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Convex Characteristics of Quaternionic Positive Definite Functions on Abelian Groups Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-25 Jingning Liu, Zeping Zhu
This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group G, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from G to the sphere group \({\mathbb {S}}\), i.e., the unit 3-sphere in the quaternion
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More About Bicomplex Möbius Transformations: Geometric, Algebraic and Analitical Aspects Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-24 M. Elena Luna–Elizarrarás, Anatoly Golberg
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Integral Formulas for Slice Cauchy–Riemann Operator and Applications Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-24 Chao Ding, Xiaoqian Cheng
The theory of slice regular functions has been developed rapidly in the past few years, and most properties are given in slices at the early stage. In 2013, Colombo et al. introduced a non-constant coefficients differential operator to describe slice regular functions globally, and this brought the study of slice regular functions in a global sense. In this article, we introduce a slice Cauchy–Riemann
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On Symmetries of Geometric Algebra Cl(3, 1) for Space-Time Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-20 Eckhard Hitzer
From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra Cl(3, 1) that can be used to describe space-time.
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Harmonic Analysis on Exceptional Domain $$E_{6(-14)}/U(1)Spin(10)$$ Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-13 Fouzia El Wassouli, Daoud Oukacha
Let $$\begin{aligned} \mathcal {D}_{16}=\left\{ Z\in \mathcal {M}_{1,2}(\mathfrak {C}^{c}):\;\begin{array}{lll} 1-\left\langle Z,Z \right\rangle +\left\langle Z^{\sharp },Z^{\sharp }\right\rangle>0,\\ 2-\left\langle Z,Z \right\rangle \; >0\end{array}\right\} \end{aligned}$$ be an exceptional domain of non-tube type and let \(\mathcal {U}_{\nu }\) and \(\mathcal {W}_{\nu }\) the associated generalized
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Short Time Quaternion Quadratic Phase Fourier Transform and Its Uncertainty Principles Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-11 Bivek Gupta, Amit K. Verma
In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion-valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff–Young inequality for QQPFT, which in particular sharpens the constant
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The Möbius Addition and Generalized Laplace–Beltrami Operator in Octonionic Space Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-08 Wei Xia, Haiyan Wang
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A Relationship Between Spin and Geometry Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-03 Peter T. J. Bradshaw
In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate
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Fourier-Poisson Transforms Associated with the Principal Series Representations of Sp(1, n) Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-05-28 Xingya Fan, Jianxun He, Xiaoke Jia
Let \(X=Sp(1,n)/Sp(n)\) be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where n is greater than or equal to 1. The Fürstenberg boundary of X is denoted as \(\Sigma \). In this paper, we focus on the Plancherel formula on X associated with the Poisson transform of vector-valued \(L^2\)-space on \(\Sigma \). Through the Fourier-Jacobi transform and the Fourier-Poisson
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Mobility Analysis of Multi-loop Coupling Mechanisms Using Geometric Algebra Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-05-27 Jinqun Guo, Yu Xiao, Qinchuan Li, Lingmin Xu, Xinxue Chai
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On SVD and Polar Decomposition in Real and Complexified Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-05-27 Dmitry Shirokov
In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures
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Distribution Function and Nonincreasing Rearrangement of $${\mathbb {B}}{\mathbb {C}}$$ -Valued Functions with $${\mathbb {B}} {\mathbb {C}}$$ -Measure Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-05-18 İlker Eryılmaz
This paper investigates the distribution function and nonincreasing rearrangement of \(\mathbb{B}\mathbb{C}\)-valued functions equipped with the hyperbolic norm. It begins by introducing the concept of the distribution function for \( \mathbb{B}\mathbb{C}\)-valued functions, which characterizes valuable insights into the behavior and structure of \(\mathbb{B}\mathbb{C}\)-valued functions, allowing
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Hausdorff–Young Inequalities for Fourier Transforms over Cayley–Dickson Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-05-10 Shihao Fan, Guangbin Ren
In this study, we extend Beckner’s seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform
