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13 pages, 295 KB

Open AccessArticle

On Corresponding Cauchy–Riemann Equations Applied to Laplace-Type Operators over Generalized Quaternions, with an Application

by Ji Eun Kim

Axioms 2025, 14(9), 700; https://doi.org/10.3390/axioms14090700 - 16 Sep 2025

Cited by 1 | Viewed by 764

Abstract

In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra

H_{α, β}

, with

α, β \in R ∖ {0}

. Starting from left/right difference quotients, we [...] Read more.

In this paper, we develop a concise differential–potential framework for the functions of a generalized quaternionic variable in the two-parameter algebra

H_{α, β}

, with

α, β \in R ∖ {0}

. Starting from left/right difference quotients, we derive complete Cauchy–Riemann (CR) systems and prove that, away from the null cone where the reduced norm N vanishes, these first-order systems are necessary and, under

C^{1}

regularity, sufficient for left/right differentiability, thereby linking classical one-dimensional calculus to a genuinely four-dimensional setting. On the potential theoretic side, the Dirac factorization

Δ_{α, β} = \bar{D} D = D \bar{D}

shows that each real component of a differentiable mapping is

Δ_{α, β}

-harmonic, yielding a clean second-order theory that separates the elliptic (Hamiltonian) and split (coquaternionic) regimes via the principal symbol. In the classical case

(α, β) = (- 1, - 1)

, we present a Poisson-type representation solving a model Dirichlet problem on the unit ball

B \subset R^{4}

, recovering mean-value and maximum principles. For computation and symbolic verification, real

4 \times 4

matrix models for left/right multiplication linearize the CR systems. Examples (polynomials, affine CR families, and split-signature contrasts) illustrate the theory, and the outlook highlights boundary integral formulations, Green kernel constructions, and discretization strategies for quaternionic PDEs. Full article

(This article belongs to the Special Issue New Perspectives in Operator Theory and Functional Analysis)

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18 pages, 2069 KB

Open AccessFeature PaperArticle

Representation of Integral Formulas for the Extended Quaternions on Clifford Analysis

by Ji Eun Kim

Mathematics 2025, 13(17), 2730; https://doi.org/10.3390/math13172730 - 25 Aug 2025

Cited by 1 | Viewed by 982

Abstract

This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a [...] Read more.

This work addresses a significant gap in the existing literature by developing integral representation formulas for extended quaternion-valued functions within the framework of Clifford analysis. While classical Cauchy-type and Borel–Pompeiu formulas are well established for complex and standard quaternionic settings, there is a lack of analogous tools for functions taking values in extended quaternion algebras such as split quaternions and biquaternions. The motivation is to extend the analytical power of Clifford analysis to these broader algebraic structures, enabling the study of more complex hypercomplex systems. The objectives are as follows: (i) to construct new Cauchy-type integral formulas adapted to extended quaternionic function spaces; (ii) to identify explicit kernel functions compatible with Clifford-algebra-valued integrands; and (iii) to demonstrate the application of these formulas to boundary value problems and potential theory. The proposed framework unifies quaternionic function theory and Clifford analysis, offering a robust analytic foundation for tackling higher-dimensional and anisotropic partial differential equations. The results not only enhance theoretical understanding but also open avenues for practical applications in mathematical physics and engineering. Full article

(This article belongs to the Special Issue Advanced Research in Complex Analysis Operators and Special Classes of Analytic Functions)

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14 pages, 580 KB

Open AccessArticle

Associative Hypercomplex Algebras Arise over a Basic Set of Subgeometric One-Dimensional Elements

by Alexander P. Yefremov

Mathematics 2025, 13(13), 2105; https://doi.org/10.3390/math13132105 - 26 Jun 2025

Cited by 1 | Viewed by 931

Abstract

An abstract set of one-dimensional (spinor-type) elements randomly oriented on a plane is introduced as a basic subgeometric object. Endowing the set with the binary operations of multiplication and invertible addition sequentially yields a specific semi-group (for which an original Cayley table is [...] Read more.

