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Titchmarsh’s Theorem in Clifford Analysis Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2021-01-07 Youssef El Haoui
The Clifford Fourier transform (CFT) has been shown to be a crucial tool in the Clifford analysis. The purpose of this paper is to derive an analog of Titchmarsh’s theorems for the CFT for functions satisfying the Lipschitz and Dini–Lipschitz conditions in the space \(L^p(\mathbb {R}^{p,q},C\ell (p,q)), 1
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Wavelet Transform of Dini Lipschitz Functions on the Quaternion Algebra Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2021-01-05 A. Bouhlal, N. Safouane, A. Achak, R. Daher
In this present work, we generalize Titchmarsh’s theorem for the complex- or hypercomplex-valued functions. Firstly, we examine the order of magnitude of the windowed linear canonical transform (WLCT) of complex-valued functions that achieved certain Lipschitz conditions on \({\mathbb {R}}\). Secondly, we studied the order of magnitude of the 2-D continuous quaternion wavelet transform (CQWT) of certain
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On the Solution of a Weinstein-Type Equation in $$\mathbb {R}^3$$ R 3 Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2021-01-04 Doan Cong Dinh
This paper deals with generalized axially symmetric potentials (GASP) which are solutions of a Weinstein-type equation in \(\mathbb {R}^3\) \(\dfrac{\partial ^2\Phi }{\partial x^2}+\dfrac{\partial ^2\Phi }{\partial y^2}+\dfrac{\partial ^2\Phi }{\partial z^2}+\dfrac{2(m+1)}{z}\dfrac{\partial \Phi }{\partial z}=0,\ m\in \mathbb {N}.\) GASP have been investigated by several generations of mathematicians
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On Elliptic Biquaternion Matrices Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2021-01-01 Cui-E Yu, Xin Liu, Yang Zhang
In this paper, the concept of the quaternionic adjoint matrix \(\chi _A\) of an elliptic biquaternion matrix A is introduced, which enable one to discuss the elliptic biquaternion problems through the quaternion ones. By this new concept, some fundamental problems, such as the right eigenvalues and eigenvectors, the singular value decomposition and the inverse can be investigated. Moreover, the least-squares
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On the Conformal Mappings and the Global Operator G Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2021-01-01 J. Oscar González Cervantes, Daniel González Campos
Some important global properties of the slice regular functions have been obtained from the global operator $$\begin{aligned}G:= \Vert \mathbf {x}\Vert ^2 \partial _0 + \mathbf {x} \sum _{i=1}^3 x_i \partial _i , \end{aligned}$$ such as a global characterization, a global Cauchy integral theorem and a global Borel–Pompeiu formula, see Colombo et al. (Trans Am Math Soc 365:303–318, 2013), González Cervantes
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Pseudo-inverses Without Matrices Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-11-26 Jacques Helmstetter
This paper explains that the definition of Moore–Penrose inverses in a given algebra A does not at all require any matrix representation of A. The pseudo-inverses y of a given element x of A (such that \(xyx=x\) and \(yxy=y\)) involve the right ideals complementary to xA and the left ideals complementary to Ax; the Moore–Penrose inverse corresponds to the complementary right and left ideals selected
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Solvability of New Constrained Quaternion Matrix Approximation Problems Based on Core-EP Inverses Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-11-26 Ivan Kyrchei, Dijana Mosić, Predrag S. Stanimirović
Based on the properties of the core-EP inverse and its dual, we investigate three variants of a novel quaternion-matrix (Q-matrix) approximation problem in the Frobenius norm: \(\min \Vert \mathbf {A}\mathbf {X}\mathbf {B}-\mathbf {C}\Vert _F\) subject to the constraints imposed to the right column space of \({\mathbf{A}}\) and the left row space of \({\mathbf{B}}\). Unique solution to the considered
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The Existence of Cauchy Kernels of Kravchenko-Generalized Dirac Operators Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-11-18 Doan Cong Dinh
This paper deals with the static Maxwell system $$\begin{aligned} \left\{ \begin{array}{ll} div(\Phi \overrightarrow{E})&{}=0,\\ \ curl\overrightarrow{E}&{}=0,\ (x_0,x_1,x_2)\in \mathbb {R}^3. \end{array} \right. \end{aligned}$$ The system is reformulated in quaternion analysis by Kravchenko in the form \(\mathcal {L}F=0\) with \(\mathcal {L}F=DF+F\alpha \). We consider special cases of the coefficient
