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1: 26.13 Permutations: Cycle Notation

… ► $\sigma \in {𝔖}_n$ is a one-to-one and onto mapping from $\{1,2,\mathrm{\dots },n\}$ to itself. … ►Cycles of length one are fixed points. … ►An element of ${𝔖}_n$ with $a_1$ fixed points, $a_2$ cycles of length $2,\mathrm{\dots },a_n$ cycles of length $n$, where $n=a_1+2a_2+\mathrm{\cdots }+n{a}_n$, is said to have cycle type $\left(a_1,a_2,\mathrm{\dots },a_n\right)$. … ►A derangement is a permutation with no fixed points. The derangement number, $d(n)$, is the number of elements of ${𝔖}_n$ with no fixed points: …

2: Bibliography G

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Algorithm 726: ORTHPOL — a package of routines for generating orthogonal polynomials and Gauss-type quadrature rules. ACM Trans. Math. Software 20 (1), pp. 21–62.

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Notes:

For remark see same journal 24(1998), p. 355.

External Links:

Document, ISSN 0098-3500, GAMS (module 726; package TOMS), ZentralBlatt

Referenced by:

2nd item, §3.5(v), §3.5(v).

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Algorithm 939: computation of the Marcum Q-function. ACM Trans. Math. Softw. 40 (3), pp. 20:1–20:21.

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Notes:

Includes Fortran program claiming 12S accuracy

External Links:

ISSN 0098-3500, Link, Document, MathReview Entry, ZentralBlatt

Referenced by:

3rd item, §10.77(ix).

Encodings:

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Computing the zeros and turning points of solutions of second order homogeneous linear ODEs. SIAM J. Numer. Anal. 41 (3), pp. 827–855.

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External Links:

ISSN 0036-1429, Document, MathReview, ZentralBlatt

Referenced by:

§10.74(vi), §3.8(v).

Encodings:

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Lectures on Integration of the Equations of Motion of a Rigid Body About a Fixed Point. Translated from the Russian by J. Shorr-Kon, Office of Technical Services, U. S. Department of Commerce, Washington, D.C..

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Notes:

Published for the National Science Foundation by the Israel Program for Scientific Translations, 1960.

External Links:

MathReview, ZentralBlatt

Referenced by:

§22.19(iv).

Encodings:

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Mutual integrability, quadratic algebras, and dynamical symmetry. Ann. Phys. 217 (1), pp. 1–20.

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External Links:

ISSN 0003-4916, Link, MathReview (A. O. Barut), ZentralBlatt

Referenced by:

§18.38(iii).

Encodings:

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3: 3.8 Nonlinear Equations

… ► … ►Let $z_1,z_2,\mathrm{\dots }$ be a sequence of approximations to a root, or fixed point, $\zeta$. … ►We have $p^{\prime }(20)=19!$ and $a_{19}=1+2+\mathrm{\cdots }+20=210$. … ►For an arbitrary starting point $z_0\in ℂ$, convergence cannot be predicted, and the boundary of the set of points $z_0$ that generate a sequence converging to a particular zero has a very complicated structure. It is called a Julia set. …

4: 36.5 Stokes Sets

… ►Stokes sets are surfaces (codimension one) in $𝐱$ space, across which ${\mathrm{\Psi }}_K(𝐱;k)$ or ${\mathrm{\Psi }}^{(\mathrm{U})}(𝐱;k)$ acquires an exponentially-small asymptotic contribution (in $k$), associated with a complex critical point of ${\mathrm{\Phi }}_K$ or . …where $j$ denotes a real critical point (36.4.1) or (36.4.2), and $\mu$ denotes a critical point with complex $t$ or $s,t$, connected with $j$ by a steepest-descent path (that is, a path where $\mathrm{\Re }\mathrm{\Phi }=\mathrm{constant}$) in complex $t$ or $(s,t)$ space. … ►The Stokes set is itself a cusped curve, connected to the cusp of the bifurcation set: … ►This consists of a cusp-edged sheet connected to the cusp-edged sheet of the bifurcation set and intersecting the smooth sheet of the bifurcation set. … ►Red and blue numbers in each region correspond, respectively, to the numbers of real and complex critical points that contribute to the asymptotics of the canonical integral away from the bifurcation sets. …

5: Bibliography D

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Complex zeros of cylinder functions. Math. Comp. 20 (94), pp. 215–222.

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Notes:

For a numerical error in one of the zeros see Amos (1985).

External Links:

ISSN 0025-5718, FullText, MathReview, ZentralBlatt

Referenced by:

2nd item.

Encodings:

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Uniform asymptotic solutions of second-order linear differential equations having a double pole with complex exponent and a coalescing turning point. SIAM J. Math. Anal. 21 (6), pp. 1594–1618.

