# dimension

(0.000 seconds)

## 1—10 of 14 matching pages

##### 1: 30.16 Methods of Computation
30.16.2 $\alpha_{j,d+1}\leq\alpha_{j,d},$
30.16.3 $\lambda^{m}_{n}\left(\gamma^{2}\right)=\lim_{d\to\infty}\alpha_{p,d},$ $p=\left\lfloor\frac{1}{2}(n-m)\right\rfloor+1$.
30.16.4 $\alpha_{p,d}-\lambda^{m}_{n}\left(\gamma^{2}\right)=O\left(\frac{\gamma^{4d}}{% 4^{2d+1}((m+2d-1)!(m+2d+1)!)^{2}}\right),$ $d\to\infty$.
30.16.7 $\sum_{j=1}^{d}e_{j,d}^{2}\frac{(n+m+2j-2p)!}{(n-m+2j-2p)!}\frac{1}{2n+4j-4p+1}% =\frac{(n+m)!}{(n-m)!}\frac{1}{2n+1}.$
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. …
##### 4: 21.2 Definitions
21.2.1 $\theta\left(\mathbf{z}\middle|\boldsymbol{{\Omega}}\right)=\sum_{\mathbf{n}\in% {\mathbb{Z}}^{g}}e^{2\pi i\left(\frac{1}{2}\mathbf{n}\cdot\boldsymbol{{\Omega}% }\cdot\mathbf{n}+\mathbf{n}\cdot\mathbf{z}\right)}.$
##### 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 $\pm\infty$, and thus a continuous spectrum on $\boldsymbol{\sigma}\geq 0$. … … Put $n_{+}=\dim N_{z}$ ($\Im z>0$) and $n_{-}=\dim N_{z}$ ($\Im z<0$), 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
• G. Pólya (1949) 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.
• ##### 8: Bibliography S
• B. Simon (1976) The Bound State of Weakly Coupled Schrödinger Operators in One and Two Dimensions. Annals of Physics 97 (2), pp. 279–288.
• D. Slepian (1964) 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.
• ##### 9: 1.2 Elementary Algebra
For matrices $\mathbf{A}$, $\mathbf{B}$ and $\mathbf{C}$ of the same dimensions, … distributive if $\mathbf{B}$ and $\mathbf{C}$ have the same dimensions