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

Dynkin isomorphism and mermin-wagner theorems for hyperbolic sigma models and recurrence of the two-dimensional vertex-reinforced jump process

We prove the vertex-reinforced jump process (VRJP) is recurrent in two dimensions for any translation invariant finite range initial rates. Our proof has two main ingredients. The first is a direct connection between the VRJP and sigma models whose target space is a hyperbolic space $\mathbb{H}^n$ or its supersymmetric counterpart $\mathbb{H}^{2|2}$. These results are analogues of well-known relations between the Gaussian free field and the local times of simple random walk. The second ingredient is a Mermin--Wagner theorem for these sigma models. This result is of intrinsic interest for the sigma models and also implies our main theorem on the VRJP. Surprisingly, our Mermin--Wagner theorem applies even though the symmetry groups of $\mathbb{H}^n$ and $\mathbb{H}^{2|2}$ are non-amenable

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

Matrix approach to hypercomplex Appell polynomials

Recently the authors presented a matrix representation approach to real Appell polynomials essentially determined by a nilpotent matrix with natural number entries. It allows to consider a set of real Appell polynomials as solution of a suitable first order initial value problem. The paper aims to confirm that the unifying character of this approach can also be applied to the construction of homogeneous Appell polynomials that are solutions of a generalized Cauchy–Riemann system in Euclidean spaces of arbitrary dimension. The result contributes to the development of techniques for polynomial approximation and interpolation in non-commutative Hypercomplex Function Theories with Clifford algebras

Upwelling in the Humboldt Coastal Current near Valparaiso, Chile

In December, 1975, an upwelling study was conducted at Punta Curaumilla near Valparaiso, Chile. Previous work (Silva, 1973) had indicated a zone of relatively intense upwelling around this point of land. In the present study , a bathythermograph survey confirmed the presence of the intensification, identified as a tongue of cool water extending seaward and equatorward of the point. Following Arthur\u27s (1965) scale analysis it is suggested that the acceleration of relative vorticity past the point of land is responsible for intensification...

A CAUCHY-KOWALEVSKI THEOREM FOR INFRAMONOGENIC FUNCTIONS

In this paper we prove a Cauchy-Kowalevski theorem for the functions satisfying the system ∂xf∂x = 0 (called inframonogenic functions)