Advances in Applied Clifford Algebras最新文献

In this paper, the (mathcal {L_C})-structure-preserving algorithms of (LDL^H) decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices based on the semi-tensor product of matrices are studied. We first propose (mathcal {L_C})-representation by using the semi-tensor product of matries and the structure matrix of the product of the quaternion. Then, (mathcal {L_C})-structure-preserving algorithms of (LDL^H) decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices are proposed by using (mathcal {L_C})-representation, and the advantages of our method are obtained by comparing the operation time and error with the real structure-preserving algorithms in Wei et al. (Quaternion matrix computations. Nova Science Publishers, Hauppauge, 2018). Finally, we apply the (mathcal {L_C})-structure-preserving algorithm of Cholesky decomposition to strict authentication of color images.

For the quaternion algebra ({mathbb {H}}) and (f:mathbb R^2rightarrow {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.

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}^2times mathbb {R}) and we obtain new spinorial characterizations of immersions in (mathbb {S}^2times mathbb {R}^2) and in (mathbb {H}^2times mathbb {R}.) We then study the theory of (H=1/2) surfaces in (mathbb {H}^2times mathbb {R}) using this spinorial approach, obtain new proofs of some of its fundamental results and give a direct relation with the theory of (H=1/2) surfaces in (mathbb {R}^{1,2}).

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 of gravity based on Cl(4, 1, R), Cl(3, 2, R) , whose bivector-generators encode the de (Anti) Sitter algebras so(4, 1), so(3, 2), respectively, and 4D conformal gravity based on Cl(3, 1, R) is reminiscent of the (AdS_{ D+1}/CFT_D) correspondence between (D+1)-dim gravity in the bulk and conformal field theory in the D-dim boundary. Although a plausible cancellation mechanism of the cosmological constant terms appearing in the real-valued curvature components associated with complex conformal gravity is possible, it does not occur simultaneously in the imaginary curvature components. Nevertheless, by including a Lagrange multiplier term in the action, it is still plausible that one might be able to find a restricted set of on-shell field configurations leading to a cancellation of the cosmological constant in curvature-squared actions due to the coupling among the real and imaginary components of the vierbein. We finalize with a brief discussion related to (U(4) times U(4)) grand-unification models with gravity based on ( Cl (5, C) = Cl(4,C) oplus Cl(4,C)). It is plausible that these grand-unification models could also be traded for models based on ( GL (4, C) times GL(4, C) ).

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 of 45-dimensional SO(10) and SO(1,9) algebras, which contain standard and additional spin operators, are introduced as well. The SO(10), SO(1,9) and the corresponding (textit{C}ell ^{texttt {R}})(0,8)(, textit{C}ell ^{texttt {R}})(1,7) representations are determined as algebras over the field of real numbers. The suggested gamma matrix representations of the Lie algebras SO(10), SO(1,9) are constructed on the basis of the Clifford algebras (textit{C}ell ^{texttt {R}})(0,8)(, textit{C}ell ^{texttt {R}})(1,7) representations. Comparison with the corresponded algebras in the space of standard 4-component Dirac spinors is demonstrated. The proposed mathematical objects allow generalization of our results, obtained earlier for the standard Dirac equation, for equations of higher spin and, especially, for equations, describing particles with spin 3/2. The maximal 84-dimensional pure matrix algebra of invariance of the 8-component Dirac equation in the Foldy–Wouthuysen representation is found. The corresponding symmetry of the Dirac equation in ordinary representation is found as well. The possible generalizations of considered Lie algebras to the arbitrary dimensional SO(n) and SO(m,n) are discussed briefly.

In this paper, we consider the polynomial Dirac equation ( left( D^m+sum _{i=0}^{m-1}a_iD^iright) u=0, (a_iin {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.

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 and the modified Cauchy–Fueter theories, and proves to have a direct impact on reformulations of quaternionic and spacetime algebra quantum theories.

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 and Kolmogorov’s inequality related to Clifford Cauchy transform. As applications, some respective results in complex plane are directly obtained. Based on the properties of the Clifford Cauchy transform and the principle of uniform boundedness, we solve existence of solutions to integral equations with Cauchy kernel in quaternionic analysis.

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,B)). We show that this function satisfies the equation

and provide a formula for the relationship between the Cayley–Dickson algebra and split Cayley–Dickson algebra, thereby giving an explicit expression for the twist function of the split Cayley–Dickson algebra. Our approach not only resolves the lack of explicit structure for the Cayley–Dickson algebra and split Cayley–Dickson algebra but also sheds light on the algebraic structure underlying this fundamental mathematical object.

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 of this basis, is it a quaternion algebra? The answer is always “yes”. If a subalgebra of dimension 8 in (mathcal {A}_n) is spanned by 8 elements of this basis, is it an octonion algebra? The answer is more often “no” than “yes”. The present article establishes the properties and the formulas that justify these two answers, and describes the fake octonion algebras.