Advances in Applied Clifford Algebras最新文献

We give estimates of the Cauchy transform in Lebesgue integral norms in Clifford analysis framework which are the generalizations of Cauchy transform in complex plane, and mainly establish the ((L^{p}, L^{q}))-boundedness of the Clifford Cauchy transform in Euclidean space ({mathbb {R}^{n+1}}) using the Clifford algebra and the Hardy–Littlewood maximal function. Furthermore, we prove Hedberg estimate and Kolmogorov’s inequality related to Clifford Cauchy transform. As applications, some respective results in complex plane are directly obtained. Based on the properties of the Clifford Cauchy transform and the principle of uniform boundedness, we solve existence of solutions to integral equations with Cauchy kernel in quaternionic analysis.

The Cayley–Dickson algebra has long been a challenge due to the lack of an explicit multiplication table. Despite being constructible through inductive construction, its explicit structure has remained elusive until now. In this article, we propose a solution to this long-standing problem by revealing the Cayley–Dickson algebra as a twisted group algebra with an explicit twist function (sigma (A,B)). We show that this function satisfies the equation

and provide a formula for the relationship between the Cayley–Dickson algebra and split Cayley–Dickson algebra, thereby giving an explicit expression for the twist function of the split Cayley–Dickson algebra. Our approach not only resolves the lack of explicit structure for the Cayley–Dickson algebra and split Cayley–Dickson algebra but also sheds light on the algebraic structure underlying this fundamental mathematical object.

Let K be a field of characteristic other than 2, and let (mathcal {A}_n) be the algebra deduced from (mathcal {A}_1=K) by n successive Cayley–Dickson processes. Thus (mathcal {A}_n) is provided with a natural basis ((f_E)) indexed by the subsets E of ({1,2,ldots ,n}). Two questions have motivated this paper. If a subalgebra of dimension 4 in (mathcal {A}_n) is spanned by 4 elements of this basis, is it a quaternion algebra? The answer is always “yes”. If a subalgebra of dimension 8 in (mathcal {A}_n) is spanned by 8 elements of this basis, is it an octonion algebra? The answer is more often “no” than “yes”. The present article establishes the properties and the formulas that justify these two answers, and describes the fake octonion algebras.

Linear coupled matrix equations are widely utilized in applications, including stability analysis of control systems and robust control. In this paper, we establish the necessary and sufficient conditions for the consistency of the system of linear coupled matrix equations and derive an expression of the corresponding general solution (where it is solvable) over quaternion. Additionally, we investigate the necessary and sufficient conditions for the system of linear coupled matrix equations with construct to have a solution and derive a formula of its general solution (where it is solvable). Finally, an algorithm and an example were provided in order to further illustrate the primary outcomes of this paper.

In this paper, we study the bicomplex version of weighted Bergman spaces and the composition operators acting on them. We also investigate the Bergman kernel, duality properties and Berezin transform. This paper is essentially based on the work of Zhu (Operator Theory in Function Spaces of Math. Surveys and Monographs, vol. 138, 2nd edn. American Mathematical Society, Providence, 2007).

The aim of this work is to show that given (uin {mathbb {H}}{setminus }{mathbb {R}}), there exists a differential operator (G^{-u}) whose solutions expand in quaternionic power series expansion ( sum _{n=0}^infty (x-u)^n a_n) in a neighborhood of (uin {mathbb {H}}). This paper also presents Stokes and Borel-Pompeiu formulas induced by (G^{-u}) and as consequence we give some versions of Cauchy’s Theorem and Cauchy’s Formula associated to these kind of regular functions.

In this paper, we introduce and study five families of Lie groups in degenerate Clifford geometric algebras. These Lie groups preserve the even and odd subspaces and some other subspaces under the adjoint representation and the twisted adjoint representation. The considered Lie groups contain degenerate spin groups, Lipschitz groups, and Clifford groups as subgroups in the case of arbitrary dimension and signature. The considered Lie groups can be of interest for various applications in physics, engineering, and computer science.

In most cases, the Lipschitz monoid (textrm{Lip}(V,Q)) is the multiplicative monoid (or semi-group) generated in the Clifford algebra (textrm{Cl}(V,Q)) by the vectors of V. But the elements of (textrm{Lip}(V,Q)) satisfy many other characteristic properties, very different from one another, which may as well be used as definitions of (textrm{Lip}(V,Q)). The present work proposes several characteristic properties, and explores some of the ways that enable us to link one property to another.

The Riemann curvature tensor is constructed using the Clifford-Dirac geometric algebra and division-algebraic operator structure.

Mean value theorems appear as fundamental tools in the analysis of harmonic functions and elliptic partial differential equations. In the present paper, we establish their bicomplex analogs for bicomplex harmonic and strongly harmonic functions with bicomplex values. Their analytical converse as well as geometrical converse characterizing open idempotent discus are also discussed.