Optimizaci��n en el c��mputos de algoritmos de cuadratura para funciones especiales. Optimization of quadrature algorithms for special functions.

En nuestro proyecto anterior aproximamos el c��lculo de una integral definida con integrandos de grandes variaciones funcionales. Nuestra aproximaci��n paraleliza el algoritmo de c��mputo de un m��todo adaptivo de cuadratura, basado en reglas de Newton-Cote. Los primeros resultados obtenidos fueron comunicados en distintos congresos nacionales e internacionales; ellos nos permintieron comenzar con una tipificaci��n de las reglas de cuadratura existentes y una clasificaci��n de algunas funciones utilizadas como funciones de prueba. Estas tareas de clasificaci��n y tipificaci��n no las hemos finalizado, por lo que pretendemos darle continuidad a fin de poder informar sobre la conveniencia o no de utilizar nuestra t��cnica. Para llevar adelante esta tarea se buscar�� una base de funciones de prueba y se ampliar�� el espectro de reglas de cuadraturas a utilizar. Adem��s, nos proponemos re-estructurar el c��lculo de algunas rutinas que intervienen en el c��mputo de la m��nima energ��a de una mol��cula. Este programa ya existe en su versi��n secuencial y est�� modelizado utilizando la aproximaci��n LCAO. El mismo obtiene resultados exitosos en cuanto a precisi��n, comparado con otras publicaciones internacionales similares, pero requiere de un tiempo de c��lculo significativamente alto. Nuestra propuesta es paralelizar el algoritmo mencionado abord��ndolo al menos en dos niveles: 1- decidir si conviene distribuir el c��lculo de una integral entre varios procesadores o si ser�� mejor distribuir distintas integrales entre diferentes procesadores. Debemos recordar que en los entornos de arquitecturas paralelas basadas en redes (t��picamente redes de ��rea local, LAN) el tiempo que ocupa el env��o de mensajes entre los procesadores es muy significativo medido en cantidad de operaciones de c��lculo que un procesador puede completar. 2- de ser necesario, paralelizar el c��lculo de integrales dobles y/o triples. Para el desarrollo de nuestra propuesta se desarrollar��n heur��sticas para verificar y construir modelos en los casos mencionados tendientes a mejorar las rutinas de c��lculo ya conocidas. A la vez que se testear��n los algoritmos con casos de prueba. La metodolog��a a utilizar es la habitual en C��lculo Num��rico. Con cada propuesta se requiere: a) Implementar un algoritmo de c��lculo tratando de lograr versiones superadoras de las ya existentes. b) Realizar los ejercicios de comparaci��n con las rutinas existentes para confirmar o desechar una mejor perfomance num��rica. c) Realizar estudios te��ricos de error vinculados al m��todo y a la implementaci��n. Se conform�� un equipo interdisciplinario integrado por investigadores tanto de Ciencias de la Computaci��n como de Matem��tica. Metas a alcanzar Se espera obtener una caracterizaci��n de las reglas de cuadratura seg��n su efectividad, con funciones de comportamiento oscilatorio y con decaimiento exponencial, y desarrollar implementaciones computacionales adecuadas, optimizadas y basadas en arquitecturas paralelas.

Quasiconformal mappings and inequalities involving special functions

This PhD thesis in Mathematics belongs to the field of Geometric Function Theory. The thesis consists of four original papers. The topic studied deals with quasiconformal mappings and their distortion theory in Euclidean <i>n</i>-dimensional spaces. This theory has its roots in the pioneering papers of F. W. Gehring and J. V��is��l�� published in the early 1960���s and it has been studied by many mathematicians thereafter. In the first paper we refine the known bounds for the so-called Mori constant and also estimate the distortion in the hyperbolic metric. The second paper deals with radial functions which are simple examples of quasiconformal mappings. These radial functions lead us to the study of the so-called p-angular distance which has been studied recently e.g. by L. Maligranda and S. Dragomir. In the third paper we study a class of functions of a real variable studied by P. Lindqvist in an influential paper. This leads one to study parametrized analogues of classical trigonometric and hyperbolic functions which for the parameter value <i>p</i> = 2 coincide with the classical functions. Gaussian hypergeometric functions have an important role in the study of these special functions. Several new inequalities and identities involving p-analogues of these functions are also given. In the fourth paper we study the generalized complete elliptic integrals, modular functions and some related functions. We find the upper and lower bounds of these functions, and those bounds are given in a simple form. This theory has a long history which goes back two centuries and includes names such as A. M. Legendre, C. Jacobi, C. F. Gauss. Modular functions also occur in the study of quasiconformal mappings. Conformal invariants, such as the modulus of a curve family, are often applied in quasiconformal mapping theory. The invariants can be sometimes expressed in terms of special conformal mappings. This fact explains why special functions often occur in this theory.

