Advances in Applied Clifford Algebras最新文献

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.

In this paper we describe the rise of global operators in the scaled quaternionic case, an important extension from the quaternionic case to the family of scaled hypercomplex numbers (mathbb {H}_t,, tin mathbb {R}^*), of which the (mathbb {H}_{-1}=mathbb {H}) is the space of quaternions and (mathbb {H}_{1}) is the space of split quaternions. We also describe the scaled Fueter-type variables associated to these operators, developing a coherent theory in this field. We use these types of variables to build different types of function spaces on (mathbb {H}_t). Counterparts of the Hardy space and of the Arveson space are also introduced and studied in the present setting. The two different adjoints in the scaled hypercomplex numbers lead to two parallel cases in each instance. Finally we introduce and study the notion of rational function.

The right-type groups are nilpotent Lie groups of step two having a pair of anticommutative operators, and many aspects of quaternionic analysis can be generalized to this kind of groups. In this paper, we use the twistor transformation to study the tangential k-Cauchy–Fueter equations and quaternionic k-regular functions on these groups. We introduce the twistor space over the ((4n+r))-dimensional complex right-type groups and use twistor transformation to construct an explicit Radon–Penrose type integral formula to solve the holomorphic tangential k-Cauchy–Fueter equation on these groups. When restricted to the real right-type group, this formula provides solutions to tangential k-Cauchy–Fueter equations. In particular, it gives us many k-regular polynomials.

We design several real representations of split quaternion matrices with the primary objective of establishing both necessary and sufficient conditions for the existence of solutions within a system of split quaternion matrix equations. This includes conditions for the general solution without any constraints, as well as (X=pm X^{eta }) solutions and (eta )-(anti-)Hermitian solutions. Furthermore, we derive the expressions for the general solutions when it is solvable. As an application, we investigate the solutions to a system of five split quaternion matrix equations involving (X^star ). Finally, we present several algorithms and numerical examples to demonstrate the results of this paper.

This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in computer science, physics, and engineering.

We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our machine learning takes the form of supervised learning, specifically multilayer perceptron neural networks. When they achieve good results in classification, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we find new invariants of braids and prove several theorems related to them. This work evolves from our experiments exploring how different types of AI cope with untangling braids with 3 strands, this is why we concentrate mostly on braids with 3 strands.

This article has two purposes. After a short reminder of classical properties of meson algebras (also called Duffin-Kemmer algebras), Sects. 4 to 7 present recent advances in the study of their algebraic structure. Then Sects. 8 to 11 explain that each meson algebra contains a Lipschitz monoid with properties quite similar to those of Lipschitz monoids in Clifford algebras.