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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 1.5) 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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Series Representation of Solutions of Polynomial Dirac Equations Adv. Appl. Clifford Algebras (IF 1.5) 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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On the Representations of Clifford and SO(1,9) Algebras for 8-Component Dirac Equation Adv. Appl. Clifford Algebras (IF 1.5) 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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A New Type of Quaternionic Regularity Adv. Appl. Clifford Algebras (IF 1.5) 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 1.5) 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 1.5) 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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Repeated Cayley–Dickson Processes and Subalgebras of Dimension 8 Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-08-09 Jacques Helmstetter
Let K be a field of characteristic other than 2, and let \(\mathcal {A}_n\) be the algebra deduced from \(\mathcal {A}_1=K\) by n successive Cayley–Dickson processes. Thus \(\mathcal {A}_n\) is provided with a natural basis \((f_E)\) indexed by the subsets E of \(\{1,2,\ldots ,n\}\). Two questions have motivated this paper. If a subalgebra of dimension 4 in \(\mathcal {A}_n\) is spanned by 4 elements
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The General Solution to a System of Linear Coupled Quaternion Matrix Equations with an Application Adv. Appl. Clifford Algebras (IF 1.5) 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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Bicomplex Weighted Bergman Spaces and Composition Operators Adv. Appl. Clifford Algebras (IF 1.5) 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 1.5) 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 1.5) 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 1.5) 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 1.5) 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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Mean Value Theorems for Bicomplex Harmonic Functions Adv. Appl. Clifford Algebras (IF 1.5) 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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Quaternion Quantum Neural Network for Classification Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-06-21 Guillermo Altamirano-Escobedo, Eduardo Bayro-Corrochano
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Propagators Beyond The Standard Model Adv. Appl. Clifford Algebras (IF 1.5) 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 1.5) 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 1.5) 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 1.5) 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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Right-Covariant Differential Calculus on Hopf Superalgebra $${{\mathbb {F}}}({\mathbb {C}}_q^{2|1})$$ Adv. Appl. Clifford Algebras (IF 1.5) 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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A Geometric Algebra Approach to Invariance Control in Sliding Regimes for Switched Systems Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-05-19 H. Sira-Ramírez, B. C. Gómez-León, M. A. Aguilar-Orduña
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Dynamical Systems of Operators Induced by Scaled Hypercomplex Rings Adv. Appl. Clifford Algebras (IF 1.5) 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 1.5) Pub Date : 2023-05-17 Vatsalya Vaibhav, Tejinder P. Singh
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On Multi-conditioned Conic Fitting in Geometric Algebra for Conics Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-05-13 Pavel Loučka, Petr Vašík
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Graded Symmetry Groups: Plane and Simple Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-05-13 Martin Roelfs, Steven De Keninck
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Biquaternionic Treatment of Inhomogeneous Time-Harmonic Maxwell’s Equations Over Unbounded Domains Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-05-10 Briceyda B. Delgado, Vladislav V. Kravchenko
We study the inhomogeneous equation \({\text {curl}}\vec {w}+\lambda \vec {w}=\vec {g},\,\lambda \in {\mathbb {C}},\,\lambda \ne 0\) over unbounded domains in \({\mathbb {R}}^{3}\), with \(\vec {g}\) being an integrable function whose divergence is also integrable. Most of the results rely heavily on the “good enough” behavior near infinity of the \(\lambda \) Teodorescu transform, which is a classical
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Least-Squares Solutions of Generalized Sylvester-Type Quaternion Matrix Equations Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-05-08 Sinem Şimşek
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The Atiyah-Singer Index Theorem for a Family of Fractional Dirac Operators on Spin Geometry Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-05-05 Rami Ahmad El-Nabulsi
The Atiyah-Singer index formula for Dirac operators acting on the space of spinors put across a kind of topological invariant (\(\hat{{A}}\) genus) of a closed spin manifold \({{\mathcal {M}}}\), hence offering a bridge between geometric and analytical aspects of the original spin manifold. In this study, we prove the index theorem for a family of fractional Dirac operators in particular for complex
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The Fourier Transform Associated to the k-Hyperbolic Dirac Operator Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-05-04 Wenxin Li, Pan Lian