Special functions of Weyl groups and their continuous and discrete orthogonality

Cette th��se s'int��resse �� l'��tude des propri��t��s et applications de quatre familles des fonctions sp��ciales associ��es aux groupes de Weyl et d��not��es $C$, $S$, $S^s$ et $S^l$. Ces fonctions peuvent ��tre vues comme des g��n��ralisations des polyn��mes de Tchebyshev. Elles sont en lien avec des polyn��mes orthogonaux �� plusieurs variables associ��s aux alg��bres de Lie simples, par exemple les polyn��mes de Jacobi et de Macdonald. Elles ont plusieurs propri��t��s remarquables, dont l'orthogonalit�� continue et discr��te. En particulier, il est prouv�� dans la pr��sente th��se que les fonctions $S^s$ et $S^l$ caract��ris��es par certains param��tres sont mutuellement orthogonales par rapport �� une mesure discr��te. Leur orthogonalit�� discr��te permet de d��duire deux types de transform��es discr��tes analogues aux transform��es de Fourier pour chaque alg��bre de Lie simple avec racines des longueurs diff��rentes. Comme les polyn��mes de Tchebyshev, ces quatre familles des fonctions ont des applications en analyse num��rique. On obtient dans cette th��se quelques formules de <<cubature>>, pour des fonctions de plusieurs variables, en liaison avec les fonctions $C$, $S^s$ et $S^l$. On fournit ��galement une description compl��te des transform��es en cosinus discr��tes de types V--VIII �� $n$ dimensions en employant les fonctions sp��ciales associ��es aux alg��bres de Lie simples $B_n$ et $C_n$, appel��es cosinus antisym��triques et sym��triques. Enfin, on ��tudie quatre familles de polyn��mes orthogonaux �� plusieurs variables, analogues aux polyn��mes de Tchebyshev, introduits en utilisant les cosinus (anti)sym��triques.

A generalization of Student���s t-distribution from the viewpoint of special functions

Student���s t-distribution has found various applications in mathematical statistics. One of the main properties of the t-distribution is to converge to the normal distribution as the number of samples tends to infinity. In this paper, by using a Cauchy integral we introduce a generalization of the t-distribution function with four free parameters and show that it converges to the normal distribution again. We provide a comprehensive treatment of mathematical properties of this new distribution. Moreover, since the Fisher F-distribution has a close relationship with the t-distribution, we also introduce a generalization of the F-distribution and prove that it converges to the chi-square distribution as the number of samples tends to infinity. Finally some particular sub-cases of these distributions are considered.

Some New Classes of Orthogonal Polynomials and Special Functions

In dieser Dissertation pr��sentieren wir zun��chst eine Verallgemeinerung der ��blichen Sturm-Liouville-Probleme mit symmetrischen L��sungen und erkl��ren eine umfassendere Klasse. Dann f��hren wir einige neue Klassen orthogonaler Polynome und spezieller Funktionen ein, welche sich aus dieser symmetrischen Verallgemeinerung ableiten lassen. Als eine spezielle Konsequenz dieser Verallgemeinerung f��hren wir ein Polynomsystem mit vier freien Parametern ein und zeigen, dass in diesem System fast alle klassischen symmetrischen orthogonalen Polynome wie die Legendrepolynome, die Chebyshevpolynome erster und zweiter Art, die Gegenbauerpolynome, die verallgemeinerten Gegenbauerpolynome, die Hermitepolynome, die verallgemeinerten Hermitepolynome und zwei weitere neue endliche Systeme orthogonaler Polynome enthalten sind. All diese Polynome k��nnen direkt durch das neu eingef��hrte System ausgedr��ckt werden. Ferner bestimmen wir alle Standardeigenschaften des neuen Systems, insbesondere eine explizite Darstellung, eine Differentialgleichung zweiter Ordnung, eine generische Orthogonalit��tsbeziehung sowie eine generische Dreitermrekursion. Au��erdem benutzen wir diese Erweiterung, um die assoziierten Legendrefunktionen, welche viele Anwendungen in Physik und Ingenieurwissenschaften haben, zu verallgemeinern, und wir zeigen, dass diese Verallgemeinerung Orthogonalit��tseigenschaft und -intervall erh��lt. In einem weiteren Kapitel der Dissertation studieren wir detailliert die Standardeigenschaften endlicher orthogonaler Polynomsysteme, welche sich aus der ��blichen Sturm-Liouville-Theorie ergeben und wir zeigen, dass sie orthogonal bez��glich der Fisherschen F-Verteilung, der inversen Gammaverteilung und der verallgemeinerten t-Verteilung sind. Im n��chsten Abschnitt der Dissertation betrachten wir eine vierparametrige Verallgemeinerung der Studentschen t-Verteilung. Wir zeigen, dass diese Verteilung gegen die Normalverteilung konvergiert, wenn die Anzahl der Stichprobe gegen Unendlich strebt. Eine ��hnliche Verallgemeinerung der Fisherschen F-Verteilung konvergiert gegen die chi-Quadrat-Verteilung. Ferner f��hren wir im letzten Abschnitt der Dissertation einige neue Folgen spezieller Funktionen ein, welche Anwendungen bei der L��sung in Kugelkoordinaten der klassischen Potentialgleichung, der W��rmeleitungsgleichung und der Wellengleichung haben. Schlie��lich erkl��ren wir zwei neue Klassen rationaler orthogonaler hypergeometrischer Funktionen, und wir zeigen unter Benutzung der Fouriertransformation und der Parsevalschen Gleichung, dass es sich um endliche Orthogonalsysteme mit Gewichtsfunktionen vom Gammatyp handelt.

