arXiv subjects
Publications and source records attributed to Ricardo Abreu Blaya.
We explore Fischer decompositions from the perspective of inframonogenic functions in Clifford analysis, a non-commutative version of harmonic functions in Euclidean spaces. We study decomposition formulas for polynomial spaces involving inframonogenic and harmonic components, providing fresh insights into their algebraic structures and further important applications.
It is well known that the spectrum of the Dirichlet Laplacian for a compact perturbation of a three-dimensional, periodically twisted tube is unstable with respect to domain deformations. This means that if the periodically twisted tube is unperturbed, then the spectrum of the Dirichlet Laplacian is purely essential. On the other hand, the perturbation of this domain produces eigenvalues below the essential spectrum. This paper considers the Dirichlet-Laplace operator with a magnetic field. We explicitly prove that the spectrum of the magnetic Laplacian is stable under small and local deformations of the domain.
The Clifford algebra language allows us to rewrite the Lamé-Navier system in terms of the Euclidean Dirac operator. In this paper, the main question we shall be concerned with is whether or not a higher order Lipschitz function on the boundary $Γ$ of a Jordan domain $Ω\subset\mathbb{R}^m$ can be decomposed into a sum of the two boundary values of a solution of the Lamé-Navier system with jump across $Γ$. Our main tool are the Hardy projections related to a singular integral operator arising in the context of Clifford analysis, which turns out to be an involution operator on the first order Lipschitz classes.
The multidimensional Cauchy-Riemann operator provides a framework for studying higher order partial differential equations in $\mathbb{R}^{m+1}$, whose solutions include polymonogenic and polyharmonic functions, among others. In this work, we aim to explicitly construct solutions to such systems, generated from families of complex valued functions which are closed under conjugation and under the action of the complex Cauchy-Riemann operator. Moreover, we prove that precisely some of these solutions also satisfy homogeneous linear differential equations involving the so-called hypercomplex derivative.
This paper is devoted to study a fundamental system of equations in Linear Elasticity Theory: the famous Lamé-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the Euclidean Dirac operator, which at the same time suggests a very natural generalization involving the so-called structural sets. Our interest lies mainly in the jump problem for these elastic systems. A generalized Teodorescu transform, to be introduced here, provides the means for obtaining the explicit solution of the jump problem for a very wide classes of regions, including those with a fractal boundary.
In this paper we find a decomposition of higher order Lipschitz functions into the traces of a polymonogenic function and solve a related Riemann-Hilbert problem. Our approach lies in using a cliffordian Cauchy-type operator, which behaves as an involution operator on higher order Lipschitz spaces. The result obtained is a multidimensional sharpened version of the Hardy decomposition of Hölder continuous functions on a simple closed curve in the complex plane.
In this paper, we study the existence of solutions to the scalar additive Jump problem and the Riemann boundary value problems in the context of vectorial Clifford analysis on domains with fractal boundaries. A reduction procedure is applied with great effectiveness to find the solution of the problems.
In this paper we solve the problem on finding a sectionally Clifford algebra-valued harmonic function, zero at infinity and satisfying certain boundary value condition related to higher order Lipschitz functions. Our main tool are the Hardy projections related to a singular integral operator arising in bimonogenic function theory, which turns out to be an involution operator on the first order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Holder continuous functions on a simple closed curve in the complex plane.
We introduce a degree-based variable topological index inspired on the Stolarsky mean (known as the generalization of the logarithmic mean). We name this new index as the Stolarsky-Puebla index: $SP_α(G) = \sum_{uv \in E(G)} d_u$, if $d_u=d_v$, and $SP_α(G) = \sum_{uv \in E(G)} \left[\left( d_u^α-d_v^α\right)/\left( α(d_u-d_v\right)\right]^{1/(α-1)}$, otherwise. Here, $uv$ denotes the edge of the network $G$ connecting the vertices $u$ and $v$, $d_u$ is the degree of the vertex $u$, and $α\in \mathbb{R} \backslash \{0,1\}$. Indeed, for given values of $α$, the Stolarsky-Puebla index reproduces well-known topological indices such as the reciprocal Randic index, the first Zagreb index, and several mean Sombor indices. Moreover, we apply these indices to random networks and demonstrate that $\left< SP_α(G) \right>$, normalized to the order of the network, scale with the corresponding average degree $\left< d \right>$.
In recent works, arbitrary structural sets in the non-commutative Clifford analysis context have been used to introduce non-trivial generalizations of harmonic Clifford algebra valued functions in $\mathbb{R}^m$. Being defined as the solutions of elliptic (generally non-strongly elliptic) partial differential equations, $(φ,ψ)$-inframonogenic and $(φ,ψ)$-harmonic functions do not share the good structure and properties of the harmonic ones. The aim of this paper it to show and clarified the relationship between these classes of functions.
In this paper we derive a Cauchy integral formula for holomorphic and hyperholomorphic functions in domains bounded by a Koch snowflake in two and three dimensional setting.
The purpose of this paper is to solve a kind of Riemann-Hilbert boundary value problem for $(φ,ψ)$-harmonic functions, which are linked with the use of two orthogonal basis of the Euclidean space $\mathbb{R}^m$. We approach this problem using the language of Clifford analysis for obtaining the explicit expression of the solution of the problem in a Jordan domain $Ω\subset\mathbb{R}^m$ with fractal boundary. One of the remarkable feature in this study is that the boundary data involves higher order Lipschitz class of functions.
This paper is devoted to a fundamental system of equations in Linear Elasticity Theory: the famous Lamé-Navier system. The Clifford algebra language allows us to rewrite this system in terms of the euclidean Dirac operator, which at the same time suggests a very natural generalization involving the so-called structural sets. We are interested in finding some structures in the solutions of these generalized Lamé-Navier systems. Using MATLAB we also implement algorithms to compute with such partial differential operators as well as to verify some theoretical results obtained in the paper.
This paper is devoted to study a fundamental system of equations in plane Linear Elasticity Theory, the two-dimensional Lamé-Navier system. We rewrite them in a compressed form in terms of the Cauchy-Riemann operators and it allows us to solve a kind of Riemann problem for this system. A generalized Teodorescu operator, to be introduced here, provides the means for obtaining the explicit solution of this problem for a very wide classes of regions, including those with a fractal boundary.
This paper is concerned with the problem of decomposing a higher order Lipschitz function on a closed Jordan curve $Γ$ into a sum of two polyanalytic functions in each open domain defined by $Γ$. Our basic tools are the Hardy projections related to a singular integral operator arising in polyanalytic function theory, which, as it is proved here, represents an involution operator on the higher order Lipschitz classes. Our result generalizes the classical Hardy decomposition of Holder continuous functions on the boundary of a domain.
In this paper, we mainly consider the Riemann boundary value problems for lower dimensional non-commutative Clifford algebras valued monogenic functions. The solutions are given in an explicit way and concrete examples are presented to illustrate the results.
In this work we propose a biquaternionic reformulation of a fractional monochromatic Maxwell system. Additionally, some examples are given to illustrate how the quaternionic fractional approach emerges in linear hydrodynamic and elasticity.