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Machine Learning Clifford Invariants of ADE Coxeter Elements Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-05-04 Siqi Chen, Pierre-Philippe Dechant, Yang-Hui He, Elli Heyes, Edward Hirst, Dmitrii Riabchenko
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Exploring Quaternion Neural Network Loss Surfaces Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-04-24 Jeremiah Bill, Bruce Cox
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Fractional Elliptic Operators with Multiple Poles on Riemannian Manifold with Clifford Bundle Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-04-23 Rami Ahmad El-Nabulsi, Waranont Anukool
We introduce new types of fractional generalized elliptic operators on a compact Riemannian manifold with Clifford bundle. The theory is applicable in well-defined differential geometry. The Connes-Moscovici theorem gives rise to dimension spectrum in terms of residues of zeta functions, applicable in the presence of multiple poles. We have discussed the problem of scalar fields over the unit co-sphere
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Multidimensional Generalized Fractional $${\pmb {S}}$$ Transform Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-04-17 Lakshmanan Subbiah, Roopkumar Rajakumar
In this paper, we introduce a new multidimensional fractional S transform \(S_{\phi ,\varvec{\alpha },\lambda }\) using a generalized fractional convolution \(\star _{\varvec{\alpha },\lambda }\) and a general window function \(\phi \) satisfying some admissibility condition. The value of \(S_{\phi ,\varvec{\alpha },\lambda }f\) is also written in the form of inner product of the input function f with
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A Note on Cohomology of Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-04-09 Bikram Banerjee, Goutam Mukherjee
In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by Clifford cohomology. We show that Clifford cohomology controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of
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Inequalities Pertaining to Quaternion Ambiguity Function Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-04-08 Imanuel Agung Sembe, Mawardi Bahri, Nasrullah Bachtiar, Muhammad Zakir
The quaternion ambiguity function is an expansion of the standard ambiguity function using quaternion algebra. Various properties such as linearity, translation, modulation, Moyal’s formula and inversion identity are studied in detail. In addition, an interesting interaction between the quaternion ambiguity function and the quaternion Fourier transform is demonstrated. Based on these facts, we seek
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Some Uncertainty Principles for the Right-Sided Multivariate Continuous Quaternion Wavelet Transform Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-04-08 Manel Hleili
For the right-sided multivariate continuous quaternion wavelet transform (CQWT), we analyse the concentration of this transform on sets of finite measure. We also establish an analogue of Heisenberg’s inequality for the quaternion wavelet transform. Finally, we extend local uncertainty principle for a set of finite measure to CQWT.
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Lipschitz Norm Estimate for a Higher Order Singular Integral Operator Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-04-08 Tania Rosa Gómez Santiesteban, Ricardo Abreu Blaya, Juan Carlos Hernández Gómez, José Luis Sánchez Santiesteban
Let \(\Gamma \) be a d-summable surface in \(\mathbb {R}^m\), i.e., the boundary of a Jordan domain in \( \mathbb {R}^m\), such that \(\int \nolimits _{0}^{1}N_{\Gamma }(\tau )\tau ^{d-1}\textrm{d}\tau <+\infty \), where \(N_{\Gamma }(\tau )\) is the number of balls of radius \(\tau \) needed to cover \(\Gamma \) and \(m-1\frac{d}{m}\), the operator \(S_\Gamma ^*\) transforms functions of the higher
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A Real Method for Solving Octonion Matrix Equation $$AXB=C$$ Based on Semi-tensor Product of Matrices Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-03-23 Xiaochen Liu, Ying Li, Wenxv Ding, Ruyu Tao
In this paper, the octonion matrix equation \(AXB=C\) is studied based on semi-tensor product of matrices. Firstly, we propose the left real element representation and the right real element representation of octonion. Then we obtain the expression of the least squares Hermitian solution to the octonion matrix equation \(AXB=C\) by combining these representations with \(\mathcal {H}\)-representation
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Common Spectral Properties of Bounded Right Linear Operators AC and BA in the Quaternionic Setting Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-03-18 Rachid Arzini, Ali Jaatit
Let X be a two-sided quaternionic Banach space and let \(A, B, C: X \longrightarrow X\) be bounded right linear quaternionic operators such that \(ACA=ABA\). Let q be a non-zero quaternion. In this paper, we investigate the common properties of \((AC)^{2}-2Re(q)AC+|q|^2I\) and \((BA)^{2}-2Re(q)BA+|q|^2I\) where I stands for the identity operator on X. In particular, we show that $$\begin{aligned} \sigma
