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The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F ˜ λ , we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F ˜ λ. [ABSTRACT FROM AUTHOR]

The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F˜λ, we construct the Mann iterative sequence and prove that the iterative sequence strongly converges to the fixed point of operator F˜λ.

In this paper we study the Dirichlet problem in the ball for the so-called inframonogenic functions, i.e. the solutions of the sandwich equation ∂ x _ f ∂ x _ = 0 , where ∂ x _ stands for the Dirac operator in R m. The main steps in deriving our results are the establishment of some interior estimates for the first order derivatives of harmonic Hölder continuous functions and the proof of certain invariance property of the higher order Lipschitz class under the action of the Poisson integral. Using Mathematica we also implement an algorithm to find explicitly the solution of such a Dirichlet problem for a much wider class of partial differential equations in the ball of R 3 with polynomial boundary data. [ABSTRACT FROM AUTHOR]

Intensive research efforts have been dedicated to the extension and development of essential aspects that resulted in the theory of one complex variable for higher-dimensional spaces. Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analyses. In this paper, first, we derive a new base of special monogenic polynomials (SMPs) in Fréchet–Cliffordian modules, named the equivalent base, and examine its convergence properties for several cases according to certain conditions applied to related constituent bases. Subsequently, we characterize its effectiveness in various convergence regions, such as closed balls, open balls, at the origin, and for all entire special monogenic functions (SMFs). Moreover, the upper and lower bounds of the order of the equivalent base are determined and proved to be attainable. This work improves and generalizes several existing results in the complex and Clifford context involving the convergence properties of the product and similar bases. [ABSTRACT FROM AUTHOR]

In this paper, we present the algebraic properties of dual quaternions and define a slice regularity of a dual quaternionic function. Since the product of dual quaternions is non-commutative, slice regularity is derived in two ways. Thereafter, we propose the Cauchy-Riemann equations and a power series corresponding to dual quaternions.

The aim of this paper is to establish a representation formula for the solutions of the Lamé-Navier system in linear elasticity theory. We also study boundary value problems for such a system in a bounded domain $ \Omega\subset {\mathbb R}^3 $, allowing a very general geometric behavior of its boundary. Our method exploits the connections between this system and some classes of second order partial differential equations arising in Clifford analysis.

Intensive research efforts have been dedicated to the extension and development of essential aspects that resulted in the theory of one complex variable for higher-dimensional spaces. Clifford analysis was created several decades ago to provide an elegant and powerful generalization of complex analyses. In this paper, first, we derive a new base of special monogenic polynomials (SMPs) in Fréchet–Cliffordian modules, named the equivalent base, and examine its convergence properties for several cases according to certain conditions applied to related constituent bases. Subsequently, we characterize its effectiveness in various convergence regions, such as closed balls, open balls, at the origin, and for all entire special monogenic functions (SMFs). Moreover, the upper and lower bounds of the order of the equivalent base are determined and proved to be attainable. This work improves and generalizes several existing results in the complex and Clifford context involving the convergence properties of the product and similar bases.

In this paper, we present the algebraic properties of dual quaternions and define a slice regularity of a dual quaternionic function. Since the product of dual quaternions is non-commutative, slice regularity is derived in two ways. Thereafter, we propose the Cauchy-Riemann equations and a power series corresponding to dual quaternions. [ABSTRACT FROM AUTHOR]

Abstract This article mainly studies the boundary value problems for hypergenic function vectors in Clifford analysis. Firstly, some properties of hypergenic quasi-Cauchy type integrals are discussed. Then, by the Schauder fixed point theorem the existence of the solution to the nonlinear boundary value problem is proved. Finally, using the compression mapping principle the existence and uniqueness of the solution to the linear boundary value problem are proved.

Using the method of normalized systems of functions, we study one representation of real analytic functions by monogenic functions (i.e., solutions of Dirac equations), which is an Almansi's formula of infinite order. As applications of the representation, we construct solutions of the inhomogeneous Dirac and poly-Dirac equations in Clifford analysis. [ABSTRACT FROM AUTHOR]

In this article, we study the steady generalized Navier-Stokes equations in a half-space in the setting of variable exponent spaces. We first establish variable exponent spaces of Clifford-valued functions in a half-space. Then, using this operator theory together with the contraction mapping principle, we obtain the existence and uniqueness of solutions to the stationary Navier-Stokes equations and Navier-Stokes equations with heat conduction in a half-space under suitable hypotheses.

Transmission boundary value problems for the time-harmonic Maxwell system, the Helmholtz operator, and the perturbed Dirac operator are formulated, using Clifford algebras, on bounded Lipschitz domains of R m with m ≥ 3. It is shown how the Dirac problem decouples into several Maxwell and Helmholtz problems. Necessary and sufficient conditions are provided for well-posedness in each case, and for the Dirac problem to be equivalent to one or several independent Maxwell problems. [ABSTRACT FROM AUTHOR]

Abstract First, a kind of generalized Teodorescu operator which is related to the k-regular functions are defined. Then the boundedness and the Hölder continuity of the generalized Teodorescu operator are discussed. Finally, the stability and error estimate of the generalized Teodorescu operator are given when the boundary surface of the integral domain Ω is perturbed.

