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

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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:

pMML, png, TeX

See also:

Annotations for §26.11 and Ch.26

… ►

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

…

2: 3.2 Linear Algebra

… ►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 ►

3.2.16 $$\kappa (𝐀)={\Vert 𝐀\Vert }_p{\Vert {𝐀}^{-1}\Vert }_p,$$

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Defines:

$\kappa (𝐀)$: condition number (locally)

Permalink:

http://dlmf.nist.gov/3.2.E16

Encodings:

pMML, png, TeX

See also:

Annotations for §3.2(iii), §3.2 and Ch.3

… ►

3.2.17 $$\frac{{\Vert {𝐱}^{\ast }-𝐱\Vert }_p}{{\Vert 𝐱\Vert }_p}\le \kappa (𝐀)\frac{{\Vert 𝐫\Vert }_p}{{\Vert 𝐛\Vert }_p}.$$

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Symbols:

$\kappa (𝐀)$: condition number

Permalink:

http://dlmf.nist.gov/3.2.E17

Encodings:

pMML, png, TeX

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Annotations for §3.2(iii), §3.2 and Ch.3

… ►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 ►

3.2.20 $$\kappa (\lambda )=\frac{1}{\left|{𝐲}^{\mathrm{T}}𝐱\right|},$$

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Symbols:

$\kappa (𝐀)$: condition number and $\lambda$: eigenvalue

Permalink:

http://dlmf.nist.gov/3.2.E20

Encodings:

pMML, png, TeX

See also:

Annotations for §3.2(v), §3.2 and Ch.3

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3: 26.10 Integer Partitions: Other Restrictions

… ►

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:

pMML, png, TeX

See also:

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

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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

… ►

26.10.7

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Symbols:

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

A&S Ref:

24.2.2

Permalink:

http://dlmf.nist.gov/26.10.E7

Encodings:

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See also:

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

… ►

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

…

4: 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. …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. …

5: 34.2 Definition: $\mathit{3}j$ Symbol

… ►They therefore satisfy the triangle conditions …The corresponding projective quantum numbers $m_1,m_2,m_3$ are given by … ►

►►

Figure 34.2.1: Angular momenta $j_r$ and projective quantum numbers $m_r$, $r=1,2,3$. Magnify

►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 …

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 $3^k=q{2}^k+r$ with , then equality holds in (27.13.2) provided $r+q\le 2^k$, a condition that is satisfied with at most a finite number of exceptions. …

8: 5.11 Asymptotic Expansions

… ►

5.11.2 $$\psi \left(z\right)\sim \ln z-\frac{1}{2z}-\sum _{k=1}^{\mathrm{\infty }}\frac{B_{2k}}{2k{z}^{2k}}.$$

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Symbols:

$B_n$: Bernoulli numbers, $\sim$: Poincaré asymptotic expansion, $\psi \left(z\right)$: psi (or digamma) function, $\ln z$: principal branch of logarithm function, $k$: nonnegative integer and $z$: complex variable

A&S Ref:

6.3.18

Referenced by:

§5.11(i), §5.11(ii), §5.4(iii)

Permalink:

http://dlmf.nist.gov/5.11.E2

Encodings:

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See also:

Annotations for §5.11(i), §5.11 and Ch.5

►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 $k\to \mathrm{\infty }$ 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 condition … ►In Condition (c) we have …

10: 1.10 Functions of a Complex Variable

… ►The last condition means that given $ϵ$ ($>0$) there exists a number $a_0\in [a,b)$ that is independent of $z$ and is such that …