The polynomial null solutions of the k-hyperbolic Dirac operator are investigated by the \(\mathfrak {osp}(1|2)\) approach. These solutions are then utilized to construct the (fractional) Fourier transform associated to the k-hyperbolic Dirac operator. The resulting integral kernels are found to be a specific kind of Dunkl kernels. Additionally, we give tight uncertainty inequalities for three distinct
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S-Spectrum of Quaternionic Right Linear Bounded Operators Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-04-20 Somayya Moulaharabbi, Mohamed Barraa
In this paper, we study the properties of the S-spectrum of right linear bounded operators on a right quaternionic Banach space. We prove some relations between the S-spectrum and most of its important parts; the approximate S-spectrum, the compression S-spectrum and the surjective S-spectrum. Among other results, we provide some properties of duality and orthogonality on a right quaternionic Banach
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Beyond the 10-fold Way: 13 Associative $$ {\mathbb Z}_2\times {\mathbb Z}_2$$ -Graded Superdivision Algebras Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-03-28 Zhanna Kuznetsova, Francesco Toppan
The “10-fold way” refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, \({\mathbb Z}_2\)-graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by
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Riemann–Hilbert Problems for Axially Symmetric Monogenic Functions in $${\mathbb {R}}^{n+1}$$ Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-03-21 Qian Huang, Fuli He, Min Ku
We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems \(\big (\)for short RHBVPs\(\big )\) for axially monogenic functions on Euclidean space \({\mathbb {R}}^{n+1},n\in {\mathbb {N}}\). With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on
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The Tangential k-Cauchy–Fueter Operator on Right-Type Groups and Its Bochner–Martinelli Type Formula Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-03-20 Yun Shi, Guangzhen Ren
The k-Cauchy–Fueter operator and the tangential k-Cauchy–Fueter operator are the quaternionic counterpart of Cauchy–Riemann operator and the tangential Cauchy–Riemann operator in the theory of several complex variables, respectively. In Wang (On the boundary complex of the k-Cauchy–Fueter complex, arXiv:2210.13656), Wang introduced the notion of right-type groups, which have the structure of nilpotent
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Rings with Centrally-Extended Higher $$*$$ -Derivations Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-03-16 O. H. Ezzat
We study the notions of centrally-extended higher \(*\)-derivations and centrally-extended generalized higher \(*\)-derivations. Both are shown to be additive in a \(*\)-ring without nonzero central ideals. Also, we prove that in semiprime \(*\)-rings with no nonzero central ideals, every centrally-extended (generalized) higher \(*\)-derivation is a (generalized) higher \(*\)-derivation.
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Bicomplex Neural Networks with Hypergeometric Activation Functions Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-03-13 Nelson Vieira
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Separable Cowreaths Over Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-03-12 Claudia Menini, Blas Torrecillas
The fundamental notion of separability for commutative algebras was interpreted in categorical setting where also the stronger notion of heavily separability was introduced. These notions were extended to (co)algebras in monoidal categories, in particular to cowreaths. In this paper, we consider the cowreath \( \left( A\otimes H_{4}^{op}, H_{4}, \psi \right) \), where \(H_{4}\) is the Sweedler 4-dimensional
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Two-sided Clifford-valued Linear Canonical Transform: Properties and Mustard Convolution Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-03-11 Aajaz A. Teali, Firdous A. Shah
The present study is the first of its kind which aims to analyse the Clifford-valued functions by introducing the notion of a two-sided Clifford-valued linear canonical transform in \(L^2({\mathbb {R}}^n, C\ell _{0,n})\), which not only embodies the classical Clifford–Fourier transform, but also yields another new variant of Clifford transforms based on the fractional Clifford–Fourier transform. To
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Simply Complete Hom-Lie Superalgebras and Decomposition of Complete Hom-Lie Superalgebras Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-02-27 Mohammed Reza Farhangdoost, Ahmad Reza Attari Polsangi, Sergei Silvestrov
Complete hom-Lie superalgebras are considered and some equivalent conditions for a hom-Lie superalgebra to be a complete hom-Lie superalgebra are established. In particular, the relation between decomposition and completeness for a hom-Lie superalgebra is described. Moreover, some conditions that the linear space of \(\alpha ^{s}\)-derivations of a hom-Lie superalgebra to be complete and simply complete
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Hayman Theorem in $${\mathcal {R}}_+^{n+1}$$ with the Clifford Analysis Setting Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-02-07 Yan Hui Zhang, Wei Wen, Kit Ian Kou
The Hayman Theorem of left-monogenic function in a Clifford Analysis Setting is established in this article. A few established conclusions regarding subharmonic functions in Euclidean half space are extended to Clifford half space.
