Advances in Applied Clifford Algebras最新文献

In this paper, we apply the semi-tensor product of matrices and the real vector representation of a quaternion matrix to find the least squares lower (upper) triangular Toeplitz solution of (AX-XB=C), (AXB-CX^{T}D=E) and (anti)centrosymmetric solution of (AXB-CYD=E). And the expressions of the least squares lower (upper) triangular Toeplitz and (anti)centrosymmetric solution for the studied equations are derived. Additionally, the necessary and sufficient conditions for the existence of solutions and general expression of the studied equations are given. Eventually, some numerical examples are provided for showing the validity and superiority of our method.

An infinite-dimensional family of exact solutions of a three-dimensional biharmonic equation was constructed by the hypercomplex method.

In this paper, we investigate the eigenvalues of quaternion tensors under Einstein Product and their applications in color video processing. We present the Ger(check{s})gorin theorem for quaternion tensors. Furthermore, we have executed some experiments to validate the efficacy of our proposed theoretical framework and algorithms. Finally, we contemplate the application of this methodology in color video compression, in which the reconstruction of an approximate original image is achieved by computing a limited number of the largest eigenvalues, yielding a favorable outcome. In summary, by utilizing block tensors in its iterations, this method converges more rapidly to the desired eigenvalues and eigentensors, which significantly reduces the time required for videos compression.

We compute and explore the full geometric product of two oriented points in conformal geometric algebra Cl(4, 1) of three-dimensional Euclidean space. We comment on the symmetry of the various components, and state for all expressions also a representation in terms of point pair center and radius vectors.

We are dedicated to addressing Riemann–Hilbert boundary value problems (RHBVPs) with variable coefficients, where the solutions are valued in the Clifford algebra of (mathbb {R}_{0,n}), for biaxially monogenic functions defined in the biaxially symmetric domains of the Euclidean space (mathbb {R}^{n}). Our research establishes the equivalence between RHBVPs for biaxially monogenic functions defined in biaxially domains and RHBVPs for generalized analytic functions on the complex plane. We derive explicit solutions and conditions for solvability of RHBVPs for biaxially monogenic functions. Additionally, we explore related Schwarz problems and RHBVPs for biaxially meta-monogenic functions.

This paper presents an approach for extracting points from conic intersections by using the concept of pencils. This method is based on QC2GA—the two-dimensional version of QCGA (Quadric Conformal Geometric Algebra)—that is demonstrated to be equivalent to GAC (Geometric Algebra for Conics). A new interpretation of QC2GA and its objects based on pencils of conics and point space elements is presented, enabling the creation, constraining, and exploitation of pencils of conics. A Geometric Algebra method for computing the discriminants and center point of a conic will also be presented, enabling the proposition of an algorithm for extracting points from a conic intersection object.

In this paper, we extend Fueter’s theorem in hypercomplex function theory to encompass a class of pseudoanalytic functions associated with the main Vekua equation. This class includes Duffin’s (mu )-regular functions as special cases, which correspond to the Yukawa equation. As the parameter (mu rightarrow 0), we recover the classical Fueter’s theorem.

We reorganize, simplify and expand the theory of contractions or interior products of multivectors, and related topics like Hodge star duality. Many results are generalized and new ones are given, like: geometric characterizations of blade contractions and regressive products, higher-order graded Leibniz rules, determinant formulas, improved complex star operators, etc. Different contractions found in the literature are discussed and compared, in special those of Clifford Geometric Algebra. Applications of the theory are developed in a follow-up paper.

The Plemelj-Sokhotski formulas, which deal with limiting values of the Bochner-Martinelli type integral, are powerful tools for analyzing boundary value problems. This article aims to study the boundary behavior of the Bochner-Martinelli type integral formula for the k-Cauchy-Fueter operator. Specifically, we consider the Plemelj-Sokhotski formulas, which will extend the corresponding results in the complex analysis of several variables.

The theory of contractions of multivectors, and star duality, was reorganized in a previous article, and here we present some applications. First, we study inner and outer spaces associated to a general multivector M via the equations (v wedge M = 0) and (v mathbin {lrcorner }M=0). They are then used to analyze special decompositions, factorizations and ‘carvings’ of M, to define generalized grades, and to obtain new simplicity criteria, including a reduced set of Plücker-like relations. We also discuss how contractions are related to supersymmetry, and give formulas for supercommutators of multi-fermion creation and annihilation operators.