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Generalized Partial-Slice Monogenic Functions: A Synthesis of Two Function Theories Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-03-09 Zhenghua Xu, Irene Sabadini
In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines
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Heron’s Formula in Higher Dimensions Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-02-17 Timothy F. Havel
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On Optimal Inequalities for Anti-invariant Riemannian Submersions from Conformal Sasakian Space Forms Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-01-21 Mehraj Ahmad Lone, Towseef Ali Wani
The aim of this paper is two-fold. First, we obtain various inequalities which involve the Ricci and scalar curvatures of horizontal and vertical distributions of anti-invariant Riemannian submersion defined from conformal Sasakian space form onto a Riemannian manifold. Second, we obtain the Chen–Ricci inequality for the said Riemannian submersion.
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Geometric Algebras of Light Cone Projective Graph Geometries Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-01-17 Garret Sobczyk
A null vector is an algebraic quantity with the property that its square is zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by \({{\mathcal {N}}}\). The rules of addition and multiplication in \({{\mathcal {N}}}\) are taken to be the same as those for real and complex square matrices. A distinct pair of null vectors is
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Algorithms for Conic Fitting Through Given Proper and Improper Waypoints in Geometric Algebra for Conics Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-01-09 Pavel Loučka, Petr Vašík
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An Extension of Slice Regular Functions in Terms of Fiber Bundle Theory Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-01-08 J. Oscar González-Cervantes
This work presents an extension, called coordinate slice extension, of the union of a finite number of axially symmetric s domains according to the fiber bundle theory and a kind of slice regular functions are defined on this coordinate slice extension.
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Concept of s-Numbers in Quaternionic Analysis and Schatten Classes Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-12-30 João Costa
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New Versions of the Plemelj–Sochocki Formula in Clifford Analysis Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-12-26 Yufeng Wang, Zhongxiang Zhang
In this paper, we give some new versions of the Plemelj–Sochocki formula under weaker condition in real Clifford Analysis which are different from the result in Luo and Du (Adv Appl Clifford Algebras 27:2531-2583, 2017). By the new versions of the Plemelj–Sochocki formula, we can give a different proof of the generalized Plemelj–Sochocki formula for the symmetric difference of boundary values, which
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Fractional Powers of the Quaternionic d-Bar Derivative Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-11-28 Arran Fernandez, Cihan Güder, Walaa Yasin
This work introduces fractional d-bar derivatives in the setting of quaternionic analysis, by giving meaning to fractional powers of the quaternionic d-bar derivative. The definition is motivated by starting from nth-order d-bar derivatives for \(n\in {\mathbb {N}}\), and further justified by various natural properties such as composition laws and its action on special functions such as Fueter polynomials
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Geometric Algebra Speaks Quantum Esperanto Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-11-11 Sebastian Xambó-Descamps
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General Right-Sided Orthogonal 2D-Planes Split Quaternionic Wave-Packet Transform Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-10-23 Hakim Monaim, Said Fahlaoui
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The $$\mathcal {L_C}$$ -Structure-Preserving Algorithms of Quaternion $$LDL^H$$ Decomposition and Cholesky Decomposition Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-10-16 Mingcui Zhang, Ying Li, Jianhua Sun, Wenxv Ding
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Dual Boas Type Results for the Quaternion Transform and Generalized Lipschitz Spaces Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-10-14 Sergey Volosivets
For the quaternion algebra \({\mathbb {H}}\) and \(f:\mathbb R^2\rightarrow {\mathbb {H}}\), we consider a two-sided quaternion Fourier transform \({\widehat{f}}\). Necessary and sufficient conditions for \({\widehat{f}}\) to belong to generalized uniform Lipschitz spaces are given in terms of behavior of f.
