}1,&\kappa=(0,\dots,0),\\ 0,&\kappa\neq(0,\dots,0).\end{cases}

ⓘ

Symbols:: $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\boldsymbol{{0}}$ : $m\times m$ matrix of zeros and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.5

Z_{\kappa}\left(\mathbf{H}\mathbf{T}\mathbf{H}^{-1}\right)=Z_{\kappa}\left(% \mathbf{T}\right),

\mathbf{H}\in\mathbf{O}(m)

.

ⓘ

Symbols:: $\in$ : element of, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $\mathbf{O}(m)$ : space of $m\times m$ orthogonal matrices, $\mathbf{H}$ : orthogonal $m\times m$ matrix, $m$ : positive integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

… ►

35.4.6

\sum_{|\kappa|=k}Z_{\kappa}\left(\mathbf{T}\right)=(\operatorname{tr}{\mathbf{% T}})^{k}.

ⓘ

Symbols:: $\operatorname{tr}\NVar{\mathbf{A}}$ : trace of matrix, $Z_{\NVar{\kappa}}\left(\NVar{\mathbf{T}}\right)$ : zonal polynomial, $\mathbf{T}$ : real symmetric $m\times m$ matrix, $k$ : nonnegative integer and $\kappa$ : vector of nonnegative integers
Permalink:: http://dlmf.nist.gov/35.4.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §35.4(ii), §35.4(ii), §35.4 and Ch.35

…

12: 3.7 Ordinary Differential Equations

… ►and

\mathbf{b}(\tau,z)

is the vector ►

3.7.7

\mathbf{b}(\tau,z)=\begin{bmatrix}b_{1}(\tau,z)\\ b_{2}(\tau,z)\end{bmatrix},

ⓘ

Defines:: $\mathbf{b}(\NVar{\tau},\NVar{z})$ : vector (locally) and $b_{\NVar{j}}(\NVar{\tau},\NVar{z})$ : vector (locally)
Permalink:: http://dlmf.nist.gov/3.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §3.7(ii), §3.7 and Ch.3

… ►Also let

\mathbf{w}

denote the

(2P+2)\times 1

vector …and

\mathbf{b}

the

(2P)\times 1

vector ►

3.7.12

\mathbf{b}=\left[b_{1}(\tau_{0},z_{0}),b_{2}(\tau_{0},z_{0}),b_{1}(\tau_{1},z_% {1}),b_{2}(\tau_{1},z_{1}),\ldots,b_{1}(\tau_{P-1},z_{P-1}),b_{2}(\tau_{P-1},z% _{P-1})\right]^{\rm T}.

ⓘ

Defines:: $\mathbf{b}(\NVar{\tau},\NVar{z})$ : vector (locally) and $\mathbf{b}(\NVar{\tau},\NVar{z})$ : vector (locally)
Symbols:: $\tau_{j}$ : change of variable and $P$ : partitioning point
Permalink:: http://dlmf.nist.gov/3.7.E12
Encodings:: pMML, png, TeX
See also:: Annotations for §3.7(iii), §3.7 and Ch.3

…

13: 21.7 Riemann Surfaces

… ►

21.7.7

\left(\frac{\partial}{\partial z_{1}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol% {{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}% \right)\Big{|}_{\mathbf{z}=\boldsymbol{{0}}},\dots,\frac{\partial}{\partial z_% {g}}\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)\Big{|}_{\mathbf{z}=% \boldsymbol{{0}}}\right)\neq\boldsymbol{{0}}.

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.7.E7
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

… ►

21.7.8

\zeta=\sum_{j=1}^{g}\omega_{j}\frac{\partial}{\partial z_{j}}\theta\genfrac{[}% {]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}% \middle|\boldsymbol{{\Omega}}\right)\Big{|}_{\mathbf{z}=\boldsymbol{{0}}}.

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\frac{\partial\NVar{f}}{\partial\NVar{x}}$ : partial derivative of $f$ with respect to $x$ , $\,\partial\NVar{x}$ : partial differential of $x$ , $g$ : positive integer, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector and $\omega$ : differential
Permalink:: http://dlmf.nist.gov/21.7.E8
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

… ►

21.7.9

E(P_{1},P_{2})=\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{% \boldsymbol{{\beta}}}\left(\int_{P_{1}}^{P_{2}}\boldsymbol{{\omega}}\middle|% \boldsymbol{{\Omega}}\right)\Bigg{/}\left(\sqrt{\zeta(P_{1})}\sqrt{\zeta(P_{2}% )}\right),

ⓘ

Defines:: $E(\NVar{P_{1}},\NVar{P_{2}})$ : prime form (locally)
Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\int$ : integral, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $P_{1},P_{2}$ : points and $\boldsymbol{{\omega}}$ : vector of differentials
Permalink:: http://dlmf.nist.gov/21.7.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §21.7(ii), §21.7 and Ch.21

►where

P_{1}

and

P_{2}

are points on

\Gamma

,

\boldsymbol{{\omega}}=(\omega_{1},\omega_{2},\dots,\omega_{g})

, and the path of integration on

\Gamma

from

P_{1}

to

P_{2}

is identical for all components. … ►Next, define an isomorphism

\boldsymbol{{\eta}}

which maps every subset

T

of

B

with an even number of elements to a

2g

-dimensional vector

\boldsymbol{{\eta}}(T)

with elements either

0

or

\tfrac{1}{2}

. …

14: 16.1 Special Notation

… ► ►►►►

$p, q$	nonnegative integers.
…
$\mathbf{a}$	vector $(a_{1},a_{2},\dots,a_{p})$ .
$\mathbf{b}$	vector $(b_{1},b_{2},\dots,b_{q})$ .
…