We give representations and properties of a hyperholomorphic function with values in commutative-quaternions. We first consider expressions of commutative-quaternions. Also, we investigate the results of derivatives and integrations for a hyperholomorphic function of commutative-quaternionic variables in Clifford analysis. [ABSTRACT FROM AUTHOR]

We introduce a one-parameter family of transforms, U ( m ) t , t > 0 , from the Hilbert space of Clifford algebra valued square integrable functions on the m –dimensional sphere, L 2 ( S m , d σ m ) ⊗ C m + 1 , to the Hilbert spaces, M L 2 ( R m + 1 ∖ { 0 } , d μ t ) , of solutions of the Euclidean Dirac equation on R m + 1 ∖ { 0 } which are square integrable with respect to appropriate measures, d μ t . We prove that these transforms are unitary isomorphisms of the Hilbert spaces and are extensions of the Segal–Bargman coherent state transform, U ( 1 ) : L 2 ( S 1 , d σ 1 ) ⟶ H L 2 ( C ∖ { 0 } , d μ ) , to higher dimensional spheres in the context of Clifford analysis. In Clifford analysis it is natural to replace the analytic continuation from S m to S C m as in (Hall, 1994; Stenzel, 1999; Hall and Mitchell, 2002) by the Cauchy–Kowalewski extension from S m to R m + 1 ∖ { 0 } . One then obtains a unitary isomorphism from an L 2 –Hilbert space to a Hilbert space of solutions of the Dirac equation, that is to a Hilbert space of monogenic functions. [ABSTRACT FROM AUTHOR]

We give a representation of the class of Cullen-regular functions in split-quaternions. We consider each Cullen's form of split-quaternions, which provides corresponding Cauchy-Riemann equations for split-quaternionic variables. Using Cullen's form, we research hyperholomorphy and the properties of functions of split-quaternionic variables which are expressed by hyperbolic coordinates. [ABSTRACT FROM AUTHOR]

In this paper, using the intertwine relations of differential operators, we study one representation of real analytic functions by Dunkl-harmonic functions, which is a generalization of the well-known Almansi formula. As an application of the representation, we construct a solution of the Dunkl-Poisson equations in Clifford analysis. Then we investigate solutions of homogeneous and inhomogeneous Dirichlet type problems for Dunkl-Poisson's equation, and inhomogeneous Dirichlet problems for Dunkl-Laplace's equation. [ABSTRACT FROM AUTHOR]

We examine corresponding Cauchy-Riemann equations by using the non-commutativity for the product on split-biquaternions. Additionally, we describe the regularity of functions and properties of their differential equations on split-biquaternions. We investigate representations and calculations of the derivatives of functions of split-biquaternionic variables. [ABSTRACT FROM AUTHOR]

In this article we show the existence of weak solutions for obstacle problems for A-Dirac equations with variable growth in the setting of variable exponent spaces of Clifford-valued functions. We also obtain the existence of weak solutions to the scalar part of A-Dirac equations in space $W_0^{1,p(x)}(Omega,Cell_n)$.

The recent growing interest in special Clifford 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 coefficients, due to their combinatoric relevance. This concerns, in particular, a generalized Appell sequence of homogeneous polynomials whose coefficient 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.

Let for s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 be the space of s-vectors in the Clifford algebra IR0,m+1 constructed over the quadratic vector space IR0,m+1 and let r, p, q, ∈ IN be such that 0 ≤ r ≤ m + 1, p < q and r + 2q ≤ m + 1. The associated linear system of first order partial differential equations derived from the equation ∂xW = 0 where W is IR(r,p,q)0,m+1 = ∑q j=p ⊕IR(r+2j)0,m+1 -valued and ∂x is the Dirac operator in IRm+1, is called a generalized Moisil-Théodoresco system of type (r, p, q) in IRm+1. For k ∈ N, k ≥ 1,MT+(m+ 1; k; IR(r,p,q)0,m+1), denotes the space of IR(r,p,q)0,m+1-valued homogeneous polynomials Wc of degree k in IRm+1 satisfying ∂xWx = 0. A characterization of Wk∈ MT+(m + 1; k;IR(r,p,q)0,m+1) is given in terms of a harmonic potential Hk+1 belonging to a subclass of IR(r,p,q)0,m -valued solid harmonics of degree (k + 1) in IRm+1. Furthermore, it is proved that each Wk∈ MT+(m+ 1; k; IR(r,p,q)0,m+1) admits a primitive Wk+1 ∈ MT+(m+ 1; k + 1; IR(r,p,q)0,m+1). Special attention is paid to the lower dimensional cases IR³ and IR4. In particular, a method is developed for constructing bases for the spaces MT+(4; k; IR(r,p,q)0,4), r being even.Para s ∈ {0, 1, ...,m+ 1} (m ≥ 2) , IR(s)0,m+1 el espacio de los s-vectors en el algebra de Clifford IR0,m+1 construida sobre el espacio de vectores cuadráticos IR0,m+1 sea r, p, q, ∈ IN tal que 0 ≤ r ≤ m + 1, p < q. El sistema lineal asociado de ecuaciones diferenciales parciales de primer orden derivado de la ecuaci´on ∂xW = 0 donde W es IR(r,p,q)0,m+1 = ∑q j=p ⊕IR(r+2j)0,m+1 1-valuada y ∂x es el operador de Dirac en IRm+1, es llamado un sistema de Moisil-Théodoresco generalizado de tipo (r, p, q) en IRm+1. Para k ∈ N, k ≥ 1,MT+(m+ 1; k; IR(r,p,q)0,m+1), denota el espacio de polinomios homogéneosWk IR(r,p,q)0,m+1- valuados de grado k en IRm+1. satisfaciendo ∂xWx = 0. Una caracterización de Wk∈ MT+(m+1; k; IR(r,p,q)0,m+1) es dada en términos de un potencial armónico Hk+1 perteneciendo a una subclase de armónicos consistentes IR(r,p,q)0,m -valuados de grado (k + 1) in IRm+1. Además es probado que todo Wk∈ MT+(m + 1; k; IR(r,p,q)0,m+1) admite una primitiva Wk+1 ∈ MT+(m + 1; k + 1; IR(r,p,q)0,m+1). Una especial atención es dada a los casos de dimensión baja IR³ y IR4. En particular, un metodo es desarrollado para construir bases para espaciosMT+(4; k; IR(r,p,q)0,4 ), r siendo par.