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Some Results for the Two-Sided Quaternionic Gabor Fourier Transform and Quaternionic Gabor Frame Operator Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-11-10 Jinxia Li, Jianxun He
In this paper, we first present some properties of the two-sided quaternionic Gabor Fourier transform (QGFT) on quaternion valued function space \(L^2({\mathbb {R}}^2,{\mathbb {H}})\), such as Parseval’s formula and the characterization of the range of the two-sided QGFT. Then, we give the definitions of quaternionic Wiener space and quaternionic Gabor frame operator (QGFO), which are the generalizations
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Hyperbolic Harmonic Functions and Hyperbolic Brownian Motion Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-10-31 Sirkka-Liisa Eriksson, Terhi Kaarakka
We study harmonic functions with respect to the Riemannian metric $$\begin{aligned} ds^{2}=\frac{dx_{1}^{2}+\cdots +dx_{n}^{2}}{x_{n}^{\frac{2\alpha }{n-2}}} \end{aligned}$$ in the upper half space \(\mathbb {R}_{+}^{n}=\{\left( x_{1},\ldots ,x_{n}\right) \in \mathbb {R}^{n}:x_{n}>0\}\). They are called \(\alpha \)-hyperbolic harmonic. An important result is that a function f is \(\alpha \)-hyperbolic
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Correction to: Clifford Wavelet Transform and the Uncertainty Principle Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-10-14 Hicham Banouh, Anouar Ben Mabrouk, Mhamed Kesri
Unfortunately, there is the slight error in the affiliation of Dr. H. Banouh.
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Generalized Fractional Cauchy–Riemann Operator Associated with the Fractional Cauchy–Riemann Operator Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-10-12 Johan Ceballos, Nicolás Coloma, Antonio Di Teodoro, Diego Ochoa–Tocachi
In this paper, we present a characterization of all linear fractional order partial differential operators with complex-valued coefficients that are associated to the generalized fractional Cauchy–Riemann operator in the Riemann–Liouville sense. To achieve our goal, we make use of the technique of an associated differential operator applied to the fractional case.
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Heisenberg’s and Hardy’s Uncertainty Principles for Special Relativistic Space-Time Fourier Transformation Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-10-01 Youssef El Haoui, Eckhard Hitzer, Said Fahlaoui
The special relativistic (space-time) Fourier transform (SFT) in Clifford algebra \(Cl_{(3,1)}\) of space-time, first introduced from a mathematical point of view in Hitzer (Adv Appl Clifford Algebras 17:497–517, 2007), extends the quaternionic Fourier transform to functions, fields and signals in space-time. The purpose of this paper is to advance the study of the SFT and investigate important properties
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Fractional Clifford–Fourier Transform and its Application Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-09-27 Haipan Shi, Heju Yang, Zunfeng Li, Yuying Qiao
In this paper, we consider a version of the fractional Clifford–Fourier transform (FrCFT) and study its several properties and applications to partial differential equations in Clifford analysis. First, we give the definition of the FrCFT and its inverse transform in the form of integral. Then, we discuss the relationship between the FrCFT and the Clifford–Fourier transform (CFT) and give some properties
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Two-Sided Fourier Transform in Clifford Analysis and Its Application Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-09-23 Haipan Shi, Heju Yang, Zunfeng Li, Yuying Qiao
In this paper, we first define a two-sided Clifford Fourier transform(CFT) and its inverse transformation on \(L^{1}\) space. Then we study the differential of the two-sided CFT, the k-th power of \(F \{h\}\), Plancherel identity and time-frequency shift of the two-sided CFT. Finally we discuss the uncertainty principle of the two-sided CFT and give an application of the two-sided CFT to a partial
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Some Fundamental Theorems of Functional Analysis with Bicomplex and Hyperbolic Scalars Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-09-15 Heera Saini; Aditi Sharma; Romesh Kumar
We discuss some properties of linear functionals on topological hyperbolic and topological bicomplex modules. The hyperbolic and bicomplex analogues of the uniform boundedness principle, the open mapping theorem, the closed graph theorem and the Hahn Banach separation theorem are proved.