…

15: 26.2 Basic Definitions

… ►Unless otherwise specified, it consists of horizontal segments corresponding to the vector

(1,0)

and vertical segments corresponding to the vector

(0,1)

. …

16: 21.5 Modular Transformations

… ►For a

g\times g

matrix

{\bf A}

we define

\operatorname{diag}\mathbf{A}

, as a column vector with the diagonal entries as elements. … ►

21.5.9

\theta\genfrac{[}{]}{0.0pt}{}{\mathbf{D}\boldsymbol{{\alpha}}-\mathbf{C}% \boldsymbol{{\beta}}+\tfrac{1}{2}\operatorname{diag}[\mathbf{C}\mathbf{D}^{% \mathrm{T}}]}{-\mathbf{B}\boldsymbol{{\alpha}}+\mathbf{A}\boldsymbol{{\beta}}+% \tfrac{1}{2}\operatorname{diag}[\mathbf{A}\mathbf{B}^{\mathrm{T}}]}\left(\left% [[\mathbf{C}\boldsymbol{{\Omega}}+\mathbf{D}]^{-1}\right]^{\mathrm{T}}\mathbf{% z}\middle|[\mathbf{A}\boldsymbol{{\Omega}}+\mathbf{B}][\mathbf{C}\boldsymbol{{% \Omega}}+\mathbf{D}]^{-1}\right)=\kappa(\boldsymbol{{\alpha}},\boldsymbol{{% \beta}},\boldsymbol{{\Gamma}})\sqrt{\det[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]}e^{\pi i\mathbf{z}\cdot\left[[\mathbf{C}\boldsymbol{{\Omega}}+% \mathbf{D}]^{-1}\mathbf{C}\right]\cdot\mathbf{z}}\theta\genfrac{[}{]}{0.0pt}{}% {\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}\left(\mathbf{z}\middle|% \boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics, $\pi$ : the ratio of the circumference of a circle to its diameter, $\det$ : determinant, $\mathrm{e}$ : base of natural logarithm, $\mathrm{i}$ : imaginary unit, $\NVar{\mathbf{A}}^{\mathrm{T}}$ : transpose of matrix, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector, $\boldsymbol{{\beta}}$ : $g$ -dimensional vector, $\operatorname{diag}\mathbf{A}$ : Transpose of $[A_{11},A_{22},\ldots,A_{gg}]$ and Matrix $\boldsymbol{{\Gamma}}$
Permalink:: http://dlmf.nist.gov/21.5.E9
Encodings:: pMML, png, TeX
See also:: Annotations for §21.5(ii), §21.5 and Ch.21

…

17: 10.77 Software

… ►

•

Boisvert and Saunders (1992). Vectorized Fortran.

…

18: 19.31 Probability Distributions

… ►If

\mathbf{x}

is a column vector with elements

x_{1},x_{2},\dots,x_{n}

and transpose

\mathbf{x}^{\mathrm{T}}

, then …

19: 21.2 Definitions

… ►

21.2.5

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in{% \mathbb{Z}}^{g}}e^{2\pi i\left(\frac{1}{2}[\mathbf{n}+\boldsymbol{{\alpha}}]% \cdot\boldsymbol{{\Omega}}\cdot[\mathbf{n}+\boldsymbol{{\alpha}}]+[\mathbf{n}+% \boldsymbol{{\alpha}}]\cdot[\mathbf{z}+\boldsymbol{{\beta}}]\right)}.

ⓘ

Defines:: $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function with characteristics
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, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.2.E5
Encodings:: pMML, png, TeX
See also:: Annotations for §21.2(ii), §21.2 and Ch.21

… ►

21.2.6

\theta\genfrac{[}{]}{0.0pt}{}{\boldsymbol{{\alpha}}}{\boldsymbol{{\beta}}}% \left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=e^{2\pi i\left(\frac{1}{2% }\boldsymbol{{\alpha}}\cdot\boldsymbol{{\Omega}}\cdot\boldsymbol{{\alpha}}+% \boldsymbol{{\alpha}}\cdot[\mathbf{z}+\boldsymbol{{\beta}}]\right)}\theta\left% (\mathbf{z}+\boldsymbol{{\Omega}}\boldsymbol{{\alpha}}+\boldsymbol{{\beta}}% \middle|\boldsymbol{{\Omega}}\right),

ⓘ

Symbols:: $\theta\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : Riemann theta function, $\theta\genfrac{[}{]}{0.0pt}{}{\NVar{\boldsymbol{{\alpha}}}}{\NVar{\boldsymbol{% {\beta}}}}\left(\NVar{\mathbf{z}}\middle|\NVar{\boldsymbol{{\Omega}}}\right)$ : 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, $\boldsymbol{{\Omega}}$ : a Riemann matrix, $\boldsymbol{{\alpha}}$ : $g$ -dimensional vector and $\boldsymbol{{\beta}}$ : $g$ -dimensional vector
Permalink:: http://dlmf.nist.gov/21.2.E6
Encodings:: pMML, png, TeX
See also:: Annotations for §21.2(ii), §21.2 and Ch.21

…

20: 21.9 Integrable Equations

… ►where

c

is a complex constant and

\mathbf{k}

,

\mathbf{l}

,

\boldsymbol{{\omega}}

, and

\boldsymbol{{\phi}}

are

g

-dimensional complex vectors; see Krichever (1976). …

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