conditioning

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1: 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. …

2: 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.

ⓘ

Symbols:: $\in$ : element of, $p\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : 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

►

Table 26.10.1: Partitions restricted by difference conditions, or equivalently with parts from

A_{j,k}

► ►►

	$p\left(\mathcal{D},n\right)$	$p\left(\mathcal{D}2,n\right)$	$p\left(\mathcal{D}2,\hbox{}\!\!\in\!T,n\right)$	$p\left(\mathcal{D}^{\prime}3,n\right)$
…

► … ►

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{\scriptsize$j$ odd}\end{subarray}}j,

ⓘ

Symbols:: $p\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : 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

\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}

ⓘ

Symbols:: $p\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : 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:: pMML, png, TeX
See also:: Annotations for §26.10(iii), §26.10 and Ch.26

… ►

26.10.12

p\left(\mathcal{D},n\right)=p\left(\mathcal{O},n\right),

ⓘ

Symbols:: $p\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : 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

…

3: 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 …

4: 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 … ►

5: 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.

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{n}\right)$ : number of compositions of $n$ , $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : 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\left(\in\!T,n\right)=F_{n-1},

n\geq 1

ⓘ

Symbols:: $\in$ : element of, $c\left(\NVar{\mathrm{condition}},\NVar{n}\right)$ : 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

…

6: 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: …

7: 34.2 Definition: $\mathit{3j}$ 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{3j}

symbol is zero. When both conditions are satisfied the

\mathit{3j}

symbol can be expressed as the finite sum …

8: 14.13 Trigonometric Expansions

… ►with conditional convergence for each. …

9: 32.5 Integral Equations

… ►satisfies

\mbox{P}_{\mbox{\scriptsize II}}

with

\alpha=0

and the boundary condition …

10: 18.32 OP’s with Respect to Freud Weights

… ►These conditions on

Q(x)

have been strengthened and also relaxed in literature. … ►with

Q(x)

satisfying similar conditions as before, see Kasuga and Sakai (2003). …