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Correction to: On the Quaternionic Quadratic Equation $$xax + bx + xc + d = 0$$ x a x + b x + x c + d = 0 Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-09-11 E. Macías-Virgós, M. J. Pereira-Sáez
Unfortunately, the author names in the reference 9 was wrongly published in the original article and the reference should be
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Polynomial Approximation in Quaternionic Bloch and Besov Spaces Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-09-10 Sorin G. Gal; Irene Sabadini
In this paper we continue our study on the density of the set of quaternionic polynomials in function spaces of slice regular functions on the unit ball by considering the case of the Bloch and Besov spaces of the first and of the second kind. Among the results we prove, we show some constructive methods based on the Taylor expansion and on the convolution polynomials. We also provide quantitative
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On Quaternionic Measures Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-31 M. E. Luna-Elizarrarás; A. Pogorui; M. Shapiro; T. Kolomiiets
In this paper we consider some basic properties of quaternionic measures.
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Variational Integral and Some Inequalities of a Class of Quasilinear Elliptic System Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-26 Yueming Lu; Pan Lian
This paper is concerned with properties for a class of degenerate elliptic equations in Clifford analysis. Here we obtain a direct proof of the existence and uniqueness for the Dirac equations by the method of variational integral. Also, we get the Poincaré inequalities for the case \(q<1\).
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Almansi Theorem and Mean Value Formula for Quaternionic Slice-regular Functions Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-25 Alessandro Perotti
We prove an Almansi Theorem for quaternionic polynomials and extend it to quaternionic slice-regular functions. We associate to every such function f, a pair \(h_1\), \(h_2\) of zonal harmonic functions such that \(f=h_1-\bar{x} h_2\). We apply this result to get mean value formulas and Poisson formulas for slice-regular quaternionic functions.
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Kanzaki’s Generalized Quadratic Spaces and Graded Salingaros Groups Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-24 Jacques Helmstetter
In 1973, Kanzaki introduced a category \(\text {Quad}_2(K)\) made of generalized quadratic spaces (E, f, g) where E was a vector space of finite dimension over a field K, f a linear form on E, and g a quadratic form on E such that the bilinear form \((a,b)\longmapsto f(a)f(b)+g(a+b)-g(a)-g(b)\) was non-degenerate on E. There are direct sums and tensor products in \(\text {Quad}_2(K)\), and its objects
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Efficient Development of Competitive Mathematica Solutions Based on Geometric Algebra with GAALOPWeb Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-24 R. Alves; D. Hildenbrand; C. Steinmetz; P. Uftring
In this paper we present a new tool for Mathematica users, based on the new web-based geometric algebra algorithm optimizer (GAALOP). GAALOPWeb for Mathematica now supports the Mathematica user with an intuitive interface for the development, testing and visualization of geometric algebra algorithms, combining the geometric intuitiveness of geometric algebra with an efficient development of algorithms
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Generalized $$(k_i)$$ ( k i ) -Monogenic Functions Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-19 Doan Cong Dinh
In this paper we introduce generalized \((k_i)\)-monogenic functions in Clifford analysis. They are the general types of the k-hypermonogenic functions founded by Leutwiler and Eriksson. Each component of a generalized \((k_i)\)-monogenic function is a solution of a generalized Weinstein’s equation. We will construct \(2^n\) generalized Cauchy kernels and give an integral representation of the generalized
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The Octonionic Bergman Kernel for the Half Space Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-19 Jinxun Wang; Xingmin Li
We obtain the octonionic Bergman kernel for half space in the octonionic analysis setting by two different methods. As a consequence, we unify the kernel forms in both complex analysis and hyper-complex analysis.