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Spinorial Representation of Submanifolds in a Product of Space Forms Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-10-11 Alicia Basilio, Pierre Bayard, Marie-Amélie Lawn, Julien Roth
We present a method giving a spinorial characterization of an immersion into a product of spaces of constant curvature. As a first application we obtain a proof using spinors of the fundamental theorem of immersion theory for such target spaces. We also study special cases: we recover previously known results concerning immersions in \(\mathbb {S}^2\times \mathbb {R}\) and we obtain new spinorial characterizations
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(Anti) de Sitter Geometry, Complex Conformal Gravity-Maxwell Theory from a Cl(4, C) Gauge Theory of Gravity and Grand Unification Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-09-18 Carlos Castro Perelman
We present the deep connections among (Anti) de Sitter geometry, and complex conformal gravity-Maxwell theory, stemming directly from a gauge theory of gravity based on the complex Clifford algebra Cl(4, C). This is attained by simply promoting the de (Anti) Sitter algebras so(4, 1), so(3, 2) to the real Clifford algebras Cl(4, 1, R), Cl(3, 2, R), respectively. This interplay between gauge theories
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On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-09-04 V. M. Simulik, I. I. Vyikon
Extended gamma matrix Clifford–Dirac and SO(1,9) algebras in the terms of \(8 \times 8\) matrices have been considered. The 256-dimensional gamma matrix representation of Clifford algebra for 8-component Dirac equation is suggested. Two isomorphic realizations \(\textit{C}\ell ^{\texttt {R}}\)(0,8) and \(\textit{C}\ell ^{\texttt {R}}\)(1,7) are considered. The corresponding gamma matrix representations
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Series Representation of Solutions of Polynomial Dirac Equations Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-09-04 Doan Cong Dinh
In this paper, we consider the polynomial Dirac equation \( \left( D^m+\sum _{i=0}^{m-1}a_iD^i\right) u=0,\ (a_i\in {\mathbb {C}})\), where D is the Dirac operator in \({\mathbb {R}}^n\). We introduce a method of using series to represent explicit solutions of the polynomial Dirac equations.
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A New Type of Quaternionic Regularity Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-08-29 A. Vajiac
I introduce a notion of quaternionic regularity using techniques based on hypertwined analysis, a refined version of general hypercomplex theory. In the quaternionic and biquaternionic cases, I show that hypertwined holomorphic (regular) functions admit a decomposition in a hypertwined sum of regular functions in certain subalgebras. The hypertwined quaternionic regularity lies in between slice regularity
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Some Estimates for the Cauchy Transform in Higher Dimensions Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-08-28 Longfei Gu
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
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The Explicit Twisted Group Algebra Structure of the Cayley–Dickson Algebra Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-08-11 Guangbin Ren, Xin Zhao
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
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The General Solution to a System of Linear Coupled Quaternion Matrix Equations with an Application Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-08-09 Long-Sheng Liu
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
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Repeated Cayley–Dickson Processes and Subalgebras of Dimension 8 Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-08-09 Jacques Helmstetter
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Bicomplex Weighted Bergman Spaces and Composition Operators Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-08-07 Stanzin Dolkar, Sanjay Kumar
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).
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On Some Quaternionic Series Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-07-31 J. Oscar González Cervantes, J. Emilio Paz Cordero, Daniel González Campos
The aim of this work is to show that given \(u\in {\mathbb {H}}{\setminus }{\mathbb {R}}\), 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 \(u\in {\mathbb {H}}\). This paper also presents Stokes and Borel-Pompeiu formulas induced by \(G^{-u}\) and as consequence we give some versions
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On Some Lie Groups in Degenerate Clifford Geometric Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-07-18 Ekaterina Filimoshina, Dmitry Shirokov
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
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Various Characteristic Properties of Lipschitzian Elements in Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-07-12 Jacques Helmstetter
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
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A New Way to Construct the Riemann Curvature Tensor Using Geometric Algebra and Division Algebraic Structure Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-06-22 Brian Jonathan Wolk
The Riemann curvature tensor is constructed using the Clifford-Dirac geometric algebra and division-algebraic operator structure.