An abstract set of one-dimensional (spinor-type) elements randomly oriented on a plane is introduced as a basic subgeometric object. Endowing the set with the binary operations of multiplication and invertible addition sequentially yields a specific semi-group (for which an original Cayley table is given) and a generic algebraic system which is shown to generate, apart from algebras of real and complex numbers, the associative hypercomplex algebras of dual numbers, split-complex numbers, and quaternions. The units of all these algebras turn out to be composed of basic 1D elements, thus ensuring the automatic fulfillment of multiplication rules (once postulated). From the standpoint of a three-dimensional space defined by a vector quaternion triad, the condition of a standard (unit) length of 1D basis elements is considered; it is shown that fulfillment of this condition provides an equation mathematically equivalent to the main equation of quantum mechanics. The similarities and differences of the proposed logical scheme with other approaches that involve abstract subgeometric objects are discussed. Full article

(This article belongs to the Section E4: Mathematical Physics)

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21 pages, 19569 KB

Open AccessArticle

The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications

by Zong-Ru Jia and Qing-Wen Wang

Mathematics 2025, 13(4), 644; https://doi.org/10.3390/math13040644 - 16 Feb 2025

Cited by 8 | Viewed by 872

Abstract

This paper presents a systematic investigation into the solvability and the general solution of a tensor equation system within the split quaternion algebra framework. As an extension of classical quaternions with distinctive pseudo-Euclidean properties, split quaternions offer unique advantages in multidimensional signal processing [...] Read more.

This paper presents a systematic investigation into the solvability and the general solution of a tensor equation system within the split quaternion algebra framework. As an extension of classical quaternions with distinctive pseudo-Euclidean properties, split quaternions offer unique advantages in multidimensional signal processing applications. We establish rigorous necessary and sufficient conditions for the existence of solutions to the proposed tensor equation system, accompanied by explicit formulations for general solutions when solvability criteria are satisfied. The theoretical framework is further strengthened by the development of computational algorithms and numerical validations through concrete examples. Notably, we demonstrate the practical implementation of our theoretical findings through encryption/decryption algorithms for color video data. Full article

(This article belongs to the Special Issue New Advances in Algebra, Ring Theory and Homological Algebra, 2nd Edition)

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23 pages, 619 KB

Open AccessArticle

A System of Tensor Equations over the Dual Split Quaternion Algebra with an Application

by Liuqing Yang, Qing-Wen Wang and Zuliang Kou

Mathematics 2024, 12(22), 3571; https://doi.org/10.3390/math12223571 - 15 Nov 2024

Cited by 9 | Viewed by 1474

Abstract

In this paper, we propose a definition of block tensors and the real representation of tensors. Equipped with the simplification method, i.e., the real representation along with the M-P inverse, we demonstrate the conditions that are necessary and sufficient for the system of [...] Read more.

In this paper, we propose a definition of block tensors and the real representation of tensors. Equipped with the simplification method, i.e., the real representation along with the M-P inverse, we demonstrate the conditions that are necessary and sufficient for the system of dual split quaternion tensor equations

(A *_{N} X, X *_{S} C) = (B, D)

, when its solution exists. Furthermore, the general expression of the solution is also provided when the solution of the system exists, and we use a numerical example to validate it in the last section. To the best of our knowledge, this is the first time that the aforementioned tensor system has been examined on dual split quaternion algebra. Additionally, we provide its equivalent conditions when its Hermitian solution

X = X^{*}

and

η

-Hermitian solutions

X = X^{η *}

exist. Subsequently, we discuss two special dual split quaternion tensor equations. Last but not least, we propose an application for encrypting and decrypting two color videos, and we validate this algorithm through a specific example. Full article

(This article belongs to the Special Issue Advances of Linear and Multilinear Algebra)

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17 pages, 294 KB

Open AccessFeature PaperArticle

The General Solution to a Classical Matrix Equation AXB = C over the Dual Split Quaternion Algebra

by Kai-Wen Si and Qing-Wen Wang

Symmetry 2024, 16(4), 491; https://doi.org/10.3390/sym16040491 - 18 Apr 2024

Cited by 16 | Viewed by 1931

Abstract

In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation

A X B = C

, and present the general solution expression when the solvability conditions are met. As an application, we delve [...] Read more.

