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## ‣ Computational aspects of harmonic wavelet Galerkin methods and an application to a precipitation front propagation model

BARROS, Saulo R. M.; PEIXOTO, Pedro S.
Fonte: PERGAMON-ELSEVIER SCIENCE LTD Publicador: PERGAMON-ELSEVIER SCIENCE LTD
Tipo: Artigo de Revista Científica
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This article is dedicated to harmonic wavelet Galerkin methods for the solution of partial differential equations. Several variants of the method are proposed and analyzed, using the Burgers equation as a test model. The computational complexity can be reduced when the localization properties of the wavelets and restricted interactions between different scales are exploited. The resulting variants of the method have computational complexities ranging from O(N(3)) to O(N) (N being the space dimension) per time step. A pseudo-spectral wavelet scheme is also described and compared to the methods based on connection coefficients. The harmonic wavelet Galerkin scheme is applied to a nonlinear model for the propagation of precipitation fronts, with the front locations being exposed in the sizes of the localized wavelet coefficients. (C) 2011 Elsevier Ltd. All rights reserved.; Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq); CNPq

## ‣ Zeros of Gegenbauer and Hermite polynomials and connection coefficients

Area, I; Dimitrov, D. K.; Godoy, E.; Ronveaux, A.
Fonte: Amer Mathematical Soc Publicador: Amer Mathematical Soc
Tipo: Artigo de Revista Científica Formato: 1937-1951
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In this paper, sharp upper limit for the zeros of the ultraspherical polynomials are obtained via a result of Obrechkoff and certain explicit connection coefficients for these polynomials. As a consequence, sharp bounds for the zeros of the Hermite polynomials are obtained.

## ‣ Wavelet-Galerkin method for computations of electromagnetic fields - Computation of connection coefficients

Yang, S. Y.; Ni, G. Z.; Ho, S. L.; Machado, J. M.; Rahman, M. A.; Wong, H. C.
Fonte: Institute of Electrical and Electronics Engineers (IEEE) Publicador: Institute of Electrical and Electronics Engineers (IEEE)
Tipo: Artigo de Revista Científica Formato: 644-648
Português
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67.63897%
One of the keg issues which makes the wavelet-Galerkin method unsuitable for solving general electromagnetic problems is a lack of exact representations of the connection coefficients. This paper presents the mathematical formulae and computer procedures for computing some common connection coefficients, the characteristic of the present formulae and procedures is that the arbitrary point values of the connection co-efficients, rather than the dyadic point values, can be determined. A numerical example is also given to demonstrate the feasibility of using the wavelet-Galerkin method to solve engineering field problems.

## ‣ Wavelet-galerkin method for computations of electromagnetic fields-computation of connection coefficients

Yang, Shiyou; Ni, Guangzheng; Ho, S. L.; Machado, Jose Marcio; Rahman, M. A.; Wong, H. C.
Tipo: Artigo de Revista Científica Formato: 644-648
Português
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One of the key issues which makes the waveletGalerkin method unsuitable for solving general electromagnetic problems is a lack of exact representations of the connection coefficients. This paper presents the mathematical formulae and computer procedures for computing some common connection coefficients. The characteristic of the present formulae and procedures is that the arbitrary point values of the connection coefficients, rather than the dyadic point values, can be determined. A numerical example is also given to demonstrate the feasibility of using the wavelet-Galerkin method to solve engineering field problems. © 2000 IEEE.

## ‣ On the Connection Coefficients of the Chebyshev-Boubaker Polynomials

Barry, Paul
Fonte: Hindawi Publishing Corporation Publicador: Hindawi Publishing Corporation
Tipo: Artigo de Revista Científica
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The Chebyshev-Boubaker polynomials are the orthogonal polynomials whose coefficient arrays are defined by ordinary Riordan arrays. Examples include the Chebyshev polynomials of the second kind and the Boubaker polynomials. We study the connection coefficients of this class of orthogonal polynomials, indicating how Riordan array techniques can lead to closed-form expressions for these connection coefficients as well as recurrence relations that define them.