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Quaternion Quantum Neural Network for Classification Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-06-21 Guillermo Altamirano-Escobedo, Eduardo Bayro-Corrochano
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Mean Value Theorems for Bicomplex Harmonic Functions Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-06-21 Abdelkader Abouricha, Aiad El Gourari, Allal Ghanmi
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.
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Propagators Beyond The Standard Model Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-06-17 Rodolfo José Bueno Rogerio, Luca Fabbri
In this paper, we explore the field propagator with a structure that is general enough to encompas both the case of newly-defined mass-dimension 1 fermions and spin-1/2 bosons. The method we employ is to define a map between spinors of different Lounesto classes, and then write the propagator in terms of the corresponding dual structures.
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Clifford Algebras, Quantum Neural Networks and Generalized Quantum Fourier Transform Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-06-13 Marco A. S. Trindade, Vinícius N. A. Lula-Rocha, S. Floquet
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Beurling’s Theorem in the Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-06-03 Othman Tyr, Radouan Daher
In this research, the Clifford–Fourier transform introduced by E. Hitzer, satisfies some uncertainty principles similar to the Euclidean Fourier transform. An analog of the Beurling–Hörmander’s theorem for the Clifford–Fourier transform is obtained. As a straightforward consequence of Beurling’s theorem, other versions of the uncertainty principle, such as the Hardy, Gelfand–Shilov and Cowling–Price
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Functional Calculus for Dual Quaternions Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-05-24 Stephen Montgomery-Smith
We give a formula for \(f(\eta ),\) where \(f:{\mathbb {C}} \rightarrow {\mathbb {C}}\) is a continuously differentiable function satisfying \(f(\bar{z}) = \overline{f(z)},\) and \(\eta \) is a dual quaternion. Note this formula is straightforward or well known if \(\eta \) is merely a dual number or a quaternion. If one is willing to prove the result only when f is a polynomial, then the methods of
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A Geometric Algebra Approach to Invariance Control in Sliding Regimes for Switched Systems Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-05-19 H. Sira-Ramírez, B. C. Gómez-León, M. A. Aguilar-Orduña
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Right-Covariant Differential Calculus on Hopf Superalgebra $${{\mathbb {F}}}({\mathbb {C}}_q^{2|1})$$ Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-05-18 Salih Celik
We define a new \({{\mathbb {Z}}}_2\)-graded quantum (2+1)-space and show that the extended \({{\mathbb {Z}}}_2\)-graded algebra of polynomials on this \({{\mathbb {Z}}}_2\)-graded quantum space, denoted by \({\mathbb F}({{\mathbb {C}}}_q^{2\vert 1 })\), is a \({{\mathbb {Z}}}_2\)-graded Hopf algebra. We construct a right-covariant differential calculus on \({{\mathbb {F}}}({{\mathbb {C}}}_q^{2\vert
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Dynamical Systems of Operators Induced by Scaled Hypercomplex Rings Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-05-17 Daniel Alpay, Ilwoo Cho
In this paper, we consider a family of the hypercomplex rings \({\mathscr {H}}=\left\{ {\mathbb {H}}_{t}\right\} _{t\in {\mathbb {R}}}\) scaled by \({\mathbb {R}}\), and the dynamical system of \({\mathbb {R}}\) acting on \({\mathscr {H}}\) via a certain action \(\theta \) of \({\mathbb {R}}\). i.e., we study an analysis on dynamical system induced by \({\mathscr {H}}\). In particular, we are interested
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Left-Right Symmetric Fermions and Sterile Neutrinos from Complex Split Biquaternions and Bioctonions Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2023-05-17 Vatsalya Vaibhav, Tejinder P. Singh