Linking Clifford analysis and combinatorics through bijective methods

The application of Clifford Analysis methods in Combinatorics has some peculiarities due to the use of noncommutative algebras. But it seems natural to expect from here some new results different from those obtained by using approaches based on several complex variable. For instances, the fact that in Clifford Analysis the point-wise multiplication of monogenic functions as well as their composition are not algebraically closed in this class of generalized holomorphic functions causes serious problems. Indeed, this is one of the reasons why in polynomial approximation almost every problem needs the construction of specially adapted polynomial bases. Our aim is to show that the analysis and comparison of different representations of the same polynomial or entire function allow to link Clifford Analysis and Combinatorics by means of bijective methods. In this context we also stress the central role of the hypercomplex derivative for power series representations in connection with the concept of Appell sequences as analytic tools for establishing this link.Fundação para a Ciência e a Tecnologia (FCT

3D-Mappings using monogenic functions

Conformal mappings of plane domains are realized by holomorphic functions with non vanishing derivative. Therefore complex differentiability plays an important role in all questions related to fundamental properties of such mapping. In contrast to the planar case, in higher dimensions the set of conformal mappings consists only of M¨obius transformations. But unfortunately M¨obius transformations are not monogenic functions and therefore also not hypercomplex differentiable. However the equivalence between both concepts - hypercomplex differentiability in the sense of [9], [11] and monogenicity - suggests the question whether monogenic functions can play or not a special role for other types of 3D-mappings, for instance, for quasi-conformal ones. Our goal is to present a case study of an approach to 3D-mappings by using particularly easy to handle monogenic homogeneous polynomials as basis for approximating the mapping function. Thereby we extend significantly the results obtained in [3]. From the numerical point of view we apply ideas from complex numerical analysis realizing the approximation via polynomials of a small real parameter

Special monogenic polynomials - properties and applications

AIP conference proceedings, vol. 936In Clifford Analysis several different methods have been developed for constructing monogenic functions as series with respect to properly chosen homogeneous monogenic polynomials. Almost all these methods rely on sets of orthogonal polynomials with their origin in classical (real) Harmonic Analysis in order to obtain the desired basis of homogeneous polynomials. We use a direct and elementary approach to this problem and construct a set of homogeneous polynomials involving only products of a hypercomplex variable and its hypercomplex conjugate. The obtained set is an Appell set of monogenic polynomials with respect to the hypercomplex derivative. Its intrinsic properties and some applications are presented.The research of the first author was partially supported by the R&D unit Matematica e Aplicacoes (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT). The research of the second author was partially supported by the Portuguese Foundation for Science and Technology (FCT) through the research program POCTI

On paravector valued homogeneous monogenic polynomials with binomial expansion

The aim of this note is to study a set of paravector valued homogeneous monogenic polynomials that can be used for a construction of sequences of generalized Appell polynomials in the context of Clifford analysis. Therefore, we admit a general form of the vector part of the first degree polynomial in the Appell sequence. This approach is different from the one presented in recent papers on this subject. We show that in the case of paravector valued polynomials of three real variables, there exist essentially two different types of such polynomials together with two other trivial types of polynomials. The proof indicates a way of obtaining analogous results in the case of polynomials of more than three variables.Fundação para a Ciência e a Tecnologia (FCT

A note on a one-parameter family of non-symmetric number triangles

The recently growing interest in special Cliff ord Algebra valued polynomial solutions of generalized Cauchy-Riemann systems in (n + 1)-dimensional Euclidean spaces suggested a detailed study of the arithmetical properties of their coe fficients, due to their combinatoric relevance. This concerns, in particular, a generalized Appell sequence of homogeneous polynomials whose coe cient's set can be treated as a one-parameter family of non-symmetric triangles of fractions. The discussion of its properties, similar to those of the ordinary Pascal triangle (which itself does not belong to the family), is carried out in this paper.Fundação para a Ciência e a Tecnologia (FCT