## ‣ On the connection and linearization problem for hypergeometric q-polynomials

Álvarez Nodarse, Renato; Arvesú, Jorge; Yáñez, Rafael José
Tipo: Artigo de Revista Científica Formato: application/pdf
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In the present paper, starting from the second-order difference hypergeometric equation on the non-uniform lattice x(s) satisfied by the set of discrete hypergeometric orthogonal q-polynomials {pn}, we find analytical expressions of the expansion coefficients of any q-polynomial rm(x(s)) on x(s) and of the product rm(x(s))qj(x(s)) in series of the set {pn}. These coefficients are given in terms of the polynomial coefficients of the second-order difference equations satisfied by the involved discrete hypergeometric q-polynomials.; This work has been partially supported by the European project INTAS-93-219-ext as well as by the Dirección General de Enseñanza Superior (DGES) of Spain under Grant BHA 2000-0206-C04-02 (R.A.N., J.A.) and PB 95-1205 (R.J.Y.) and by the Junta de Andalucía (R.J.Y.) under Grant FQM207.; 27 pages, no figures.-- MSC2000 code: 33D45.; MR#: MR1824666 (2002f:33025); Zbl#: Zbl 0983.33009

## ‣ Linearization and connection formulae involving squares of Gegenbauer polynomials

Sánchez-Ruiz, Jorge
Tipo: Artigo de Revista Científica Formato: text/html
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Several linearization-like and connection-like formulae relating the classical Gegenbauer polynomials and their squares are obtained using a theorem of the theory of generalized hypergeometric functions.; This work has been partially supported by the Junta de Andalucía, under the Research Grant FQM0207, and by the Spanish DGES Project PB96-0170.; 7 pages, no figures.-- MSC2000 code: 33C45.; MR#: MR1820610 (2003c:33014); Zbl#: Zbl 0978.33003

## ‣ Linearization and connection coefficients for hypergeometric-type polynomials

López Artés, Pedro; Sánchez Dehesa, Jesús; Martínez-Finkelshtein, Andrei; Sánchez-Ruiz, Jorge
Tipo: Artigo de Revista Científica Formato: text/html
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We consider the problem of finding closed analytical formulas for both the linearization and connection coefficients for hypergeometric-type polynomials, directly in terms of the corresponding differential equations. We illustrate the method by producing explicit formulas for Hermite polynomials.; This work has been partially supported by the Junta de Andalucía, under the research grants FQM0229 (P. L. A. and A. M. F.) and FQM0207 (J. S. D. and J. S. R.), and by the Spanish DGES projects PB96-0170 (J. S. R.) and PB95-1205. Also, support from the European project INTAS-93-219-EXT is acknowledged.; 12 pages, no figures.-- MSC2000 codes: 33C45, 26C05.-- Issue title: "Proceedings of the VIIIth Symposium on Orthogonal Polynomials and Their Application" (Seville, Spain, Sep 1997).; MR#: MR1662679 (99k:33006); Zbl#: Zbl 0927.33005

## ‣ Connection coefficients for Laguerre-Sobolev orthogonal polynomials

Marcellán, Francisco; Sánchez-Ruiz, Jorge
Tipo: Artigo de Revista Científica Formato: application/pdf
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^aLaguerre–Sobolev polynomials are orthogonal with respect to an inner product of the form $$\langle p,q\rangle=\int_0^{\infty}p(x)q(x)x^{\alpha}e^{-x}\,dx +\lambda\int_0^{\infty}p'(x)q'(x)\,d\mu(x),$$ with $\alpha>-1,\ \lambda>0$, and p,q in P, the linear space of polynomials with real coefficients.; For each of these two families of Laguerre–Sobolev polynomials [see attached full-text paper], here we give the explicit expression of the connection coefficients in their expansion as a series of standard Laguerre polynomials. The inverse connection problem of expanding Laguerre polynomials in series of Laguerre–Sobolev polynomials, and the connection problem relating two families of Laguerre–Sobolev polynomials with different parameters, are also considered.; This work has been supported by Dirección General de Investigación (MCyT) of Spain under Grant BFM2000-0206-C04-01 and INTAS Project 2000-272. J. Sánchez-Ruiz was also partially supported by the Junta de Andalucía, under the research Grant FQM0207.; 19 pages, no figures.-- MSC2000 codes: 33C45, 42C05.; MR#: MR1991819 (2004h:33020); Zbl#: Zbl 1033.42027

## ‣ Direct bijective computation of the generating series for 2 and 3-connection coefficients of the symmetric group

Morales, Alejandro H.; Vassilieva, Ekaterina A.
Tipo: Artigo de Revista Científica
Português
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We evaluate combinatorially certain connection coefficients of the symmetric group that count the number of factorizations of a long cycle as a product of three permutations. Such factorizations admit an important topological interpretation in terms of unicellular constellations on orientable surfaces. Algebraic computation of these coefficients was first done by Jackson using irreducible characters of the symmetric group. However, bijective computations of these coefficients are so far limited to very special cases. Thanks to a new bijection that refines the work of Schaeffer and Vassilieva, and Vassilieva, we give an explicit closed form evaluation of the generating series for these coefficients. The main ingredient in the bijection is a modified oriented tricolored tree tractable to enumerate. Finally, reducing this bijection to factorizations of a long cycle into two permutations, we get the analogue formula for the corresponding generating series.; Comment: 23 pages, 9 figures