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On the Bundle of Clifford Algebras Over the Space of Quadratic Forms Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-02-06 Arkadiusz Jadczyk
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The Rules of 4-Dimensional Perspective: How to Implement Lorentz Transformations in Relativistic Visualization Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-02-01 Andrew J. S. Hamilton
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The Supergeometric Algebra Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-02-01 Andrew J. S. Hamilton
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On the Mean Ergodic Theorem in Bicomplex Banach Modules Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-02-02 Panagiotis N. Koumantos
In this paper the mean ergodic theorem in bicomplex Banach modules is studied. Under appropriate conditions of boundness for the iterates compositions of a bicomplex linear and bounded operator on a bicomplex Banach module, and of weak compactness of the average sequence on the idempotent components, analogous to that of the classical case on Banach spaces, strong convergence of the mean sequence is
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On the Jackson–Stechkin Theorems for the Best Approximations of Functions in Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-01-26 Othman Tyr, Radouan Daher
In this research, we look at problems in the theory of approximation of functions in real Clifford algebras. We prove analogues of direct and inverse approximation theorems in terms of best approximations of functions with bounded spectrum and the moduli of smoothness of all orders constructed by the generalized Steklov operators.
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Necessity of Tensorial Connections for Spinorial Systems Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-01-04 Luca Fabbri
We consider spinorial fields in polar form to deduce their respective tensorial connection in various physical situations: we show that in some cases the tensorial connection is a useful tool, instead in other cases it arises as a necessary object. The comparative analysis of the different cases possessing a tensorial connection is done, investigating the analogies between space-time structures. Eventual
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B-Spline Pythagorean Hodograph Curves in Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2023-01-03 Michal Bizzarri, Kryštof Kadlec, Miroslav Lávička, Zbyněk Šír
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Adaptive Fourier Decomposition of Slice Regular Functions Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-12-19 Ming Jin, Ieng Tak Leong, Tao Qian, Guangbin Ren
In the slice Hardy space over the unit ball of quaternions, we introduce the slice hyperbolic backward shift operator \(\mathcal S_a\) with the decomposition process $$\begin{aligned} f=e_a\langle f, e_a\rangle +B_{a}*\mathcal S_a f, \end{aligned}$$ where \(e_a\) denotes the slice normalized Szegö kernel and \( B_a \) the slice Blaschke factor. Iterating the above decomposition process, a corresponding
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Equations Satisfied by the Coordinates of Lipschitzian Elements in Clifford Algebras Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-12-15 Jacques Helmstetter
In every Clifford algebra \(\textrm{Cl}(V,Q)\) over a field, there is a Lip-schitz monoid (or semi-group) \(\textrm{Lip}(V,Q)\) that satisfies a lot of remarkable properties. In general, it is the multiplicative monoid generated by the vectors of the space V. Each of its two components is an irreducible algebraic submanifold. The present article is devoted to the equations that are satisifed by the
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Less is More: Efficient Networked VR Transformation Handling Using Geometric Algebra Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-12-08 Manos Kamarianakis, Ilias Chrysovergis, Nick Lydatakis, Mike Kentros, George Papagiannakis
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Geometric Invariant Decomposition of $$\text {SU}(\textbf{3})$$ Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-12-06 Martin Roelfs
A novel invariant decomposition of diagonalizable \(n \times n\) matrices into n commuting matrices is presented. This decomposition is subsequently used to split the fundamental representation of \(\mathfrak {su}({3})\) Lie algebra elements into at most three commuting elements of \(\mathfrak {u}({3})\). As a result, the exponential of an \(\mathfrak {su}({3})\) Lie algebra element can be split into