Special Functions

Exam questions and solutions in LaTex

Special Functions pdf

Exam questions and solutions in PDF

Partial Quasi-Bilateral Generating Function Involving some Special Functions

A group-theoretic method of obtaining more general class of generating functions from a given class of partial quasi-bilateral generating functions involving Hermite, Laguerre and Gegenbaur polynomials are discussed.

Special Functions and Pathways for Problems in Astrophysics: An Essay in Honor of A.M. Mathai

The paper provides a review of A.M. Mathai's applications of the theory of special functions, particularly generalized hypergeometric functions, to problems in stellar physics and formation of structure in the Universe and to questions related to reaction, diffusion, and reaction-diffusion models. The essay also highlights Mathai's recent work on entropic, distributional, and differential pathways to basic concepts in statistical mechanics, making use of his earlier research results in information and statistical distribution theory. The results presented in the essay cover a period of time in Mathai's research from 1982 to 2008 and are all related to the thematic area of the gravitationally stabilized solar fusion reactor and fractional reaction-diffusion, taking into account concepts of non-extensive statistical mechanics. The time period referred to above coincides also with Mathai's exceptional contributions to the establishment and operation of the Centre for Mathematical Sciences, India, as well as the holding of the United Nations (UN)/European Space Agency (ESA)/National Aeronautics and Space Administration (NASA) of the United States/ Japanese Aerospace Exploration Agency (JAXA) Workshops on basic space science and the International Heliophysical Year 2007, around the world. Professor Mathai's contributions to the latter, since 1991, are a testimony for his social con-science applied to international scientific activity.

Generalized Fractional Calculus, Special Functions and Integral Transforms: What is the Relation?

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Some results on holomorphic generalized functions

In this note, we present three independent results within generalized complex analysis (in the Colombeau sense). The first of them deals with non-removable singularities; we construct a generalized function u on an open subset Omega of C(n), which is not a holomorphic generalized function on Omega but it is a holomorphic generalized function on Omega\S, where S is a hypersurface contained in Omega. The second result shows the existence of a holomorphic generalized function with prescribed values in the zero-set of a classical holomorphic function. The last result states the existence of a compactly supported solution to the (partial derivative) over bar operator.

Families of orthogonal functions defined by the Weyl groups of compact Lie groups

Plusieurs familles de fonctions sp��ciales de plusieurs variables, appel��es fonctions d'orbites, sont d��finies dans le contexte des groupes de Weyl de groupes de Lie simples compacts/d'alg��bres de Lie simples. Ces fonctions sont ��tudi��es depuis pr��s d'un si��cle en raison de leur lien avec les caract��res des repr��sentations irr��ductibles des alg��bres de Lie simples, mais ��galement de par leurs sym��tries et orthogonalit��s. Nous sommes principalement int��ress��s par la description des relations d'orthogonalit�� discr��te et des transformations discr��tes correspondantes, transformations qui permettent l'utilisation des fonctions d'orbites dans le traitement de donn��es multidimensionnelles. Cette description est donn��e pour les groupes de Weyl dont les racines ont deux longueurs diff��rentes, en particulier pour les groupes de rang $2$ dans le cas des fonctions d'orbites du type $E$ et pour les groupes de rang $3$ dans le cas de toutes les autres fonctions d'orbites.

On E-functions of semisimple Lie groups

We develop and describe continuous and discrete transforms of class functions on a compact semisimple, but not simple, Lie group G as their expansions into series of special functions that are invariant under the action of the even subgroup of the Weyl group of G. We distinguish two cases of even Weyl groups-one is the direct product of even Weyl groups of simple components of G and the second is the full even Weyl group of G. The problem is rather simple in two dimensions. It is much richer in dimensions greater than two-we describe in detail E-transforms of semisimple Lie groups of rank 3.

An approach to generalized holomorphic functions

A study of the generalized holomorphic functions, HG(Omega), having in mind its strict elements, i.e. those which are in HG(Omega) - H(Omega), as well as the possibility of the existence of hybrid elements, i.e. elements which have, in a part of a domain Omega subset of C-n, the strict behaviour and, in another part of the same domain, the classical behaviour, is carried out in this work. The study of hybrid elements is important in the approach of a concept of generalized domain of holomorphy.

On extension theorems for holomorphic generalized functions

We present two extension theorems for holomorphic generalized functions. The first one is a version of the classic Hartogs extension theorem. In this, we start from a holomorphic generalized function on an open neighbourhood of the bounded open boundary, extending it, holomorphically, to a full open. In the second theorem a generalized version of a classic result is obtained, done independently, in 1943, by Bochner and Severi. For this theorem, we start from a function that is holomorphic generalized and has a holomorphic representative on the bounded domain boundary, we extend it holomorphically the function, for the whole domain.