We aim to investigate the differentiability of multidual functions and the notion of the hyperholomor-phicity to multidual-valued functions. Also, we provide the basic statements which extend holomorphic functions to the higher multidual generalized Clifford analysis. [ABSTRACT FROM AUTHOR]

First, a kind of generalized Teodorescu operator which is related to the k-regular functions are defined. Then the boundedness and the Hölder continuity of the generalized Teodorescu operator are discussed. Finally, the stability and error estimate of the generalized Teodorescu operator are given when the boundary surface of the integral domain Ω is perturbed. [ABSTRACT FROM AUTHOR]

In this paper we prove that a generalization of complex Π-operator in Clifford analysis, obtained by the use of two orthogonal bases of a Euclidean space, possesses several mapping and invertibility properties, as studied before for quaternion-valued functions as well as in the standard Clifford analysis setting. We improve and generalize most of those previous results in this direction and additionally other consequent results are presented. In particular, the expression of the jump of the generalized Π-operator across the boundary of the domain is obtained as well as an estimate for the norm of the Π-operator is given. At the end an application of the generalized Π-operator to the solution of Beltrami equations is studied where we give conditions for a solution to realize a local and global homeomorphism. [ABSTRACT FROM AUTHOR]

In this paper, we establish a Hodge-type decomposition of variable exponent Lebesgue spaces of Clifford-valued functions, where one of the subspaces is the space of all monogenic L p ( x ) -functions. Using this decomposition, we obtain the existence and uniqueness of solutions to the homogeneous A -Dirac equations with variable growth under certain appropriate conditions and to the Stokes equations in the setting of variable exponent spaces of Clifford-valued functions. [ABSTRACT FROM AUTHOR]

In this paper we study the stationary generalized Navier-Stokes equations when the viscosity is not only a constant but also a function which depends on the position and the shear-velocity. For this we establish an improved decomposition of variable exponent Lebesgue spaces of Clifford-valued functions. Using this decomposition together with Clifford operator calculus, we obtain the existence, uniqueness and representation of solutions for the generalized Stokes equations and the generalized Navier-Stokes equations with variable viscosity in the setting of variable exponent spaces of Clifford-valued functions. Furthermore, the equivalences of solutions and weak solutions for the aforementioned equations are justified. [ABSTRACT FROM AUTHOR]

The idea of multiplexing motivates us to develop the theory on the Fourier transform (FT) of multivector-valued functions. In this paper, in the framework of Clifford analysis, we establish a Heisenberg-Pauli-Weyl type uncertainty principle for the FT of multivector-valued functions. [ABSTRACT FROM AUTHOR]

Abstract: The classical Hadamard three-circles theorem (1896) gives a relation between the maximum absolute values of an analytic function on three concentric circles. More precisely, it asserts that if f is an analytic function in the annulus , , and if , , and are the maxima of f on the three circles corresponding, respectively, to , , and r then In this paper we introduce a Hadamardʼs three-hyperballs type theorem in the framework of Clifford analysis. As a concrete application, we obtain an overconvergence property of special monogenic simple series. [Copyright &y& Elsevier]

Abstract: In this paper, we give the Hörmanderʼs theorem for the Dirac operator over an open subset with Clifford algebra. Some sufficient condition on the existence of the weak solutions for the Dirac operator has been obtained in the sense of Clifford analysis. In particular, if Ω is bounded, then we prove that for any f in space with value in Clifford algebra, there exists a weak solution of the Dirac operator such that with u in the space as well. The method is based on Hörmanderʼs existence theorem in complex analysis and the weighted space is utilized. [Copyright &y& Elsevier]