dimension

SearchAdvancedHelp

(0.000 seconds)

1—10 of 14 matching pages

1: 30.16 Methods of Computation

… ►

30.16.2 $${\alpha }_{j,d+1}\le {\alpha }_{j,d},$$

ⓘ

Symbols:

$d$: dimension and ${\alpha }_{j,k}$: real eigenvalues

Permalink:

http://dlmf.nist.gov/30.16.E2

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(i), §30.16 and Ch.30

… ►

30.16.3 $${\lambda }_n^m\left({\gamma }^2\right)=\underset{d\to \mathrm{\infty }}{lim}{\alpha }_{p,d},$$ $p=\lfloor \frac{1}{2}(n-m)\rfloor +1$.

ⓘ

Symbols:

$\lfloor x\rfloor$: floor of $x$ , ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\alpha }_{j,k}$: real eigenvalues and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.16.E3

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(i), §30.16 and Ch.30

… ►

30.16.4 $${\alpha }_{p,d}-{\lambda }_n^m\left({\gamma }^2\right)=O\left(\frac{{\gamma }^{4d}}{4^{2d+1}{((m+2d-1)!(m+2d+1)!)}^2}\right),$$ $d\to \mathrm{\infty }$.

ⓘ

Symbols:

$O\left(x\right)$: order not exceeding, $!$: factorial (as in $n!$), ${\lambda }_n^m\left({\gamma }^2\right)$: eigenvalues of the spheroidal differential equation, $m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\alpha }_{j,k}$: real eigenvalues and ${\gamma }^2$: real parameter

Permalink:

http://dlmf.nist.gov/30.16.E4

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(i), §30.16 and Ch.30

… ►

30.16.7 $$\sum _{j=1}^d{e}_{j,d}^2\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1}=\frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}.$$

ⓘ

Symbols:

$!$: factorial (as in $n!$), $m$: nonnegative integer, $n\ge m$: integer degree and $d$: dimension

Permalink:

http://dlmf.nist.gov/30.16.E7

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(ii), §30.16 and Ch.30

… ►

30.16.8 $$a_{n,k}^m({\gamma }^2)=\underset{d\to \mathrm{\infty }}{lim}{e}_{k+p,d},$$

ⓘ

Symbols:

$m$: nonnegative integer, $n\ge m$: integer degree, $d$: dimension, ${\gamma }^2$: real parameter and $a_{n,k}^m({\gamma }^2)$: coefficients

Permalink:

http://dlmf.nist.gov/30.16.E8

Encodings:

pMML, png, TeX

See also:

Annotations for §30.16(ii), §30.16 and Ch.30

…

2: About Color Map

… ►We use color to augment these vizualizations, either to reinforce the recognition of the height, or to convey an extra dimension to represent the phase of complex valued functions. …

3: 1.6 Vectors and Vector-Valued Functions

… ►

Green’s Theorem

… ►

Green’s Theorem (for Volume)

…

4: 21.2 Definitions

… ►

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

ⓘ

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

Encodings:

pMML, png, TeX

See also:

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

…

5: 1.18 Linear Second Order Differential Operators and Eigenfunction Expansions

… ►

Example 1: In one and two dimensions any $q(x)$ with a ‘Dip, or Well’ has a partly discrete spectrum

►Suppose that $X$ is the whole real line in one dimension, and that $q(x)$, in (1.18.28) has (non-oscillatory) limits of $0$ at both $\pm \mathrm{\infty }$, and thus a continuous spectrum on $𝝈\ge 0$. … … ►Put $n_{+}=dim{N}_z$ ($\mathrm{\Im }z>0$) and $n_{-}=dim{N}_z$ (), the deficiency indices for $T$. … ►Thus $N_z$ has dimension 0, 1 or 2. …

6: 10.73 Physical Applications

… ►Accordingly, the spherical Bessel functions appear in all problems in three dimensions with spherical symmetry involving the scattering of electromagnetic radiation. …

7: Bibliography P

… ►

Remarks on computing the probability integral in one and two dimensions. In Proceedings of the Berkeley Symposium on Mathematical Statistics and Probability, 1945, 1946, pp. 63–78.

ⓘ

External Links:

MathReview (L. A. Aroian), ZentralBlatt

Referenced by:

(7.8.8), §7.8, Erratum (V1.0.17) for Equation (7.8.8).

Encodings:

BibTeX

…

8: Bibliography S

… ►

The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.

ⓘ

External Links:

Document, MathReview Entry

Referenced by:

§1.18(viii).

Encodings:

BibTeX

… ►

Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions. Bell System Tech. J. 43, pp. 3009–3057.

ⓘ

External Links:

ISSN 0005-8580, MathReview (J. Meixner), ZentralBlatt

Referenced by:

§30.12.

Encodings:

BibTeX

…

9: 1.2 Elementary Algebra

… ►For matrices $𝐀$, $𝐁$ and $𝐂$ of the same dimensions, … ►distributive if $𝐁$ and $𝐂$ have the same dimensions …

10: 18.38 Mathematical Applications

… ►

Quadrature “Extended” to Pseudo-Spectral (DVR) Representations of Operators in One and Many Dimensions

…