60 research outputs found
Monogenic polynomials of four variables with binomial expansion
In the recent past one of the main concern of research in the field of Hypercomplex Function Theory in Clifford Algebras was the development of a variety of new tools for a deeper understanding about its true elementary roots in the Function Theory of one Complex Variable. Therefore the study of the space of monogenic (Clifford holomorphic) functions by its stratification via homogeneous monogenic polynomials is a useful tool. In this paper we consider
the structure of those polynomials of four real variables with binomial expansion. This allows a complete characterization of sequences of 4D generalized monogenic Appell polynomials by three different types of polynomials. A particularly important case is that of monogenic
polynomials which are simply isomorphic to the integer powers of one complex variable and therefore also called pseudo-complex powers.Fundação para a Ciência e a Tecnologia (FCT
On Vietoris' number sequence and combinatorial identities with quaternions
Ruscheweyh and Salinas showed in 2004 the relationship of a celebrated theorem of Vietoris (1958) about the positivity of certain sine and cosine sums with the function theoretic concept of stable holomorphic functions in the unit disc. The present paper reveals that the coefficient sequence in Vietoris' theorem is identical to a number sequence obtained by a new combinatorial identity which involves generators of quaternions. In this sense Vietoris' sequence of rational numbers combines seemingly disperse subjects in Real, Complex and Hypercomplex Analysis. Thereby we show that a non-standard application of Clifford algebra tools is able to reveal new insights in objects of combinatorial nature.The work of the first and third authors was supported by Portuguese funds
through the CIDMA - Center for Research and Development in Mathematics
and Applications, and the Portuguese Foundation for Science and Technology
(“FCT-Fundação para a Ciência e Tecnologia”), within project PEstOE/MAT/UI4106/2013.
The work of the second author was supported by Portuguese funds through the CMAT - Centre of Mathematics and FCT within the Project UID/MAT/00013/2013.info:eu-repo/semantics/publishedVersio
Quaternions : a mathematica package for quaternionic analysis
This paper describes new issues of the Mathematica standard package Quaternions for implementing Hamilton's Quaternion Algebra.
This work attempts to endow the original package with the ability to perform operations on symbolic expressions involving quaternion-valued functions.
A collection of new functions is introduced in order to provide basic mathematical tools necessary for dealing with regular functions in , for . The performance of the package is illustrated by presenting several examples and applications.Centro de Matemática da Universidade do MinhoFundação para a Ciência e a Tecnologia (FCT)Centro de Investigação e Desenvolvimento em Matemática e Aplicações da Universidade de Aveir
q-deformed harmonic and Clifford analysis and the q-Hermite and Laguerre polynomials
We define a q-deformation of the Dirac operator, inspired by the one
dimensional q-derivative. This implies a q-deformation of the partial
derivatives. By taking the square of this Dirac operator we find a
q-deformation of the Laplace operator. This allows to construct q-deformed
Schroedinger equations in higher dimensions. The equivalence of these
Schroedinger equations with those defined on q-Euclidean space in quantum
variables is shown. We also define the m-dimensional q-Clifford-Hermite
polynomials and show their connection with the q-Laguerre polynomials. These
polynomials are orthogonal with respect to an m-dimensional q-integration,
which is related to integration on q-Euclidean space. The q-Laguerre
polynomials are the eigenvectors of an su_q(1|1)-representation
Infinite-order Differential Operators Acting on Entire Hyperholomorphic Functions
Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper, we investigate the continuity of a class of infinite-order differential operators acting on spaces of entire hyperholomorphic functions. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different from each other, in both cases, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite-order differential operators acting on these two classes of functions. This is particularly remarkable since the exponential function is not in the kernel of the Dirac operator, but it plays an important role in the theory of entire monogenic functions with growth conditions
Infinite-order Differential Operators Acting on Entire Hyperholomorphic Functions
Infinite-order differential operators appear in different fields of mathematics and physics and in the past decade they turned out to be of fundamental importance in the study of the evolution of superoscillations as initial datum for Schrödinger equation. Inspired by the operators arising in quantum mechanics, in this paper, we investigate the continuity of a class of infinite-order differential operators acting on spaces of entire hyperholomorphic functions. We will consider two classes of hyperholomorphic functions, both being natural extensions of holomorphic functions of one complex variable. We show that, even though these two notions of hyperholomorphic functions are quite different from each other, in both cases, entire hyperholomorphic functions with exponential bounds play a crucial role in the continuity of infinite-order differential operators acting on these two classes of functions. This is particularly remarkable since the exponential function is not in the kernel of the Dirac operator, but it plays an important role in the theory of entire monogenic functions with growth conditions
Aplicações numéricas e combinatórias de polinómios de Appell generalizados
Doutoramento em MatemáticaThis thesis studies properties and applications of different generalized Appell
polynomials in the framework of Clifford analysis.
As an example of 3D-quasi-conformal mappings realized by generalized Appell
polynomials, an analogue of the complex Joukowski transformation of order
two is introduced. The consideration of a Pascal n-simplex with hypercomplex
entries allows stressing the combinatorial relevance of hypercomplex Appell
polynomials.
