permutation%20symmetry

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11: Bibliography C

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B. C. Carlson (2004) Symmetry in c, d, n of Jacobian elliptic functions. J. Math. Anal. Appl. 299 (1), pp. 242–253.

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External Links:: ISSN 0022-247X, Document, MathReview Entry, ZentralBlatt
Referenced by:: §19.25(v), §19.25(v), §22.13(i), §22.6(ii), §22.6(iii), §22.8(i), §22.8(ii), Profile Bille C. Carlson.
Encodings:: BibTeX

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B. C. Carlson (2011) Permutation symmetry for theta functions. J. Math. Anal. Appl. 378 (1), pp. 42–48.

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External Links:: ISSN 0022-247X, Document, FullText, MathReview (Mohammad Ahmad), ZentralBlatt
Referenced by:: §20.11(v), §20.7(i), §20.7(i), §20.7(ii), §20.7(ii), §20.7(ii), §20.7(iii), §20.7(iii), §20.7(vii), §20.7(vii), Profile Bille C. Carlson.
Encodings:: BibTeX

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B. C. Carlson (1998) Elliptic Integrals: Symmetry and Symbolic Integration. In Tricomi’s Ideas and Contemporary Applied Mathematics (Rome/Turin, 1997), Atti dei Convegni Lincei, Vol. 147, pp. 161–181.

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External Links:: MathReview (S. L. Kalla), ZentralBlatt
Referenced by:: §19.26(iii), §19.29(i).
Encodings:: BibTeX

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P. A. Clarkson (1991) Nonclassical Symmetry Reductions and Exact Solutions for Physically Significant Nonlinear Evolution Equations. In Nonlinear and Chaotic Phenomena in Plasmas, Solids and Fluids (Edmonton, AB, 1990), W. Rozmus and J. A. Tuszynski (Eds.), pp. 72–79.

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External Links:: MathReview
Referenced by:: §29.19(ii).
Encodings:: BibTeX

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Combinatorial Object Server (website) Department of Computer Science, University of Victoria, Canada.

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Notes:: Permutations, set and integer partitions, sequences, graphs, and trees.
External Links:: Combinatorial Object Server
Referenced by:: 1st item.
Encodings:: BibTeX

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12: 20.7 Identities

… ►The symmetry, applicable also to §§20.7(iii) and 20.7(vii), is obtained by modifying traditional theta functions in the manner recommended by Carlson (2011) and used for further purposes by Fukushima (2012). … ►See Lawden (1989, pp. 19–20). This reference also gives the eleven additional identities for the permutations of the four theta functions. … ►

20.7.34

\frac{\theta_{1}\left(z,q^{2}\right)\theta_{3}\left(z,q^{2}\right)}{\theta_{1}% \left(z,iq\right)}=\frac{\theta_{2}\left(z,q^{2}\right)\theta_{4}\left(z,q^{2}% \right)}{\theta_{2}\left(z,iq\right)}=i^{-1/4}\sqrt{\frac{\theta_{2}\left(0,q^% {2}\right)\theta_{4}\left(0,q^{2}\right)}{2}}.

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Symbols:: $\theta_{\NVar{j}}\left(\NVar{z},\NVar{q}\right)$ : theta function, $\mathrm{i}$ : imaginary unit, $z$ : complex and $q$ : nome
Referenced by:: §20.7(iv), Erratum (V1.0.5) for Chapters 8, 20, 36
Permalink:: http://dlmf.nist.gov/20.7.E34
Encodings:: pMML, png, TeX
Addition (effective with 1.0.5):: This equation has been added. See Walker (2012).
See also:: Annotations for §20.7(ix), §20.7 and Ch.20

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13: 34.3 Basic Properties: $\mathit{3j}$ Symbol

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§34.3(ii) Symmetry

►Even permutations of columns of a

\mathit{3j}

symbol leave it unchanged; odd permutations of columns produce a phase factor

(-1)^{j_{1}+j_{2}+j_{3}}

, for example, … ►Equations (34.3.11) and (34.3.12) are called Regge symmetries. Additional symmetries are obtained by applying (34.3.8)–(34.3.10) to (34.3.11)) and (34.3.12). …

