# condition numbers

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##### 1: 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 $c\left(\in\!T,n\right)=F_{n-1},$ $n\geq 1$.
##### 2: 3.2 Linear Algebra
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
3.2.16 $\kappa(\mathbf{A})=\|\mathbf{A}\|_{p}\;\|\mathbf{A}^{-1}\|_{p},$
3.2.17 $\frac{\|\mathbf{x}^{*}-\mathbf{x}\|_{p}}{\|\mathbf{x}\|_{p}}\leq\kappa(\mathbf% {A})\frac{\|\mathbf{r}\|_{p}}{\|\mathbf{b}\|_{p}}.$
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
3.2.20 $\kappa(\lambda)=\frac{1}{\left|\mathbf{y}^{\rm T}\mathbf{x}\right|},$
##### 3: 26.10 Integer Partitions: Other Restrictions
26.10.1 $p\left(\mathcal{D},0\right)=p\left(\mathcal{D}k,0\right)=p\left(\in\!S,0\right% )=1.$
26.10.6 $p\left(\mathcal{D},n\right)=\frac{1}{n}\sum_{t=1}^{n}p\left(\mathcal{D},n-t% \right)\sum_{\begin{subarray}{c}j\mathbin{|}t\\ \mbox{\scriptsizej odd}\end{subarray}}j,$
26.10.7 $\sum(-1)^{k}p\left(\mathcal{D},n-\tfrac{1}{2}(3k^{2}\pm k)\right)=\begin{cases% }(-1)^{r},&n=3r^{2}\pm r,\\ 0,&\mbox{otherwise},\end{cases}$
26.10.8 $\sum(-1)^{k}p\left(\mathcal{D},n-(3k^{2}\pm k)\right)=\begin{cases}1,&n=\tfrac% {1}{2}(r^{2}\pm r),\\ 0,&\mbox{otherwise},\end{cases}$
##### 4: 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. …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. …
##### 5: 34.2 Definition: $\mathit{3j}$ Symbol
They therefore satisfy the triangle conditions …The corresponding projective quantum numbers $m_{1},m_{2},m_{3}$ are given by … If 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 …
##### 6: 26.12 Plane Partitions
Then the number of plane partitions in $B(r,s,t)$ is … The number of symmetric plane partitions in $B(r,r,t)$ is … The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. The number of descending plane partitions in $B(r,r,r)$ is …
##### 7: 27.13 Functions
If $3^{k}=q2^{k}+r$ with $0, then equality holds in (27.13.2) provided $r+q\leq 2^{k}$, a condition that is satisfied with at most a finite number of exceptions. …
##### 8: 5.11 Asymptotic Expansions
For the Bernoulli numbers $B_{2k}$, see §24.2(i). With the same conditions, …For explicit formulas for $g_{k}$ in terms of Stirling numbers see Nemes (2013a), and for asymptotic expansions of $g_{k}$ as $k\to\infty$ see Boyd (1994) and Nemes (2015a). … Next, and again with the same conditions, …
##### 9: 2.10 Sums and Sequences
Sufficient conditions for the validity of this second result are: … First, the conditions can be weakened. …For example, Condition (b) can be replaced by: … Furthermore, (2.10.31) remains valid with the weaker conditionIn Condition (c) we have …
##### 10: 1.10 Functions of a Complex Variable
The last condition means that given $\epsilon$ ($>0$) there exists a number $a_{0}\in[a,b)$ that is independent of $z$ and is such that …