The concept of totally regular variables and its relation to generalized Appell
polynomials leads to the construction of new bases for the space of
homogeneous holomorphic polynomials whose elements are all isomorphic to
the integer powers of the complex variable. For this reason, such polynomials
are called pseudo-complex powers (PCP). Different variants of them are
subject of a detailed investigation.
Special attention is paid to the numerical aspects of PCP. An efficient algorithm
based on complex arithmetic is proposed for their implementation. In this
context a brief survey on numerical methods for inverting Vandermonde
matrices is presented and a modified algorithm is proposed which illustrates
advantages of a special type of PCP.
Finally, combinatorial applications of generalized Appell polynomials are
emphasized. The explicit expression of the coefficients of a particular type of
Appell polynomials and their relation to a Pascal simplex with hypercomplex
entries are derived. The comparison of two types of 3D Appell polynomials
leads to the detection of new trigonometric summation formulas and
combinatorial identities of Riordan-Sofo type characterized by their expression
in terms of central binomial coefficients.Esta tese estuda propriedades e aplicações de diferentes polinómios de Appell
generalizados no contexto da análise de Clifford.
Exemplificando uma transformação realizada por polinómios de Appell
generalizados, é introduzida uma transformação análoga à transformação de
Joukowski complexa de ordem dois. A análise de um n- simplex de Pascal com
entradas hipercomplexas permite sublinhar a relevância combinatória de
polinómios hipercomplexos de Appell.
O conceito de variáveis totalmente regulares e a sua relação com polinómios
de Appell generalizados conduz à construção de novas bases para o espaço
dos polinómios homogéneos holomorfos cujos elementos são todos isomorfos
às potências inteiras da variável complexa. Por este motivo, tais polinómios
são chamados de potências pseudo-complexas (PCP). Diferentes variantes de
PCP são objeto de uma investigação detalhada.
É dada especial atenção aos aspectos numéricos de PCP. Um algoritmo
eficiente baseado em aritmética complexa é proposto para a sua
implementação. Neste contexto, é apresentado um breve resumo de métodos
numéricos para inverter matrizes de Vandermonde e é proposto um algoritmo
modificado para ilustrar as vantagens de um tipo especial de PCP.
Finalmente, são enfatizadas aplicações combinatórias de polinómios de Appell
generalizados. A expressão explícita dos coeficientes de um tipo particular de
polinómios de Appell e a sua relação com um simplex de Pascal com entradas
hipercomplexas são obtidas. A comparação de dois tipos de polinómios de
Appell tridimensionais leva à deteção de novas fórmulas envolvendo somas
trigonométricas e de identidades combinatórias do tipo de Riordan – Sofo,
caracterizadas pela sua expressão em termos de coeficientes binomiais
centrais
Discrete Clifford analysis
Doutoramento em MatemáticaEsta tese estuda os fundamentos de uma teoria discreta de funções em dimensões superiores usando a linguagem das Álgebras de Clifford. Esta abordagem combina as ideias do Cálculo Umbral e Formas Diferenciais. O potencial desta abordagem assenta essencialmente da osmose entre ambas as linguagens. Isto permitiu a construção de operadores de entrelaçamento entre estruturas contínuas e discretas, transferindo resultados conhecidos do contínuo para o discreto. Adicionalmente, isto resultou numa transcrição mimética de bases de polinómios, funções geradoras, Decomposição de Fischer, Lema de Poincaré, Teorema de Stokes, fórmula de Cauchy e fórmula de Borel-Pompeiu. Esta teoria também inclui a descrição dos homólogos discretos de formas diferenciais, campos vectores e integração discreta. De facto, a construção resultante de formas diferenciais, campos vectores e integração discreta em termos de coordenadas baricêntricas conduz à correspondência entre a teoria de Diferenças Finitas e a teoria de Elementos Finitos, dando um núcleo de aplicações desta abordagem promissora em análise numérica. Algumas ideias preliminares deste ponto de vista foram apresentadas nesta tese. Também foram apresentados resultados preliminares na teoria discreta de funções em complexos envolvendo simplexes. Algumas ligações com Combinatória e Mecânica Quântica foram também apresentadas ao longo desta tese.This thesis studies the fundamentals of a higher dimensional discrete function theory using the Clifford Algebra setting. This approach combines the ideas of Umbral Calculus and Differential Forms. Its powerful rests mostly on the interplay between both languages. This allowed the construction of intertwining operators between continuous and discrete structures, lifting the well known results from continuum to discrete. Furthermore, this resulted in a mimetic transcription of basis polynomial, generating functions, Fischer Decomposition, Poincaré and dual-Poincaré lemmata, Stokes theorem and Cauchy’s formula. This theory also includes the description discrete counterparts of differential forms, vector-fields and discrete integration. Indeed the resulted construction of discrete differential forms, discrete vector-fields and discrete integration in terms of barycentric coordinates leads to the correspondence between the theory of Finite Differences and the theory of Finite Elements, which gives a core of promising applications of this approach in numerical analysis. Some preliminary ideas on this point of view were presented in this thesis. We also developed some preliminary results in the theory of discrete monogenic functions on simplicial complexes. Some connections with Combinatorics and Quantum Mechanics were also presented along this thesis
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