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Leveraging Prior Known Vector Green Functions in Solving Perturbed Dirac Equation in Clifford Algebra Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-14 Morteza Shahpari; Andrew Seagar
Solving boundary value problems with boundary element methods requires specific Green functions suited to the boundary conditions of the problem. Using vector algebra, one often needs to use a Green function for the Helmholtz equation whereas it is a solution of the perturbed Dirac equation that is required for solving electromagnetic problems using Clifford algebra. A wealth of different Green functions
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The Unified Standard Model Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-11 Brage Gording; Angnis Schmidt-May
The aim of this work is to find a simple mathematical framework for our established description of particle physics. We demonstrate that the particular gauge structure, group representations and charge assignments of the Standard Model particles are all captured by the algebra M(8, \(\mathbb {C})\) of complex \(8 \times 8\) matrices. This algebra is well motivated by its close relation to the normed
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Correction to: A Geometric Algebra Based Higher Dimensional Approximation Method for Statics and Kinematics of Robotic Manipulators Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-07 Sudharsan Thiruvengadam, Karol Miller
Typographic error in Eq. (41c): the subscript.
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On the Clifford Algebraic Description of Transformations in a 3D Euclidean Space Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-08-02 Jayme Vaz; Stephen Mann
We discuss how transformations in a three dimensional euclidean space can be described in terms of the Clifford algebra \({\mathcal {C}}\ell _{3,3}\) of the quadratic space \({\mathbb {R}}^{3,3}\). We show that this algebra describes in a unified way the operations of reflection, rotation (circular and hyperbolic), translation, shear and non-uniform scale. Moreover, using Hodge duality, we define an
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GAALOPWeb for Matlab: An Easy to Handle Solution for Industrial Geometric Algebra Implementations Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-07-30 Dietmar Hildenbrand; Christian Steinmetz; Radek Tichý
We present GAALOPWeb for Matlab, a new easy to handle solution for Geometric Algebra implementations for Matlab. We demonstrate its usability for industrial applications based on a forward kinematics algorithm of a serial robot arm and illustrate it with the help of high run-time performance.
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Differential Topological Aspects in Octonionic Monogenic Function Theory Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-07-24 Rolf Sören Kraußhar
In this paper we apply a homologous version of the Cauchy integral formula for octonionic monogenic functions to introduce for this class of functions the notion of multiplicity of zeroes and a-points in the sense of the topological mapping degree. As a big novelty we also address the case of zeroes lying on certain classes of compact zero varieties. This case has not even been studied in the associative
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Self-Reverse Elements and Lines in an Algebra for 3D Space Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-07-15 Robert J. Cripps; Ben Cross; Glen Mullineux
A geometric algebra provides a single environment in which geometric entities can be represented and manipulated and in which transforms can be applied to these entities. A number of versions of geometric algebra have been proposed and the aim of the paper is to investigate one of these as it has a number of advantageous features. Points, lines and planes are presented naturally by element of grades
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The Symplectic Fueter–Sce Theorem Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-07-14 David Eelbode; Sonja Hohloch; Guner Muarem
In this paper we present a symplectic analogue of the Fueter theorem. This allows the construction of special (polynomial) solutions for the symplectic Dirac operator \(D_s\), which is defined as the first-order \(\mathfrak {sp}(2n)\)-invariant differential operator acting on functions on \(\mathbb {R}^{2n}\) taking values in the metaplectic spinor representation.
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Spacetime Geometry with Geometric Calculus Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-07-11 David Hestenes
Geometric Calculus is developed for curved-space treatments of General Relativity and comparison is made with the flat-space gauge theory approach by Lasenby, Doran and Gull. Einstein’s Principle of Equivalence is generalized to a gauge principle that provides the foundation for a new formulation of General Relativity as a Gauge Theory of Gravity on a curved spacetime manifold. Geometric Calculus provides
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On the Supports of Functions Associated to the Radially Deformed Fourier Transform Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-07-01 Shanshan Li; Jinsong Leng; Minggang Fei
In recent work a radial deformation of the Fourier transform in the setting of Clifford analysis was introduced. The key idea behind this deformation is a family of new realizations of the Lie superalgebra \({\mathfrak {osp}}(1|2)\) in terms of a so-called radially deformed Dirac operator \({\mathbf {D}}\) depending on a deformation parameter c such that for \(c=0\) the classical Dirac operator is
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A Discrete Version of Plane Wave Solutions of the Dirac Equation in the Joyce Form Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-30 Volodymyr Sushch
We construct a discrete version of the plane wave solution to a discrete Dirac-Kähler equation in the Joyce form. A geometric discretisation scheme based on both forward and backward difference operators is used. The conditions under which a discrete plane wave solution satisfies a discrete Joyce equation are discussed.
