compatibility condition

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1: 32.4 Isomonodromy Problems

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§32.4(i) Definition

… ►(32.4.3) is the compatibility condition of (32.4.1). … … ►The compatibility condition of (32.4.1) with … ►

2: Browsers

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3: 34.10 Zeros

… ►In a $\mathit{3}j$ symbol, if the three angular momenta $j_1,j_2,j_3$ do not satisfy the triangle conditions (34.2.1), or if the projective quantum numbers do not satisfy (34.2.3), then the $\mathit{3}j$ symbol is zero. Similarly the $\mathit{6}j$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3}j$ symbols in the summation. …However, the $\mathit{3}j$ and $\mathit{6}j$ symbols may vanish for certain combinations of the angular momenta and projective quantum numbers even when the triangle conditions are fulfilled. …

4: 26.10 Integer Partitions: Other Restrictions

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26.10.1 $$p(𝒟,0)=p(𝒟k,0)=p(\in S,0)=1.$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $k$: nonnegative integer and $S$: set

Permalink:

http://dlmf.nist.gov/26.10.E1

Encodings:

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Annotations for §26.10(i), §26.10 and Ch.26

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Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from $A_{j,k}$.

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	$p(𝒟,n)$	$p(𝒟2,n)$	$p(𝒟2,\in T,n)$	$p({𝒟}^{\prime }3,n)$
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26.10.5 $$\sum _{n=0}^{\mathrm{\infty }}p(\in S,n)q^n=\prod _{j\in S}\frac{1}{1-q^j}.$$

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Symbols:

$\in$: element of, $p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer, $n$: nonnegative integer, $S$: set and $q$: parameter

Permalink:

http://dlmf.nist.gov/26.10.E5

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(ii), §26.10 and Ch.26

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26.10.6 $$p(𝒟,n)=\frac{1}{n}\sum _{t=1}^{n}p(𝒟,n-t)\sum _{\begin{array}{c}j|t\\ j\text{ odd}\end{array}}j,$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ , $j$: nonnegative integer and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.10.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(iii), §26.10 and Ch.26

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26.10.12 $$p(𝒟,n)=p(𝒪,n),$$

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Symbols:

$p(\mathrm{condition},n)$: restricted number of partitions of $n$ and $n$: nonnegative integer

Permalink:

http://dlmf.nist.gov/26.10.E12

Encodings:

pMML, png, TeX

See also:

Annotations for §26.10(iv), §26.10 and Ch.26

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5: 3.2 Linear Algebra

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§3.2(iii) Condition of Linear Systems

… ►The sensitivity of the solution vector $𝐱$ in (3.2.1) to small perturbations in the matrix $𝐀$ and the vector $𝐛$ is measured by the condition number …The larger the value $\kappa (𝐀)$, the more ill-conditioned the system. … ►

§3.2(v) Condition of Eigenvalues

►If $𝐀$ is nondefective and $\lambda$ is a simple zero of $p_n(\lambda )$, then the sensitivity of $\lambda$ to small perturbations in the matrix $𝐀$ is measured by the condition number …

6: 26.11 Integer Partitions: Compositions

… ► $c\left(n\right)$ denotes the number of compositions of $n$, and $c_m\left(n\right)$ is the number of compositions into exactly $m$ parts. … ►

26.11.1 $$c\left(0\right)=c(\in T,0)=1.$$

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Symbols:

$\in$: element of, $c\left(n\right)$: number of compositions of $n$ , $c(\mathrm{condition},n)$: restricted number of compositions of $n$ and $T$: set

Permalink:

http://dlmf.nist.gov/26.11.E1

Encodings:

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See also:

Annotations for §26.11 and Ch.26

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26.11.6 $$c(\in T,n)=F_{n-1},$$ $n\ge 1$.

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Symbols:

$\in$: element of, $c(\mathrm{condition},n)$: restricted number of compositions of $n$ , $n$: nonnegative integer, $T$: set and $F_n$: Fibonacci numbers

Permalink:

http://dlmf.nist.gov/26.11.E6

Encodings:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

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7: 14.27 Zeros

… ► $P_{\nu }^{\mu }\left(x\pm \mathrm{i}0\right)$ (either side of the cut) has exactly one zero in the interval $(-\mathrm{\infty },-1)$ if either of the following sets of conditions holds: …

8: 34.2 Definition: $\mathit{3}j$ Symbol

… ►They therefore satisfy the triangle conditions … ►If either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the $\mathit{3}j$ symbol is zero. When both conditions are satisfied the $\mathit{3}j$ symbol can be expressed as the finite sum …

9: 14.13 Trigonometric Expansions

… ►with conditional convergence for each. …

10: 32.5 Integral Equations

… ►satisfies ${\text{P}}_{\text{II}}$ with $\alpha =0$ and the boundary condition …