14: 32.14 Combinatorics

… ►Let

S_{N}

be the group of permutations

\boldsymbol{\pi}

of the numbers

1,2,\dots,N

(§26.2). … ►

32.14.1

\lim_{N\to\infty}\mathrm{Prob}\left(\frac{\ell_{N}(\boldsymbol{\pi})-2\sqrt{N}% }{N^{1/6}}\leq s\right)=F(s),

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Symbols:: $\boldsymbol{\pi}(m)$ : permutations, $\ell_{N}(\boldsymbol{\pi})$ : length and $F(s)$ : distribution function
Permalink:: http://dlmf.nist.gov/32.14.E1
Encodings:: pMML, png, TeX
See also:: Annotations for §32.14 and Ch.32

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15: 26.17 The Twelvefold Way

§26.17 The Twelvefold Way

…

16: 19.16 Definitions

… ►which is homogeneous and of degree

-a

in the

z

’s, and unchanged when the same permutation is applied to both sets of subscripts

1,\dots,n

. … …

17: 20 Theta Functions

Chapter 20 Theta Functions

…

18: 19.25 Relations to Other Functions

… ►The transformations in §19.7(ii) result from the symmetry and homogeneity of functions on the right-hand sides of (19.25.5), (19.25.7), and (19.25.14). …Thus the five permutations induce five transformations of Legendre’s integrals (and also of the Jacobian elliptic functions). … ►With

0\leq k^{2}\leq 1

and

\mathrm{p,q,r}

any permutation of the letters

\mathrm{c,d,n}

, define … ►In (19.25.38) and (19.25.39)

j

k

\ell

is any permutation of the numbers

1,2,3

. … ►(

{F_{1}}

and

F_{D}

are equivalent to the

R

-function of 3 and

n

variables, respectively, but lack full symmetry.) …

19: Bibliography N

… ►

D. Naylor (1989) On an integral transform involving a class of Mathieu functions. SIAM J. Math. Anal. 20 (6), pp. 1500–1513.

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External Links:: ISSN 0036-1410, Document, MathReview (K. C. Gupta), ZentralBlatt
Referenced by:: §28.26(ii).
Encodings:: BibTeX

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W. J. Nellis and B. C. Carlson (1966) Reduction and evaluation of elliptic integrals. Math. Comp. 20 (94), pp. 223–231.

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External Links:: FullText, MathReview (I. A. Stegun), ZentralBlatt
Referenced by:: §19.37(iv).
Encodings:: BibTeX

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G. Nemes and A. B. Olde Daalhuis (2016) Uniform asymptotic expansion for the incomplete beta function. SIGMA Symmetry Integrability Geom. Methods Appl. 12, pp. 101, 5 pages.

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External Links:: ISSN 1815-0659, Document, Link, MathReview Entry, ZentralBlatt
Referenced by:: §8.18(ii), §8.18(ii), Erratum (V1.0.14) for Subsections 8.18(ii)–8.11(v).
Encodings:: BibTeX

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M. Noumi and Y. Yamada (1999) Symmetries in the fourth Painlevé equation and Okamoto polynomials. Nagoya Math. J. 153, pp. 53–86.

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External Links:: ISSN 0027-7630, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.8(iv).
Encodings:: BibTeX

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M. Noumi (2004) Painlevé Equations through Symmetry. Translations of Mathematical Monographs, Vol. 223, American Mathematical Society, Providence, RI.

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Notes:: Translated from the 2000 Japanese original by the author
External Links:: ISBN 0-8218-3221-2, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.2(v).
Encodings:: BibTeX

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20: Peter A. Clarkson

… ►Clarkson has published numerous papers on integrable systems (primarily Painlevé equations), special functions, and symmetry methods for differential equations. … Kruskal, he developed the “direct method” for determining symmetry solutions of partial differential equations in New similarity reductions of the Boussinesq equation (with M. …