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External Links:

ISSN 0036-1410, Document, MathReview (F. W. J. Olver), ZentralBlatt

Referenced by:

§14.26, §2.8(vi).

Encodings:

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Uniform asymptotic solutions of second-order linear differential equations having a simple pole and a coalescing turning point in the complex plane. SIAM J. Math. Anal. 25 (2), pp. 322–353.

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External Links:

ISSN 0036-1410, Document, MathReview (H. C. Howard), ZentralBlatt

Referenced by:

§2.8(vi).

Encodings:

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Convergent expansions for solutions of linear ordinary differential equations having a simple turning point, with an application to Bessel functions. Stud. Appl. Math. 107 (3), pp. 293–323.

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External Links:

ISSN 0022-2526, Document, MathReview, ZentralBlatt

Referenced by:

§10.20(ii), §2.8(i).

Encodings:

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Olver’s error bound methods applied to linear ordinary differential equations having a simple turning point. Anal. Appl. (Singap.) 12 (4), pp. 385–402.

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External Links:

ISSN 0219-5305, Document, Link, MathReview (Thomas Mbah Acho), ZentralBlatt

Referenced by:

5th item, §14.32.

Encodings:

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6: 36.4 Bifurcation Sets

§36.4 Bifurcation Sets

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§36.4(i) Formulas

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Critical Points for Cuspoids

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Critical Points for Umbilics

… ►Elliptic umbilic bifurcation set (codimension three): for fixed $z$, the section of the bifurcation set is a three-cusped astroid …

7: 26.12 Plane Partitions

… ►A plane partition, $\pi$, of a positive integer $n$, is a partition of $n$ in which the parts have been arranged in a 2-dimensional array that is weakly decreasing (nonincreasing) across rows and down columns. … ►It is useful to be able to visualize a plane partition as a pile of blocks, one block at each lattice point $(h,j,k)\in \pi$. … ►The plane partition in Figure 26.12.1 is an example of a cyclically symmetric plane partition. … ►A plane partition is totally symmetric if it is both symmetric and cyclically symmetric. … ►The example of a strict shifted plane partition also satisfies the conditions of a descending plane partition. …

8: 12.11 Zeros

… ►If $a>-\frac{1}{2}$, then $V(a,x)$ has no positive real zeros, and if $a=\frac{3}{2}-2n$, $n\in \mathbb{Z}$, then $V(a,x)$ has a zero at $x=0$. … ►When $a>-\frac{1}{2}$ the zeros are asymptotically given by $z_{a,s}$ and $\overline{z_{a,s}}$, where $s$ is a large positive integer and … ►When $a=\frac{1}{2}$ these zeros are the same as the zeros of the complementary error function $\mathrm{erfc}(z/\sqrt{2})$; compare (12.7.5). … ►For large negative values of $a$ the real zeros of $U(a,x)$, $U^{\prime }(a,x)$, $V(a,x)$, and $V^{\prime }(a,x)$ can be approximated by reversion of the Airy-type asymptotic expansions of §§12.10(vii) and 12.10(viii). For example, let the $s$th real zeros of $U(a,x)$ and $U^{\prime }(a,x)$, counted in descending order away from the point $z=2\sqrt{-a}$, be denoted by $u_{a,s}$ and $u_{a,s}^{\prime }$, respectively. …

9: 19.36 Methods of Computation

… ►If (19.36.1) is used instead of its first five terms, then the factor ${(3r)}^{-1/6}$ in Carlson (1995, (2.2)) is changed to ${(3r)}^{-1/8}$. … ►If $x$, $y$, and $z$ are permuted so that , then the computation of $R_F(x,y,z)$ is fastest if we make $c_0^2\le a_0^2/2$ by choosing $\theta =1$ when or $\theta =-1$ when $y\ge (x+z)/2$. … ►We compute $R_F(1,2,4)$ by setting $\theta =1$, $t_0=c_0=1$, and $a_0=\sqrt{3}$. … ►Near these points there will be loss of significant figures in the computation of $R_J$ or $R_D$. … ►For computation of Legendre’s integral of the third kind, see Abramowitz and Stegun (1964, §§17.7 and 17.8, Examples 15, 17, 19, and 20). …

10: 23.9 Laurent and Other Power Series

… ►Let $z_0(\ne 0)$ be the nearest lattice point to the origin, and define … ►

$c_2={\displaystyle \frac{1}{20}}{g}_2,$

… ►Also, Abramowitz and Stegun (1964, (18.5.25)) supplies the first 22 terms in the reverted form of (23.9.2) as $1/\mathrm{\wp }\left(z\right)\to 0$. … ►where $a_{0,0}=1$, $a_{m,n}=0$ if either $m$ or , and …For $a_{m,n}$ with $m=0,1,\mathrm{\dots },12$ and $n=0,1,\mathrm{\dots },8$, see Abramowitz and Stegun (1964, p. 637).