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3-Hom-Lie Algebras Based on $$\sigma $$ σ -Derivation and Involution Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-27 Viktor Abramov; Sergei Silvestrov
We show that, having a Hom-Lie algebra and an element of its dual vector space that satisfies certain conditions, one can construct a ternary totally skew-symmetric bracket and prove that this ternary bracket satisfies the Hom-Filippov-Jacobi identity, i.e. this ternary bracket determines the structure of 3-Hom-Lie algebra on the vector space of a Hom-Lie algebra. Then we apply this construction to
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Sedeonic Equations in Field Theory Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-25 Victor L. Mironov; Sergey V. Mironov
We consider the factorization of differential operator of Klein–Gordon equation on the base of space-time algebra of sixteen-component sedeons. It is shown that generalized sedeonic wave equations can be used both to describe quantum particles and the force fields responsible for the interaction of particles. In particular, we discuss the first-order and second-order wave equations for sedeonic potentials
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Projection Factors and Generalized Real and Complex Pythagorean Theorems Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-25 André L. G. Mandolesi
Projection factors describe the contraction of Lebesgue measures in orthogonal projections between subspaces of a real or complex inner product space. They are connected to Grassmann and Clifford algebras and to the Grassmann angle between subspaces, and lead to generalized Pythagorean theorems, relating measures of subsets of real or complex subspaces and their orthogonal projections on certain families
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Drinfeld Double for Infinitesimal BiHom-bialgebras Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-23 Tianshui Ma; Haiyan Yang
The BiHom-version of infinitesimal bialgebra was studied in [22]. In this paper, the notion of covariant BiHom-bialgebra is introduced generalizing infinitesimal BiHom-bialgebra above. Especially, we give the characterizations of quasitriangular covariant BiHom-bialgebra, then we obtain the concepts of associative BiHom-Yang-Baxter pair and associative BiHom-Yang-Baxter equation. Furthermore, we present
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Generalized Clifford Algebras Associated to Certain Partial Differential Equations Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-09 Doan Cong Dinh
In the classical Clifford analysis the Laplace operator is factorized by the Cauchy–Riemann operator \(\Delta =\overline{D}D\). The consequence is all components of a monogenic function are harmonic functions. In more general situation, suppose that we are given a linear partial differential equation. We wish to find a generalized Clifford algebra such that all components of a generalized monogenic
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Kato S-Spectrum in the Quaternionic Setting Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-06 K. Thirulogasanthar; B. Muraleetharan
In a right quaternionic Hilbert space, for a bounded right linear operator, the Kato S-spectrum is introduced and studied to a certain extent. In particular, it is shown that the Kato S-spectrum is a non-empty compact subset of the S-spectrum and it contains the boundary of the S-spectrum. Using right-slice regular functions, local S-spectrum, at a point of a right quaternionic Hilbert space, and the
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Octonion Measures for Solutions of PDEs Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-05 S. V. Ludkowski
The article is devoted to a new type of octonion measures. The considered such measures are related with solutions of high order hyperbolic PDEs and related Markov processes. Their characteristic functionals are investigated. Cylindrical distributions of these measures are studied.