Clifford analysis between continuous and discrete

Some decades ago D. Knuth et al. have coined concrete mathematics as the blending of CONtinuous and disCRETE math, taking into account that problems of standard discrete mathematics can often be solved by methods based on continuous mathematics together with a controlled manipulation of mathematical formulas. Of course, it was not a new idea, but due to the ongoing emergence of computer aided algebraic manipulation tools of that time it emphasized their use for elegant solutions of old problems or even the detection of new important relationships. Our aim is to show that the same philosophy can be successfully applied to Clifford Analysis by taking advantages of its inherent non-commutative algebra to obtain results or develop methods that are di erent from other ones. In particular, we determine new binomial sums by using a hypercomplex generating function for a special type of monogenic polynomials and develop an algorithm for the determination of their scalar and vector part which illustrates well the diifferences to the corresponding complex case.The research of the first author was partially supported by the R&D Unit Matemdtica e Aplicagoes (UIMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT)

On special functions in the context of clifford analysis

Considering the foundation of Quaternionic Analysis by R. Fueter and his collaborators in the beginning of the 1930s as starting point of Clifford Analysis, we can look back to 80 years of work in this eld. However the interest in multivariate analysis using Clifford algebras only started to grow significantly in the 70s. Since then a great amount of papers on Clifford Analysis referring different classes of Special Functions have appeared. This situation may have been triggered by a more systematic treatment of monogenic functions by their multiple series development derived from Gegenbauer or associated Legendre polynomials (and not only by their integral representation). Also approaches to Special Functions by means of algebraic methods, either Lie algebras or through Lie groups and symmetric spaces gained by that time importance and infuenced their treatment in Clifford Analysis. In our talk we will rely on the generalization of the classical approach to Special Functions through differential equations with respect to the hypercomplex derivative, which is a more recently developed tool in Clifford Analysis. In this context special attention will be payed to the role of Special Functions as intermediator between continuous and discrete mathematics. This corresponds to a more recent trend in combinatorics, since it has been revealed that many algebraic structures have hidden combinatorial underpinnings.The research of the first author was partially supported by the Centro de Investigacao e Desenvolvimento em Matematica e Aplicacoes (CIDMA) of the University of Aveiro, through the Portuguese Foundation for Science and Technology (FCT)

Special functions versus elementary functions in hypercomplex function theory

In recent years special hypercomplex Appell polynomials have been introduced by several authors and their main properties have been studied by different methods and with different objectives. Like in the classical theory of Appell polynomials, the generating function of hypercomplex Appell polynomials is a hypercomplex exponential function. The observation that this generalized exponential function has, for example, a close relationship with Bessel functions confirmed the practical significance of investigation on special classes of hypercomplex differentiable functions. Its usefulness for combinatorial studies has also been investigated. Moreover, an extension of those ideas led to the construction of complete sets of hypercomplex Appell polynomial sequences. Here we show how this opens the way for a more systematic study of the relation between some classes of Special Functions and Elementary Functions in Hypercomplex Function Theory.FC

A Pascal-like triangle with quaternionic entries

In this paper we consider a Pascal-like triangle as result of the expansion of a binomial in terms of the generators e1, e2 of the non-commutative Clifford algebra Cℓ0, 2 over R. The study of various patterns in such structure and the discussion of its properties are carried out.This work was supported by Portuguese funds through the Research Centre of Mathematics of University of Minho - CMAT, and the Center of Research and Development in Mathematics and Applications - CIDMA (University of Aveiro), and the Portuguese Foundation for Science and Technology ("FCT -Fundacao para a Ciencia e Tecnologia"), within projects UIDB/00013/2020, UIDP/00013/2020, and UIDB/04106/2020

Isolamento de novas estirpes de Bacillus thuringiensis e Bacillus thuringiensis para o controle de Culex quinquefasciatus e Aedes aegypti.

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