Advances in Applied Clifford Algebras最新文献

We introduce the Cayley–Dickson Fourier transform (CDFT), a novel framework for harmonic analysis of functions valued in the non-associative Cayley–Dickson algebras ( mathcal {C}_m ). The central challenge lies in the failure of associativity and alternativity for ( m geqslant 4 ), which obstructs classical Fourier analytic methods. To overcome this, we develop a two-stage approach: first, we construct the transform on real-valued Schwartz-type spaces, establishing continuity, inversion, and isometric properties; second, we extend the theory to fully ( mathcal {C}_m )-valued functions by leveraging intrinsic algebraic structures, such as slice-wise multiplicativity and weak commutativity. Key innovations include a modified duality between differentiation and multiplication, governed by twisted sign involutions that precisely compensate for non-associative distortions, and a restricted convolution theorem for Gaussian-type functions that exploits the real scalar structure of their transforms. We prove that the CDFT admits an explicit inverse via symmetrization over coordinate reflections, acts isometrically on ( L^2(mathbb {R}^m, mathcal {C}_m) ), and exhibits a period-four symmetry that generalizes classical Fourier periodicity. These results collectively establish the CDFT as a rigorous and structurally faithful extension of Fourier analysis to the full Cayley–Dickson hierarchy.

We consider a bicomplex analogue of the Landau Hamiltonian defined via its idempotent representation as a couple of the classical Landau Hamiltonians on two separate complex discs. We provide a complete characterization of its (L^2)-eigenspaces when acting on the so-called bicomplex p-Hilbert space, which next employed to explore the common eigenfunction problem associated with the magnetic bc-Laplacian and its (dagger )-conjugate. The corresponding eigenspaces give rise to the polyanalytic version of the bicomplex Bergman spaces for which we provide the explicit expressions for their reproducing kernels.

This article uses Clifford algebra of positive definite signature to derive octonions and the Lie exceptional algebra (textrm{G2}) from calibrations using (mathrm{Pin(7)}). This is simpler than the usual exterior algebra derivation and uncovers a subalgebra of (mathrm{Spin(}7)) that enables (textrm{G2}) and an invertible element used to classify six new power-associative algebras, which are found to be related to the symmetries of (textrm{G2}) in a way that breaks the symmetry of octonions. The 4-form calibration terms of (mathrm{Spin(7)}) are related to an ideal with three idempotents and provides a direct construction of (textrm{G2}) for each of the 480 representations of the octonions. Clifford algebra thus provides a new construction of (textrm{G2}) without using the Lie bracket. A calibration in 15 dimensions is shown to generate the sedenions and to include one of the power-associative algebras, a result previously found by Cawagas.

Let V be a vector space of finite dimension over a field K,  and Q a quadratic form on V. A bilinear form compatible with Q is a bilinear form (varphi ) defined on any subspace S of V such that (varphi (s,s)=Q(s)) for all (sin S). The bilinear forms compatible with Q,  together with an exceptional empty element, constitute an associative and unital monoid (textrm{Cbf}(V,Q)). In the first part of this work, the main purpose is a surjective homomorphism from the Lipschitz monoid (textrm{Lip}(V,Q)) onto this monoid (textrm{Cbf}(V,Q)). In the second part, V is provided with an alternating bilinear form (Omega ,) and some analogous properties are established for the monoid of bilinear forms compatible with (Omega ). When K is the field of real numbers, the controversy about an eventual Lipschitz monoid for (Omega ) is recalled.

The Dirac equation is mapped to a first order differential equation for complex quaternionic functions to benefit from potential theory and quaternionic analysis. This method provides solutions which do not depend on particular matrix representations for Clifford algebras. We additionally deduce zero-mode solutions for the Dirac–Weyl equation, when non-vanishing vector potentials are presupposed.

In this paper, we study an effective algorithm for the Schur decomposition of a quaternion matrix. Firstly, we give a complex structure-preserving algorithm for the Hessenberg decomposition of a quaternion matrix by Householder transformation. Secondly, we implement the implicit double shift QR strategy on the obtained Hessenberg matrix, and design the corresponding complex structure-preserving algorithm. Moreover, the effectiveness of the newly proposed algorithm is verified by numerical experiments. At last, the proposed algorithm is used to deal with a blind color image watermarking problem.

We present the ways of constructing special subsets of conics present in the pencils of conics using Geometric Algebra for Conics (GAC). In particular, we offer geometrically oriented approaches to obtain the line-pairs and generalised parabolas that can be found in the pencils, by applying the tools of GAC and the classical theory of projective conics. In addition, we also describe the construction of a conic passing through five points and, throughout the work, we demonstrate the usage of points at infinity in the mentioned problems as well. The text is accompanied by examples with corresponding figures and includes a partial classification of some of the cases one may encounter in the topic.

Motion polynomials are a specific type of polynomial over a Clifford algebra that can conveniently describe rational motions. There exists an algorithm for the factorization of motion polynomials that works in generic cases. It hinges on the invertibility of a certain coefficient occurring in the algorithm. If this coefficient is not invertible, factorizations may or may not exist. In the case of existence we call this an irregular factorization. We characterize quadratic motion polynomials with irregular factorizations in terms of algebraic equations and present examples whose number of unique factorizations range from one to infinitely many. For two special sub-cases we show the unique existence of such polynomials. In case of commuting factors we obtain the conformal Villarceau motion, in case of rigid body motions the circular translation.

This paper extends the concepts of weighted Drazin-star (WDS) and weighted star-Drazin (WSD) matrices to domain of quaternion matrices. We develop determinantal representations for these matrices, leveraging the theory of noncommutative row-column determinants, considering both general and Hermitian cases. As specific instances, we derive the determinantal representations of the complex WDS and WSD matrices by employing minors of appropriately constructed complex matrices. Furthermore, we investigate two-sided quaternion equations, along with one-sided particular types, where the unique solutions are expressed using WDS and WSD matrices. Explicit solutions for these quaternion matrix equations are obtained using Cramer-type methods. Finally, a numerical example is provided to confirm applicability and efficacy of our findings.

The Algebra of Physical Space (APS) is used to explore the Constructive Standard Model (CSM) of particle physics. Namely, this paper connects the spinor formalism of the APS to massive amplitudes in the CSM. A novel equivalency between traditional CSM and APS-CSM formalisms is introduced, called the Scattering Algebra (SA), with example calculations confirming the consistency of results between both frameworks. Through this all, two significant insights are revealed: The identification of traditional CSM spin spinors with Lorentz rotors in the APS, and the connection of the CSM to various formalisms through ray spinor structure. The CSM’s results are replicated in massive cases, showcasing the power of the index-free, matrix-free, coordinate-free, geometric approach and paving the way for future research into massless cases, amplitude-construction, and Wigner little group methods within the APS.