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Clifford Valued Shearlet Transform Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-04 Jyoti Sharma; Shivam Kumar Singh
This paper deals with the construction of \(n=3 \text{ mod } 4\) Clifford algebra \(Cl_{n,0}\)-valued admissible shearlet transform using the shearlet group \((\mathbb {R}^* < imes \mathbb {R}^{n-1}) < imes \mathbb {R}^n\), a subgroup of affine group of \({\mathbb {R}}^n\). The admissibility conditions for a nonzero Clifford valued square integrable function have been obtained. Various properties such
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Time Series, Hidden Variables and Spatio-Temporal Ordinality Networks Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-06-03 Sudharsan Thiruvengadam; Jei Shian Tan; Karol Miller
In this work, a novel methodology for the modelling and forecasting of time series using higher-dimensional networks in \(\mathbf{R }^{4,1}\) space is presented. Time series data is partitioned, transformed and mapped into five-dimensional conformal space as a network which we call the ‘Spatio-Temporal Ordinality Network’ (STON). These STONs are characterised using specific Clifford Algebraic multivector
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Conformal Mappings Revisited in the Octonions and Clifford Algebras of Arbitrary Dimension Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-05-28 Rolf Sören Kraußhar
In this paper we revisit the concept of conformality in the sense of Gauss in the context of octonions and Clifford algebras. We extend a characterization of conformality in terms of a system of partial differential equations and differential forms using special orthonormal sets of continuous functions that have been used before in the particular quaternionic setting. The aim is to describe to which
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BiHom–Lie Superalgebra Structures and BiHom–Yang–Baxter Equations Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-05-19 Shengxiang Wang; Shuangjian Guo
In this paper, we first introduce the notion of BiHom–Lie superalgebras, which is a generalization of both BiHom–Lie algebras and Hom–Lie superalgebras. Also, we explore some general classes of BiHom–Lie admissible superalgebras and describe all these classes via G-BiHom-associative superalgebras, where G is a subgroup of the symmetric group \(S_{3}\). Finally, we obtain a method to construct the solutions
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Brauer–Clifford Group of ( $$\varvec{S,{{\mathcal {G}}},H}$$ S , G , H )-Azumaya Algebras Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-04-28 Thomas Guédénon
In this paper we extend the notion of the Brauer–Clifford group to the case of an \((S,{{\mathcal {G}}},H)\)-algebra, when H is a cocommutative Hopf algebra, \({{\mathcal {G}}}\) is a Lie algebra in the symmetric monoidal category of left H-modules, and S is a commutative algebra which is an H-module algebra, a \({\mathcal G}\)-module algebra and the H-action is compatible with the \({\mathcal {G}}\)-action
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The Moore–Penrose Inverse and Singular Value Decomposition of Split Quaternions Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-04-24 Rafał Abłamowicz
Using the concept of a transposition anti-involution in Clifford algebra \(C \, \ell _{1,1}\) and the isomorphisms \({\mathbb {H}}_s \cong C \, \ell _{1,1} \cong \text {Mat}(2,{\mathbb {R}}),\) where \({\mathbb {H}}_s\) is the algebra of split quaternions and \(\text {Mat}(2,{\mathbb {R}})\) is the algebra of \(2 \times 2\) real matrices, one can find the Moore–Penrose inverse \(q^{+}\) of a non-zero
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Conjugate Harmonic Functions of Fueter Type Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-04-16 Xingya Fan
Let \({\mathcal {H}}\) be an oriented three-dimensional manifold and \({\mathbb {H}}_+={\mathbb {R}}_+\oplus {\mathcal {H}}\). The author introduces non-abelian vector valued Fourier transforms on \({\mathcal {H}}\) and Poisson integrals on \({\mathbb {H}}_+\). Through the boundary behaviour of Poisson integral, the author obtains the characterization of conjugate harmonic functions of Fueter type
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Split Quaternion-Valued Twin-Multistate Hopfield Neural Networks Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-04-07 Masaki Kobayashi
Complex-valued Hopfield neural networks have been extended to 4-dimensional models using quaternions and commutative quaternions. The algebra of split quaternions is another 4-dimensional hypercomplex number system. In this work, a split quaternion-valued Hopfield neural network is defined. By the computer simulations using the twin-multistate activation function and projection rule, we evaluate the
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Algorithms in Linear Algebraic Groups Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-04-07 Sushil Bhunia; Ayan Mahalanobis; Pralhad Shinde; Anupam Singh
