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

Fonte: PERGAMON-ELSEVIER SCIENCE LTD
Publicador: PERGAMON-ELSEVIER SCIENCE LTD

Tipo: Artigo de Revista Científica

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#Harmonic wavelets#Connection coefficients#Pseudo-spectral#Burgers equation#Computational complexity#Front propagation#Precipitation fronts#Wavelet Galerkin method#BURGERS-EQUATION#SCHEME#Computer Science, Interdisciplinary Applications

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

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

Fonte: Amer Mathematical Soc
Publicador: Amer Mathematical Soc

Tipo: Artigo de Revista Científica
Formato: 1937-1951

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#Orthogonal polynomials#zeros of Gegenbauer polynomials#zeros of Hermite polynomials#connection coefficients

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.

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

Fonte: Institute of Electrical and Electronics Engineers (IEEE)
Publicador: Institute of Electrical and Electronics Engineers (IEEE)

Tipo: Artigo de Revista Científica
Formato: 644-648

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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.

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

Fonte: Universidade Estadual Paulista
Publicador: Universidade Estadual Paulista

Tipo: Artigo de Revista Científica
Formato: 644-648

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#Connection coefficients#Wavelet bases#Waveletgalerkin method#Wavelet-Galerkin method#Galerkin methods#Wavelet transforms#Electromagnetic field theory

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.

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

Fonte: Hindawi Publishing Corporation
Publicador: Hindawi Publishing Corporation

Tipo: Artigo de Revista Científica

Publicado em 04/08/2013
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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.

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

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica
Formato: application/pdf

Publicado em 01/05/2001
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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

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

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em /04/2001
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#Orthogonal polynomials#Linearization and connection coefficients#Generalized hypergeometric function#Matemáticas

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

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

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica
Formato: text/html

Publicado em 16/11/1998
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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

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## ‣ Connection coefficients for Laguerre-Sobolev orthogonal polynomials

Fonte: Elsevier
Publicador: Elsevier

Tipo: Artigo de Revista Científica
Formato: application/pdf

Publicado em 15/07/2003
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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## ‣ On Stokes Matrices in terms of Connection Coefficients

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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## ‣ A recurrence formula for Jack connection coefficients

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/09/2014
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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 11/10/2007
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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/11/2011
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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)].

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 15/06/2015
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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/11/2010
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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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...

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 06/10/2009
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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.

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## ‣ Connection coefficients for basic Harish-Chandra series

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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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

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

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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#Connection coefficients#Wavelet bases#Waveletgalerkin method#Wavelet-Galerkin method#Galerkin methods#Wavelet transforms#Electromagnetic field theory

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.

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