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A Note on Centralizers and Twisted Centralizers in Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-09-17 Ekaterina Filimoshina, Dmitry Shirokov
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
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Machine Learning Discovers Invariants of Braids and Flat Braids Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-09-11 Alexei Lisitsa, Mateo Salles, Alexei Vernitski
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The Bessel–Clifford Function Associated to the Cayley–Laplace Operator Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-09-09 David Eelbode
In this paper the Cayley–Laplace operator \(\Delta _{xu}\) is considered, a rotationally invariant differential operator which can be seen as a generalisation of the classical Laplace operator for functions depending on wedge variables \(X_{ab}\) (the minors of a matrix variable). We will show that the Bessel–Clifford function appears naturally in the framework of two-wedge variables, and explain how
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Recent Advances for Meson Algebras and their Lipschitz Monoids Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-09-10 Jacques Helmstetter
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On Octonionic Submodules Generated by One Element Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-09-09 Qinghai Huo, Guangbin Ren
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Parametrizing Clifford Algebras’ Matrix Generators with Euler Angles Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-09-02 Manuel Beato Vásquez, Melvin Arias Polanco
A parametrization, given by the Euler angles, of Hermitian matrix generators of even and odd non-degenerate Clifford algebras is constructed by means of the Kronecker product of a parametrized version of Pauli matrices and by the identification of all possible anticommutation sets. The internal parametrization of the matrix generators allows a straightforward interpretation in terms of rotations, and
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Higher Order Geometric Algebras and Their Implementations Using Bott Periodicity Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-08-31 Marek Stodola, Jaroslav Hrdina
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Quaternion Convolutional Neural Networks: Current Advances and Future Directions Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-08-28 Gerardo Altamirano-Gomez, Carlos Gershenson
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Hypercomplex Representation of Finite-Dimensional Unital Archimedean f-Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-08-28 Sayed Kossentini
In this paper, we characterize all N-dimensional hypercomplex numbers having unital Archimedean f-algebra structure. We use matrix representation of hypercomplex numbers to define an order structure on the matrix spectra. We prove that the unique (up to isomorphism) unital Archimedean f-algebra of hypercomplex numbers of dimension \(N \ge 1\) is that with real and simple spectrum. We also show that
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Geometric Structures on the Quaternionic Unit Ball and Slice Regular Möbius Transformations Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-08-17 Raul Quiroga-Barranco
Building from ideas of hypercomplex analysis on the quaternionic unit ball, we introduce Hermitian, Riemannian and Kähler-like structures on the latter. These are built from the so-called regular Möbius transformations. Such geometric structures are shown to be natural generalizations of those from the complex setup. Our structures can be considered as more natural, from the hypercomplex viewpoint
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Bounds for the Zeros of a Quaternionic Polynomial with Restricted Coefficients Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-08-07 Abdullah Mir, Abrar Ahmad
In this paper, we are concerned with the problem of locating the zeros of polynomials and regular functions with quaternionic coefficients when their real and imaginary parts are restricted. The extended Schwarz’s lemma, the maximum modulus theorem, and the structure of the zero sets defined in the newly constructed theory of regular functions and polynomials of a quaternionic variable are used to
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On the Construction of Beltrami Fields and Associated Boundary Value Problems Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-08-01 Pablo E. Moreira, Briceyda B. Delgado
In this paper, we present two simple methods for constructing Beltrami fields. The first one consists of a composition of operators, including a quaternionic transmutation operator as well as the computation of formal powers for the function \(f(x)=e^{\textbf{i}\lambda x}\). For the second method, we generate Beltrami fields from harmonic functions, and using the intrinsic relation between the normal
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Quaternionic Subspace Gabor Frames and Their Duals Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-07-14 Yun-Zhang Li, Xiao-Li Zhang
Due to its potential application in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention. This paper addresses quaternionic subspace Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. We characterize subspace quaternionic Gabor frames in terms of quaternionic Zak transformation matrices. For an
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On the Geometry of Quantum Spheres and Hyperboloids Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-07-13 Giovanni Landi, Chiara Pagani
We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are \(*\)-quantum spaces for the quantum orthogonal group \(\mathcal {O}(SO_q(3))\). We construct line bundles over the quantum homogeneous space associated with the quantum subgroup SO(2) of \(SO_q(3)\). The line bundles are associated to the quantum principal bundle via representations of SO(2)
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Models of CR Manifolds and Their Symmetry Algebras Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-07-05 Jan Gregorovič, Martin Kolář, Francine Meylan, David Sykes
In this paper we give an exposition of several recent results on local symmetries of real submanifolds in complex space, featuring new examples and important corollaries. Departing from Levi non-degenerate hypersurfaces, treated in the classical Chern–Moser theory, we explore three important classes of manifolds, which naturally extend the classical case. We start with quadratic models for real submanifolds
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The Clifford Algebra of the Density Matrix: An Elementary Approach Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-06-29 Pedro Amao, Hernan Castillo
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A Multi-dimensional Unified Concavity and Convexity Detection Method Based on Geometric Algebra Adv. Appl. Clifford Algebras (IF 1.1) Pub Date : 2024-07-02 Jiyi Zhang, Huanhuan Liu, Tianzi Wei, Ruitong Liu, Chunwang Jia, Fan Yang
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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