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Computational Aspects of Geometric Algebra Products of Two Homogeneous Multivectors Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-11-14 Stephane Breuils, Vincent Nozick, Akihiro Sugimoto
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Symmetric Functions and 3D Fermion Representation of $$\pmb {W_{1+\infty }}$$ Algebra Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-11-14 Wang Na, Bai Yang, Cui Zhennan, Wu Ke
In this paper, we consider the actions of affine Yangian and \(W_{1+\infty }\) algebra on three cases of symmetric functions. The first one is Schur functions of 2D Young diagrams. It is known that affine Yangian and \(W_{1+\infty }\) algebra can be represented by 1 Boson field with center 1 in this case. The second case is the symmetric functions \(Y_\lambda ({\mathbf{p}})\) of 2D Young diagrams which
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Pitt’s Inequality and Logarithmic Uncertainty Principle for the Clifford-Fourier Transform Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-11-12 Shanshan Li, Minggang Fei
In this paper, we prove the sharp Pitt’s inequality for a generalized Clifford-Fourier transform which is given by a similar operator exponential as the classical Fourier transform but containing generators of Lie superalgebra. As an application, the Beckner’s logarithmic uncertainty principle for the Clifford-Fourier transform is established.
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Supersymmetric Schur Q-functions and Super BKP Hierarchy Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-11-12 Fang Huang, Chuanzhong Li
In this paper, we first define supersymmetric Schur Q-functions and give their vertex operators realization. By means of the vertex operator, we obtain a series of non-linear partial differential equations of infinite order, called the super BKP hierarchy and the super BKP hierarchy governs the supersymmetric Schur Q-functions as the tau functions. Moreover, we prove that supersymmetric Schur Q-functions
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Rock Classification with Features Based on Higher Order Riesz Transform Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-10-07 Martin Reinhardt, Swanhild Bernstein, Johanna Richter
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Special Affine Fourier Transform for Space-Time Algebra Signals in Detail Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-10-07 Eckhard Hitzer
We generalize the space-time Fourier transform (SFT) (Hitzer in Adv Appl Clifford Algebras 17(3):497–517, 2007) to a special affine Fourier transform (SASFT, also known as offset linear canonical transform) for 16-dimensional space-time multivector Cl(3, 1)-valued signals over the domain of space-time (Minkowski space) \(\mathbb {R}^{3,1}.\) We establish how it can be computed in terms of the SFT,
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Zeon and Idem-Clifford Formulations of Hypergraph Problems Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-10-07 Samuel Ewing, G. Stacey Staples
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Wigner–Yanase Skew Information and Uncertainty Relations for Quaternionic Mixed States Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-10-06 Wenxin Li, Pan Lian, Yuxia Liang
In this note, we first derive a Schödinger uncertainty relation for any pair of quaternionic observables and a mixed state. Then, the Wigner–Yanase skew information is introduced in the quaternion setting. Based on the skew information, we establish a new quantum uncertainty inequality for the non-Hermitian quaternionic observables.
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Basis-Free Formulas for Characteristic Polynomial Coefficients in Geometric Algebras Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-09-09 Kamron Abdulkhaev, Dmitry Shirokov
In this paper, we discuss characteristic polynomials in (Clifford) geometric algebras \(\mathcal {G}_{p,q}\) of vector space of dimension \(n=p+q\). We present basis-free formulas for all characteristic polynomial coefficients in the cases \(n\le 6\), alongside with a method to obtain general form of these formulas. The formulas involve only the operations of geometric product, summation, and operations
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Intrinsic General Relativity and Clifford Algebra Adv. Appl. Clifford Algebras (IF 1.5) Pub Date : 2022-09-05 Thierry Socroun, Dominique Girardot
This paper proposes to go beyond the Einstein General Relativity theory in a noncommutative geometric framework. As a first step, we rewrite the General Relativity theory intrinsically (coordinate-free formulation). As a second step, we rewrite the first and the second Bianchi identities, including torsion, within minimal algebraic hypotheses. Then, in order to extend the General Relativity theory