This paper presents some algorithms in linear algebraic groups. These algorithms solve the word problem and compute the spinor norm for orthogonal groups. This gives us an algorithmic definition of the spinor norm. We compute the double coset decomposition with respect to a Siegel maximal parabolic subgroup, which is important in computing infinite-dimensional representations for some algebraic groups
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Factorization and Generalized Roots of Dual Complex Matrices with Rodrigues’ Formula Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-04-02 Danail Brezov
The paper provides an efficient method for obtaining powers and roots of dual complex \(2\times 2\) matrices based on a far reaching generalization of De Moivre’s formula. We also resolve the case of normal \(3\times 3\) and \(4\times 4\) matrices using polar decomposition and the direct sum structure of \(\mathfrak {so}_4\). The compact explicit expressions derived for rational powers formally extend
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Berberian Extension and its S -spectra in a Quaternionic Hilbert Space Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-03-17 B. Muraleetharan; K. Thirulogasanthar
For a bounded right linear operators A, in a right quaternionic Hilbert space \(V_\mathbb {H}^R\), following the complex formalism, we study the Berberian extension \(A^\circ \), which is an extension of A in a right quaternionic Hilbert space obtained from \(V_\mathbb {H}^R\). In the complex setting, the important feature of the Berberian extension is that it converts approximate point spectrum of
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TbGAL: A Tensor-Based Library for Geometric Algebra Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-03-16 Eduardo Vera Sousa; Leandro A. F. Fernandes
Geometric algebra is a powerful mathematical framework that allows us to use geometric entities (encoded by blades) and orthogonal transformations (encoded by versors) as primitives and operate on them directly. In this work, we present a high-level C++ library for geometric algebra. By manipulating blades and versors decomposed as vectors under a tensor structure, our library achieves high performance
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Correction to: Algebraic Construction of Near-Bent and APN Functions Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-03-12 Prasanna Poojary, Harikrishnan Panackal, Vadiraja G. R. Bhatta
Remark 4.1 and Remark 4.5 in Section 4 will be true only if a is a power of two.
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The Quartet of Eigenvectors for Quaternionic Lorentz Transformation Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-03-12 Mikhail Kharinov
In this paper the Lorentz transformation, considered as the composition of a rotation and a Lorentz boost, is decomposed into a linear combination of two orthogonal transforms. In this way a two-term expression of the Lorentz transformation by means of quaternions is proposed. An analytical solution to the problem of finding eigenvectors is given. The conditions for the existence of eigenvectors are
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A Low-Memory Time-Efficient Implementation of Outermorphisms for Higher-Dimensional Geometric Algebras Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-02-22 Ahmad Hosny Eid
From the beginning of David Hestenes rediscovery of geometric algebra in the 1960s, outermorphisms have been a cornerstone in the mathematical development of GA. Many important mathematical formulations in GA can be expressed as outermorphisms such as versor products, linear projection operators, and mapping between related coordinate frames. Over the last two decades, GA-based mathematical models
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Polar Decomposition of Complexified Quaternions and Octonions Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-02-22 Stephen J. Sangwine; Eckhard Hitzer
We present a hitherto unknown polar representation of complexified quaternions (also known as biquaternions), also applicable to complexified octonions. The complexified quaternion is factored into the product of two exponentials, one trigonometric or circular, and one hyperbolic. The trigonometric exponential is a real quaternion, the hyperbolic exponential has a real scalar part and imaginary vector
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A 1d Up Approach to Conformal Geometric Algebra: Applications in Line Fitting and Quantum Mechanics Adv. Appl. Clifford Algebras (IF 1.066) Pub Date : 2020-02-22 Anthony N. Lasenby
We discuss an alternative approach to the conformal geometric algebra (CGA) in which just a single extra dimension is necessary, as compared to the two normally used. This is made possible by working in a constant curvature background space, rather than the usual Euclidean space. A possible benefit, which is explored here, is that it is possible to define cost functions for geometric object matching
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