In this paper, we investigate the necessary and sufficient conditions for solving a dual split quaternion matrix equation

A X B = C

, and present the general solution expression when the solvability conditions are met. As an application, we delve into the necessary and sufficient conditions for the existence of a Hermitian solution to this equation by using a newly defined real representation method. Furthermore, we obtain the solutions for the dual split quaternion matrix equations

A X = C

and

X B = C

. Finally, we provide a numerical example to demonstrate the findings of this paper. Full article

(This article belongs to the Section Mathematics)

15 pages, 316 KB

Open AccessArticle

Quadratic Equation in Split Quaternions

by Wensheng Cao

Axioms 2022, 11(5), 188; https://doi.org/10.3390/axioms11050188 - 20 Apr 2022

Cited by 2 | Viewed by 2794

Abstract

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing [...] Read more.

Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of

x^{2} + b x + c = 0

in split quaternion algebra. Full article

(This article belongs to the Section Algebra and Number Theory)

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27 pages, 797 KB

Open AccessArticle

A Generalization of Quaternions and Their Applications

by Hong-Yang Lin, Marc Cahay, Badri N. Vellambi and Dennis Morris

Symmetry 2022, 14(3), 599; https://doi.org/10.3390/sym14030599 - 17 Mar 2022

Cited by 7 | Viewed by 5461

Abstract

There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, [...] Read more.

There are a total of 64 possible multiplication rules that can be defined starting with the generalized imaginary units first introduced by Hamilton. Of these sixty-four choices, only eight lead to non-commutative division algebras: two are associated to the left- and right-chirality quaternions, and the other six are generalizations of the split-quaternion concept first introduced by Cockle. We show that the

4 \times 4

matrix representations of both the left- and right-chirality versions of the generalized split-quaternions are algebraically isomorphic and can be related to each other by

4 \times 4

permutation matrices of the

C_{2} \times C_{2}

group. As examples of applications of the generalized quaternion concept, we first show that the left- and right-chirality quaternions can be used to describe Lorentz transformations with a constant velocity in an arbitrary spatial direction. Then, it is shown how each of the generalized split-quaternion algebras can be used to solve the problem of quantum-mechanical tunneling through an arbitrary one-dimensional (1D) conduction band energy profile. This demonstrates that six different spinors (

4 \times 4

matrices) can be used to represent the amplitudes of the left and right propagating waves in a 1D device. Full article

(This article belongs to the Section Physics)

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18 pages, 299 KB

Open AccessArticle

Fixed Points Results in Algebras of Split Quaternion and Octonion

by Mobeen Munir, Asim Naseem, Akhtar Rasool, Muhammad Shoaib Saleem and Shin Min Kang

Symmetry 2018, 10(9), 405; https://doi.org/10.3390/sym10090405 - 17 Sep 2018

Cited by 2 | Viewed by 3339

Abstract

Fixed points of functions have applications in game theory, mathematics, physics, economics and computer science. The purpose of this article is to compute fixed points of a general quadratic polynomial in finite algebras of split quaternion and octonion over prime fields

Z_{p}

[...] Read more.

Fixed points of functions have applications in game theory, mathematics, physics, economics and computer science. The purpose of this article is to compute fixed points of a general quadratic polynomial in finite algebras of split quaternion and octonion over prime fields

Z_{p}

. Some characterizations of fixed points in terms of the coefficients of these polynomials are also given. Particularly, cardinalities of these fixed points have been determined depending upon the characteristics of the underlying field. Full article

(This article belongs to the Special Issue Discrete Mathematics and Symmetry)

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