application to combinatorics

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1: 26.20 Physical Applications

… ►An English translation of Pólya (1937) on applications of combinatorics to chemistry has been published as Pólya and Read (1987). …The latter reference also describes chemical applications of other combinatorial techniques. ►Applications of combinatorics, especially integer and plane partitions, to counting lattice structures and other problems of statistical mechanics, of which the Ising model is the principal example, can be found in Montroll (1964), Godsil et al. (1995), Baxter (1982), and Korepin et al. (1993). For an application of statistical mechanics to combinatorics, see Bressoud (1999). …

2: 26.19 Mathematical Applications

… ►Combinatorics has applications to analysis, algebra, and geometry. …Partitions and plane partitions have applications to representation theory (Bressoud (1999), Macdonald (1995), and Sagan (2001)) and to special functions (Andrews et al. (1999) and Gasper and Rahman (2004)). …

3: Mourad E. H. Ismail

… ►Ismail has published numerous papers on special functions, orthogonal polynomials, approximation theory, combinatorics, asymptotics, and related topics. … ► Suslov), Kluwer Academic Publishers, 2001; Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics (with F. … Garvan), Kluwer Academic Publishers, 2001; and Theory and Applications of Special Functions: A volume dedicated to Mizan Rahman (with E. … ►Ismail serves on several editorial boards including the Cambridge University Press book series Encyclopedia of Mathematics and its Applications, and on the editorial boards of 9 journals including Proceedings of the American Mathematical Society (Integrable Systems and Special Functions Editor); Constructive Approximation; Journal of Approximation Theory; and Integral Transforms and Special Functions. …

4: George E. Andrews

… ►An expert on

q

-series, he is the author of

q

-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra. … ►Andrews was elected to the American Academy of Arts and Sciences in 1997, and to the National Academy of Sciences (USA) in 2003. …Andrews served as President of the AMS from February 1, 2009 to January 31, 2011, and became a Fellow of the AMS in 2012. …

5: Bibliography T

… ►

N. M. Temme (1990b) Uniform asymptotic expansions of a class of integrals in terms of modified Bessel functions, with application to confluent hypergeometric functions. SIAM J. Math. Anal. 21 (1), pp. 241–261.

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External Links:: ISSN 0036-1410, Document, MathReview (Pablo Martín), ZentralBlatt
Referenced by:: §13.8(iii), §13.8(iii).
Encodings:: BibTeX

… ►

N. M. Temme (1992b) Asymptotic inversion of the incomplete beta function. J. Comput. Appl. Math. 41 (1-2), pp. 145–157.

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Notes:: Asymptotic methods in analysis and combinatorics
External Links:: ISSN 0377-0427, Document, MathReview (Dumitru Acu), ZentralBlatt
Referenced by:: §8.18(ii).
Encodings:: BibTeX

… ►

C. A. Tracy and H. Widom (1997) On exact solutions to the cylindrical Poisson-Boltzmann equation with applications to polyelectrolytes. Phys. A 244 (1-4), pp. 402–413.

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External Links:: ISSN 0378-4371, Document, MathReview (Vitor R. Vieira)
Referenced by:: §32.11(iv).
Encodings:: BibTeX

… ►

A. Tucker (2006) Applied Combinatorics. 5th edition, John Wiley and Sons, New York.

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Referenced by:: §26.15.
Encodings:: BibTeX

►

S. A. Tumarkin (1959) Asymptotic solution of a linear nonhomogeneous second order differential equation with a transition point and its application to the computations of toroidal shells and propeller blades. J. Appl. Math. Mech. 23, pp. 1549–1565.

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External Links:: Document, MathReview (F. W. J. Olver), ZentralBlatt
Referenced by:: §9.1, Notations E, Notations E.
Encodings:: BibTeX

…

6: 32.14 Combinatorics

§32.14 Combinatorics

… ►With

1\leq m_{1}<\cdots<m_{n}\leq N

\boldsymbol{\pi}(m_{1}),\boldsymbol{\pi}(m_{2}),\dots,\boldsymbol{\pi}(m_{n})

is said to be an increasing subsequence of

\boldsymbol{\pi}

of length

n

when

\boldsymbol{\pi}(m_{1})<\boldsymbol{\pi}(m_{2})<\cdots<\boldsymbol{\pi}(m_{n})

. … ►

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

… ►

32.14.3

w(x)\sim\operatorname{Ai}\left(x\right),

x\to+\infty

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Symbols:: $\operatorname{Ai}\left(\NVar{z}\right)$ : Airy function, $\sim$ : asymptotic equality and $x$ : real
Permalink:: http://dlmf.nist.gov/32.14.E3
Encodings:: pMML, png, TeX
See also:: Annotations for §32.14 and Ch.32

►

32.14.4

w(x)\sim\sqrt{-\tfrac{1}{2}x},

x\to-\infty

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Symbols:: $\sim$ : asymptotic equality and $x$ : real
Permalink:: http://dlmf.nist.gov/32.14.E4
Encodings:: pMML, png, TeX
See also:: Annotations for §32.14 and Ch.32

…

7: Bibliography B

… ►

E. Bannai and T. Ito (1984) Algebraic Combinatorics. I: Association Schemes. The Benjamin/Cummings Publishing Co., Inc., Menlo Park, CA.

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External Links:: ISBN 0-8053-0490-8, MathReview Entry, ZentralBlatt
Referenced by:: §18.28(xi).
Encodings:: BibTeX

►

E. Bannai (1990) Orthogonal Polynomials in Coding Theory and Algebraic Combinatorics. In Orthogonal Polynomials (Columbus, OH, 1989), NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., Vol. 294, pp. 25–53.