## ‣ On Stokes Matrices in terms of Connection Coefficients

Guzzetti, Davide
Tipo: Artigo de Revista Científica
Português
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The classical problem of computing a complete system of Stokes multipliers of a linear system of ODEs of rank one in terms of some connection coefficients of an associated hypergeometric system of ODEs, is solved with no genericness assumptions on the residue matrix at zero, by an extension of the method of [3].; Comment: 53 pages, 10 figures

## ‣ A recurrence formula for Jack connection coefficients

Kanunnikov, Andrei L.; Vassilieva, Ekaterina A.
Tipo: Artigo de Revista Científica
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This article is devoted to the study of Jack connection coefficients, a generalization of the connection coefficients of the classical commutative subalgebras of the group algebra of the symmetric group closely related to the theory of Jack symmetric functions. First introduced by Goulden and Jackson (1996) these numbers indexed by three partitions of a given integer $n$ and the Jack parameter $\alpha$ are defined as the coefficients in the power sum expansion of the Cauchy sum for Jack symmetric functions. While very little is known about them, examples of computations for small values of $n$ tend to show that the nice properties of the special cases $\alpha =1$ (connection coefficients of the class algebra) and $\alpha = 2$ (connection coefficients of the double coset algebra) extend to general $\alpha$. Goulden and Jackson conjectured that Jack connection coefficients are polynomials in $\beta = \alpha-1$ with non negative integer coefficients given by some statistics on matchings on a set of $2n$ elements, the so called Matchings-Jack conjecture. In this paper we look at the case when two of the integer partitions are equal to the single part $(n)$ and use a framework by Lasalle (2008) for Jack symmetric functions to show that the coefficients satisfy a simple recurrence formula that makes their computation very effective and allow a better understanding of their properties. In particular we prove the Matchings-Jack conjecture in this case. Furthermore...

## ‣ The connection problem associated with a Selberg type integral and the $q$-Racah polynomials

Mimachi, Katsuhisa
Tipo: Artigo de Revista Científica
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The connection problem associated with a Selberg type integral is solved. The connection coefficients are given in terms of the $q$-Racah polynomials. As an application of the explicit expression of the connection coefficients, examples of the monodromy-invariant Hermitian form of non-diagonal type are presented. It is noteworthy that such Hermitian forms are intimately related with the correlation functions of non-diagonal type in $\hat{sl_2}$-confromal field theory.

## ‣ Explicit monomial expansions of the generating series for connection coefficients

Vassilieva, Ekaterina A.
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
47.09326%
This paper is devoted to the explicit computation of generating series for the connection coefficients of two commutative subalgebras of the group algebra of the symmetric group, the class algebra and the double coset algebra. As shown by Hanlon, Stanley and Stembridge (1992), these series gives the spectral distribution of some random matrices that are of interest to statisticians. Morales and Vassilieva (2009, 2011) found explicit formulas for these generating series in terms of monomial symmetric functions by introducing a bijection between partitioned hypermaps on (locally) orientable surfaces and some decorated forests and trees. Thanks to purely algebraic means, we recover the formula for the class algebra and provide a new simpler formula for the double coset algebra. As a salient ingredient, we derive a new explicit expression for zonal polynomials indexed by partitions of type [a,b,1^(n-a-b)].

## ‣ Connection coefficients for classical orthogonal polynomials of several variables

Iliev, Plamen; Xu, Yuan
Tipo: Artigo de Revista Científica
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Connection coefficients between different orthonormal bases satisfy two discrete orthogonal relations themselves. For classical orthogonal polynomials whose weights are invariant under the action of the symmetric group, connection coefficients between a basis consisting of products of hypergeometric functions and another basis obtained form the first one by applying a permutation are studied. For the Jacobi polynomials on the simplex, it is shown that the connection coefficients can be expressed in terms of Tratnik's multivariable Racah polynomials and their weights. This gives, in particular, a new interpretation of the hidden duality between the variables and the degree indices of the Racah polynomials, which lies at the heart of their bispectral properties. These techniques also lead to explicit formulas for connection coefficients of Hahn and Krawtchouk polynomials of several variables, as well as for orthogonal polynomials on balls and spheres.

