# compatibility conditions

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##### 1: 32.4 Isomonodromy Problems
###### §32.4(i) Definition
(32.4.3) is the compatibility condition of (32.4.1). … … The compatibility condition of (32.4.1) with …
##### 3: 34.10 Zeros
In a $\mathit{3j}$ 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 $\mathit{3j}$ symbol is zero. Similarly the $\mathit{6j}$ symbol (34.4.1) vanishes when the triangle conditions are not satisfied by any of the four $\mathit{3j}$ symbols in the summation. …However, the $\mathit{3j}$ and $\mathit{6j}$ 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
26.10.1 $p\left(\mathcal{D},0\right)=p\left(\mathcal{D}k,0\right)=p\left(\in\!S,0\right% )=1.$
26.10.5 $\sum_{n=0}^{\infty}p\left(\in\!S,n\right)q^{n}=\prod_{j\in S}\frac{1}{1-q^{j}}.$
26.10.6 $p\left(\mathcal{D},n\right)=\frac{1}{n}\sum_{t=1}^{n}p\left(\mathcal{D},n-t% \right)\sum_{\begin{subarray}{c}j\mathbin{|}t\\ \mbox{\scriptsizej odd}\end{subarray}}j,$
##### 5: 3.2 Linear Algebra
###### §3.2(iii) Condition of Linear Systems
The sensitivity of the solution vector $\mathbf{x}$ in (3.2.1) to small perturbations in the matrix $\mathbf{A}$ and the vector $\mathbf{b}$ is measured by the condition number …The larger the value $\kappa(\mathbf{A})$, the more ill-conditioned the system. …
###### §3.2(v) Condition of Eigenvalues
If $\mathbf{A}$ is nondefective and $\lambda$ is a simple zero of $p_{n}(\lambda)$, then the sensitivity of $\lambda$ to small perturbations in the matrix $\mathbf{A}$ 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 $c\left(0\right)=c\left(\in\!T,0\right)=1.$
26.11.6 $c\left(\in\!T,n\right)=F_{n-1},$ $n\geq 1$.
##### 7: Bibliography Y
• A. J. Yee (2004) Partitions with difference conditions and Alder’s conjecture. Proc. Natl. Acad. Sci. USA 101 (47), pp. 16417–16418.
• ##### 8: 14.27 Zeros
$P^{\mu}_{\nu}\left(x\pm i0\right)$ (either side of the cut) has exactly one zero in the interval $(-\infty,-1)$ if either of the following sets of conditions holds: …
##### 9: 34.2 Definition: $\mathit{3j}$ Symbol
They therefore satisfy the triangle conditionsIf either of the conditions (34.2.1) or (34.2.3) is not satisfied, then the $\mathit{3j}$ symbol is zero. When both conditions are satisfied the $\mathit{3j}$ symbol can be expressed as the finite sum …
##### 10: 14.13 Trigonometric Expansions
with conditional convergence for each. …