3D-mappings by means of monogenic functions and their approximation

We consider quasi-conformal 3D-mappings realized by hypercomplex di erentiable (monogenic) functions and their polynomial approximation. Main tools are the series development of monogenic functions in terms of hypercomplex variables and the generalization of L. V. Kantorovich's approach for approximating conformal mappings by powers of a small parameter. .This work was partially supported by the R&D Unit Matematica e Aplicacoes (UIMA) of the University of Aveiro, through Portuguese Foundation for Science and Technology (FCT

Fischer decomposition by inframonogenic functions

Let D denote the Dirac operator in the Euclidean space R^m. In this paper, we present a refinement of the biharmonic functions and at the same time an extension of the monogenic functions by considering the equation DfD=0. The solutions of this "sandwich" equation, which we call inframonogenic functions, are used to obtain a new Fischer decomposition for homogeneous polynomials in R^m.Comment: 10 pages, accepted for publication in CUBO, A Mathematical Journa

Quaternions in Applied Sciences

After more than hundred years of arguments in favour and against quaternions, of exciting odysseys with new insights as well as disillusions about their usefulness the mathematical world saw in the last 40 years a burst in the application of quaternions and its generalizations in almost all disciplines that are dealing with problems in more than two dimensions. Our aim is to sketch some ideas - necessarily in a very concise and far from being exhaustive manner - which contributed to the picture of the recent development. With the help of some historical reminiscences we firstly try to draw attention to quaternions as a special case of Clifford Algebras which play the role of a unifying language in the Babylon of several different mathematical languages. Secondly, we refer to the use of quaternions as a tool for modelling problems and at the same time for simplifying the algebraic calculus in almost all applied sciences. Finally, we intend to show that quaternions in combination with classical and modern analytic methods are a powerful tool for solving concrete problems thereby giving origin to the development of Quaternionic Analysis and, more general, of Clifford Analysis

Bernoulli polynomials and Pascal matrices in the context of Clifford analysis

AbstractThis paper describes an approach to generalized Bernoulli polynomials in higher dimensions by using Clifford algebras. Due to the fact that the obtained Bernoulli polynomials are special hypercomplex holomorphic (monogenic) functions in the sense of Clifford Analysis, they have properties very similar to those of the classical polynomials. Hypercomplex Pascal and Bernoulli matrices are defined and studied, thereby generalizing results recently obtained by Zhang and Wang (Z. Zhang, J. Wang, Bernoulli matrix and its algebraic properties, Discrete Appl. Math. 154 (11) (2006) 1622–1632)

(Discrete) Almansi Type Decompositions: An umbral calculus framework based on osp(1∣2)\mathfrak{osp}(1|2) symmetries

We introduce the umbral calculus formalism for hypercomplex variables starting from the fact that the algebra of multivariate polynomials \BR[\underline{x}] shall be described in terms of the generators of the Weyl-Heisenberg algebra. The extension of \BR[\underline{x}] to the algebra of Clifford-valued polynomials $\mathcal{P}$ gives rise to an algebra of Clifford-valued operators whose canonical generators are isomorphic to the orthosymplectic Lie algebra $\mathfrak{osp}(1|2)$. This extension provides an effective framework in continuity and discreteness that allow us to establish an alternative formulation of Almansi decomposition in Clifford analysis (c.f. \cite{Ryan90,MR02,MAGU}) that corresponds to a meaningful generalization of Fischer decomposition for the subspaces $\ker (D')^k$. We will discuss afterwards how the symmetries of \mathfrak{sl}_2(\BR) (even part of $\mathfrak{osp}(1|2)$) are ubiquitous on the recent approach of \textsc{Render} (c.f. \cite{Render08}), showing that they can be interpreted in terms of the method of separation of variables for the Hamiltonian operator in quantum mechanics.Comment: Improved version of the Technical Report arXiv:0901.4691v1; accepted for publication @ Math. Meth. Appl. Sci http://www.mat.uc.pt/preprints/ps/p1054.pdf (Preliminary Report December 2010

A mitochondrial perspective on early land plants : new loci in evolving chondriomes

Mitochondrial DNA in land plants is characterised by very slowly evolving gene sequences in contrast to a very variable genome structure displaying frequent gene relocations and transfers. Studies in this thesis address different aspects of chondriome evolution with a focus on early land plants. The mitochondrial nad4 gene was established as a novel phylogenetic marker for liverworts, including an analysis of the secondary structure of the group II intron nad4i548 conserved in these plants. A review of the already well known mitochondrial nad5 gene is accompanied by a revision of the folding structure of the included group I intron nad5i753, revealing peculiarities unique for the liverwort genus Pellia. A phylogenetic study on liverworts is reported, which combines the novel nad4 sequences with nad5 data from several labs, and the chloroplast genes rbcL and rps4, resulting in a well supported topology. The gene continuities of the gene clusters nad5 nad4-nad2 and trnA-trnT-nad7 were of special interest regarding the variability of the mitochondrial genome in the earliest diverging land plant lineages, and revealed very different patterns of evolution in the analysed spacers and among different plant groups. Loss and potential regain of the trnT gene was found in liverworts. Finally, studies on the degeneration of nad7 into a pseudogene in liverworts identified different modes of sequence degeneration in the major liverwort subclades

Laguerre polynomials in several hypercomplex variables and their matrix representation

Recently the creation matrix, intimately related to the Pascal matrix and its generalizations, has been used to develop matrix representations of special polynomials, in particular Appell polynomials. In this paper we describe a matrix approach to polynomials in several hypercomplex variables based on special block matrices whose structures simulate the creation matrix and the Pascal matrix. We apply the approach to hypercomplex Laguerre polynomials, although it can be used for other Appell sequences, too

A note on a generalized Joukowski transformation

AbstractIt is well known that the Joukowski transformation plays an important role in physical applications of conformal mappings, in particular in the study of flows around airfoils. We present, for n≥2, an n-dimensional hypercomplex analogue of the Joukowski transformation and describe in some detail the 3D case. A generalized 3D Joukowski profile, produced with Maple, is included