## ‣ Bijective evaluation of the connection coefficients of the double coset algebra

Morales, Alejandro H.; Vassilieva, Ekaterina A.
Tipo: Artigo de Revista Científica
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This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partition $\nu$, gives the spectral distribution of some random real matrices that are of interest in random matrix theory. We provide an explicit evaluation of this series when $\nu=(n)$ in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between locally orientable partitioned hypermaps and some permuted forests.; Comment: 12 pages, 5 figures

## ‣ Polynomial properties of Jack connection coefficients and generalization of a result by D\'enes

Vassilieva, Ekaterina A.
Tipo: Artigo de Revista Científica
Português
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47.9069%
This article is devoted to the computation of Jack connection coefficients, a generalization of the connection coefficients of two classical commutative subalgebras of the group algebra of the symmetric group: the class algebra and the double coset algebra. The connection coefficients of these two algebraic structures are of significant interest in the study of Schur and zonal polynomials as well as the irreducible characters of the symmetric group and the zonal spherical functions. Furthermore they play an important role in combinatorics as they give the number of factorizations of a permutation into a product of permutations with given cyclic properties. Usually studied separately, these two families of coefficients share strong similar properties. First (partially) introduced by Goulden and Jackson in 1996, Jack connection coefficients provide a natural unified approach closely related to the theory of Jack polynomials, a family of bases in the ring of symmetric functions indexed by a parameter \alpha that generalizes both Schur (case \alpha = 1) and zonal polynomials (case \alpha = 2). Jack connection coefficients are also directly linked to Jack characters, a general view of the characters of the symmetric group and the zonal spherical functions. Goulden and Jackson conjectured that these coefficients are polynomials in \alpha with nice combinatorial properties...

## ‣ Finsleroid gives rise to the angle-preserving connection

Asanov, G. S.
Tipo: Artigo de Revista Científica
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37.44391%
The Finslerian unit ball is called the {\it Finsleroid} if the covering indicatrix is a space of constant curvature. We prove that Finsler spaces with such indicatrices possess the remarkable property that the tangent spaces are conformally flat with the conformal factor of the power dependence on the Finsler metric function. It is amazing but the fact that in such spaces the notion of the two-vector angle defined by the geodesic arc on the indicatrix can readily be induced from the Riemannian space obtained upon the conformal transformation, which opens up the straightforward way to induce also the connection coefficients and the concomitant curvature tensor. Thus, we are successfully inducing the Levi-Civita connection from the Riemannian space into the Finsleroid space, obtaining the isometric connection. The resultant connection coefficients are not symmetric. However, the metricity condition holds fine, that is, the produced covariant derivative of the Finsleroid metric tensor vanishes identically. The particular case underlined by the axial Finsleroid of the ${\mathbf\cF\cF^{PD}_{g}}$-type is explicitly evaluated in detail. Keywords: Finsler metrics, connection, curvature, conformal properties.

## ‣ Connection coefficients for basic Harish-Chandra series

Stokman, Jasper V.
Tipo: Artigo de Revista Científica
Português
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Basic Harish-Chandra series are asymptotically free meromorphic solutions of the system of basic hypergeometric difference equations associated to root systems. The associated connection coefficients are explicitly computed in terms of Jacobi theta functions. We interpret the connection coefficients as the transition functions for asymptotically free meromorphic solutions of Cherednik's root system analogs of the quantum Knizhnik-Zamolodchikov equations. They thus give rise to explicit elliptic solutions of root system analogs of dynamical Yang-Baxter and reflection equations. Applications to quantum c-functions, basic hypergeometric functions, reflectionless difference operators and multivariable Baker-Akhiezer functions are discussed.; Comment: 34 pages. In the second version some additional references are included. In third version a typo in formula (1.10) is corrected, and references are updated

## ‣ Wavelet-galerkin method for computations of electromagnetic fields-computation of connection coefficients

Yang, Shiyou; Ni, Guangzheng; Ho, S. L.; Machado, Jose Marcio; Rahman, M. A.; Wong, H. C.
Fonte: Institute of Electrical and Electronics Engineers (IEEE) Publicador: Institute of Electrical and Electronics Engineers (IEEE)
Tipo: Artigo de Revista Científica Formato: 644-648
Português
Relevância na Pesquisa
67.700986%
One of the key issues which makes the waveletGalerkin method unsuitable for solving general electromagnetic problems is a lack of exact representations of the connection coefficients. This paper presents the mathematical formulae and computer procedures for computing some common connection coefficients. The characteristic of the present formulae and procedures is that the arbitrary point values of the connection coefficients, rather than the dyadic point values, can be determined. A numerical example is also given to demonstrate the feasibility of using the wavelet-Galerkin method to solve engineering field problems. © 2000 IEEE.