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External Links:: MathReview, ZentralBlatt
Referenced by:: §18.38(iii).
Encodings:: BibTeX

… ►

A. P. Bassom, P. A. Clarkson, C. K. Law, and J. B. McLeod (1998) Application of uniform asymptotics to the second Painlevé transcendent. Arch. Rational Mech. Anal. 143 (3), pp. 241–271.

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External Links:: ISSN 0945-8396, Document, MathReview (Andrei A. Kapaev), ZentralBlatt
Referenced by:: §32.11(ii).
Encodings:: BibTeX

… ►

C. Bingham, T. Chang, and D. Richards (1992) Approximating the matrix Fisher and Bingham distributions: Applications to spherical regression and Procrustes analysis. J. Multivariate Anal. 41 (2), pp. 314–337.

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External Links:: ISSN 0047-259X, Document, MathReview (A. L. Rukhin), ZentralBlatt
Referenced by:: §35.10, §35.9.
Encodings:: BibTeX

… ►

F. Bowman (1953) Introduction to Elliptic Functions with Applications. English Universities Press, Ltd., London.

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Notes:: Reprinted by Dover Publications Inc., New York, 1961.
External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §19.30(iii), §19.32, §19.33(i), §22.15(i), §22.15(ii), §22.15(ii), §22.18(ii), §22.19(v), Ch.22.
Encodings:: BibTeX

…

8: 15.17 Mathematical Applications

§15.17 Mathematical Applications

… ►This topic is treated in §§15.10 and 15.11. … ►

§15.17(iii) Group Representations

… ►

§15.17(iv) Combinatorics

►In combinatorics, hypergeometric identities classify single sums of products of binomial coefficients. …

9: Bibliography M

… ►

T. M. MacRobert (1967) Spherical Harmonics. An Elementary Treatise on Harmonic Functions with Applications. 3rd edition, International Series of Monographs in Pure and Applied Mathematics, Vol. 98, Pergamon Press, Oxford.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: Ch.14.
Encodings:: BibTeX

►

Magma (website) Computational Algebra Group, School of Mathematics and Statistics, University of Sydney, Australia.

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Notes:: A software package (the successor to CAYLEY) designed to solve computationally hard problems in algebra, number theory, geometry, and combinatorics, and to provide a mathematically rigorous environment for computing with algebraic, number-theoretic, combinatoric, and geometric objects.
External Links:: Magma
Referenced by:: 4th item.
Encodings:: BibTeX

… ►

L. C. Maximon (1955) On the evaluation of indefinite integrals involving the special functions: Application of method. Quart. Appl. Math. 13, pp. 84–93.

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External Links:: MathReview (A. Erdélyi), ZentralBlatt
Referenced by:: §14.17(i).
Encodings:: BibTeX

… ►

N. Michel and M. V. Stoitsov (2008) Fast computation of the Gauss hypergeometric function with all its parameters complex with application to the Pöschl-Teller-Ginocchio potential wave functions. Comput. Phys. Comm. 178 (7), pp. 535–551.

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External Links:: ISSN 0010-4655, Document, FullText, MathReview Entry, ZentralBlatt
Referenced by:: §15.19(i), §15.19(i).
Encodings:: BibTeX

… ►

R. E. Moore (1979) Methods and Applications of Interval Analysis. SIAM Studies in Applied Mathematics, Vol. 2, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA.

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External Links:: ISBN 0-89871-161-4, MathReview (H. Ratschek), ZentralBlatt
Referenced by:: §3.1(ii).
Encodings:: BibTeX

…

10: Bibliography G

… ►

F. G. Garvan and M. E. H. Ismail (Eds.) (2001) Symbolic Computation, Number Theory, Special Functions, Physics and Combinatorics. Developments in Mathematics, Vol. 4, Kluwer Academic Publishers, Dordrecht.

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External Links:: ISBN 1-4020-0101-0, MathReview Entry
Referenced by:: Profile Mourad E.H. Ismail.
Encodings:: BibTeX

… ►

C. D. Godsil, M. Grötschel, and D. J. A. Welsh (1995) Combinatorics in Statistical Physics. In Handbook of Combinatorics, Vol. 2, R. L. Graham, M. Grötschel, and L. Lovász (Eds.), pp. 1925–1954.

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External Links:: MathReview Entry, ZentralBlatt
Referenced by:: §26.20.
Encodings:: BibTeX

… ►

R. L. Graham, M. Grötschel, and L. Lovász (Eds.) (1995) Handbook of Combinatorics. Vols. 1, 2. Elsevier Science B.V., Amsterdam.

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External Links:: ISBN 0-444-88002-X, MathReview, ZentralBlatt
Referenced by:: §26.19, §26.20, Ch.26.
Encodings:: BibTeX

… ►

T. V. Gramtcheff (1981) An application of Airy functions to the Tricomi problem. Math. Nachr. 102 (1), pp. 169–181.

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External Links:: ISSN 0025-584X, Document, MathReview (John M. S. Rassias), ZentralBlatt
Referenced by:: §9.16.
Encodings:: BibTeX

… ►

A. Gray, G. B. Mathews, and T. M. MacRobert (1922) A Treatise on Bessel Functions and their Applications to Physics. 2nd edition, Macmillan and Co., London.

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Notes:: Table errata: Math. Comp. v. 24 (1970), p. 240, v. 31 (1978), pp. 806 and 1046, and v. 32 (1978), pp. 318. Reprinted by Dover Publications Inc., New York, 1966.
External Links:: MathReview, ZentralBlatt
Referenced by:: §10.21(x), §10.73(i), §10.73(i).
Encodings:: BibTeX

…