Advances in Applied Clifford Algebras最新文献

Quaternion algebra (mathbb {H}) is a noncommutative associative algebra, and recently quaternionic Fourier analysis has become the focus of an active research due to their potentials in signal analysis and color image processing. The problems related to quaternions are nontrivial and challenging due to noncommutativity of quaternion multiplication. This paper is devoted to establishing the framework of quaternionic generalized norm retrieval (QGNR) in quaternion Euclidean spaces (mathbb {H}^{M}). We introduce the concept of QGNR in (mathbb {H}^{M}) that is defined for general quaternionic self-adjoint matrix sequences. Recall that, even in (mathbb {C}^{M}) ((mathbb {R}^{M}))-setting, the existing literature on norm retrieval problems is only for orthogonal projection matrix sequences instead of general self-adjoint matrix sequences. We characterize QGNR-sequences in terms of their phaselift operators and induced real matrices, present an Edidin type theorem on QGNR for (mathbb {H}^{M}), and investigate the topological property of QGNR-sequences. Finally, we turn to constructing more QGNR-sequences. We prove that a quaternionic self-adjoint matrix sequence (mathcal {F}={F_{n}}_{nin mathbb {N}_{N}}) is such that all ({TF_{n}T^{*}}_{nin mathbb {N}_{N}}) with quaternionic invertible matrices T allow QGNR for (mathbb {H}^{M}) if and only if (mathcal {F}) allows quaternionic generalized phase retrieval, and characterize quaternionic generalized norm retrieval multipliers that transform every QGNR-sequence into another QGNR-sequence.

We consider square matrices over (mathbb {C}) satisfying an identity relating their eigenvalues and the corresponding eigenvectors re-proved and discussed by Denton, Parker, Tao and Zhang, called the eigenvector-eigenvalue identity. We prove that for an eigenvalue (lambda ) of a given matrix, the identity holds if and only if the geometric multiplicity of (lambda ) equals its algebraic multiplicity. We do not make any other assumptions on the matrix and allow the multiplicity of the eigenvalue to be greater than 1, which provides a substantial generalization of the identity. In the proof, we use exterior algebra, particularly the properties of higher adjugates of a matrix.

We consider an extension of Jackson calculus into higher dimensions and specifically into Clifford analysis for the case of commuting variables. In this case, Dirac is the operator of the first q-partial derivatives (or q-differences) ({_{q}}mathbf {mathcal {D}}= sum _{i=1}^n e_i,{_{q}}partial _i), where ({_{q}}partial _i) denotes the q-partial derivative with respect to (x_i). This Dirac operator factorizes the q-deformed Laplace operator. Similar to the case of classical Clifford analysis, we then consider the q-deformed Euler and Gamma operators and their relations to each other. Nullsolutions of this q-Dirac equation are called q-monogenic. Using the Fischer decomposition, we can decompose the space of homogeneous polynomials into spaces of q-monogenic polynomials. Using the q-deformed Cauchy–Kovalevskaya extension theorem, we can construct q-monogenic functions. Overall, we show the analogies and the differences between classical Clifford and Jackson-Clifford analysis. In particular, q-monogenic functions need not be monogenic and vice versa.

This paper presents a discussion on the branching problem that arises in the Weil representation of the exceptional Lie group of type (G_2). The focus is on its decomposition under the threefold cover of (SL(2,, {mathbb {R}})) associated with the short root of (G_2).

We construct the q-deformed Clifford algebra of (mathfrak {sl}_2) and study its properties. This allows us to define the q-deformed noncommutative Weil algebra (mathcal {W}_q(mathfrak {sl}_2)) for (U_q(mathfrak {sl}_2)) and the corresponding cubic Dirac operator (D_q). In the classical case this was done by Alekseev and Meinrenken in 2000. We show that the cubic Dirac operator (D_q) is invariant with respect to the (U_q({mathfrak {sl}}_2))-action and (*)-structures on (mathcal {W}_q(mathfrak {sl}_2)), moreover, the square of (D_q) is central in (mathcal {W}_q(mathfrak {sl}_2)). We compute the spectrum of the cubic element on finite-dimensional and Verma modules of (U_q(mathfrak {sl}_2)) and the corresponding Dirac cohomology.

This paper presents the Geometric Algebra approach to the Wigner little group for photons using the Spacetime Algebra, incorporating a mirror-based view for physical interpretation. The shift from a point-based view to a mirror-based view is a modern movement that allows for a more intuitive representation of geometric and physical entities, with vectors and their higher-grade counterparts viewed as hyperplanes. This reinterpretation simplifies the implementation of homogeneous representations of geometric objects within the Spacetime Algebra and enables a relative view via projective geometry. Then, after utilizing the intrinsic properties of Geometric Algebra, the Wigner little group is seen to induce a projective geometric algebra as a subalgebra of the Spacetime Algebra. However, the dimension-agnostic nature of Geometric Algebra enables the generalization of induced subalgebras to ((1+n))-dimensional Minkowski geometric algebras, termed little photon algebras. The lightlike transformations (translations) in these little photon algebras are seen to leave invariant the (pseudo)canonical electromagetic field bivector. Geometrically, this corresponds to Lorentz transformations that do not change the intersection of the spacelike polarization hyperplane with the lightlike wavevector hyperplane while simultaneously not affecting the lightlike wavevector hyperplane. This provides for a framework that unifies the analysis of symmetries and substructures of point-based Geometric Algebra with mirror-based Geometric Algebra.

In this article, we verify the boundedness of the Cauchy type integral operators under the generalized Hölder norm in Clifford analysis, which are called H-B theorems of the Cauchy integral operators in Clifford analysis. We first demonstrate the generalized 2P theorems and the generalized Muskhelishvili theorem in Clifford analysis by Du’s method derived from Du (J Math (PRC) 2(2):115–12, 1982) and Lu (Boundary value problems of analytic functions. World Scientific, Singapore, 1993), which greatly refines the coefficients estimate of inequality in Du et al. (Acta Math Sci 29B(1):210–224, 2009) and Zhang (Complex Var Elliptic Equ 52(6):455–473, 2007). Then, we obtain the H-B theorems which extend and improve the corresponding results in Du et al. (2009) and Wang and Du (Z Anal Anwend, 2024).

In this article we study some algebraic aspects of multicomplex numbers ({mathbb {M}}_n). For (nge 2) a canonical representation is defined in terms of the multiplication of (n-1) idempotent elements. This representation facilitates computations in this algebra and makes it possible to introduce a generalized conjugacy (Lambda _n), i.e. a composition of the n multicomplex conjugates (Lambda _n:=dagger _1cdots dagger _n), as well as a multicomplex norm. The ideals of the ring of multicomplex numbers are then studied in details, free ({mathbb {M}}_n)-modules and their linear operators are considered and, finally, we develop Hilbert spaces on the multicomplex algebra.

In the present article, the research work of many years is summarized in an interim report. This concerns the connection between logic, space, time and matter. The author always had in mind two things, namely 1. The discovery/construction of an interface between matter and mind, and 2. some entry points for the topos view that concern graphs, grade rotations and contravariant involutions in geometric Boolean lattices. In this part of the MiTopos theme I follow the historic approach to mathematical physics and remain with the Clifford algebra of the Minkowski space. It turns out that this interface is a basic morphogenetic structure inherent in both matter and thought. It resides in both oriented spaces and logic, and most surprisingly is closely linked to the symmetries of elementary particle physics.

The study of complex functions is based around the study of holomorphic functions, satisfying the Cauchy-Riemann equations. The relatively recent field of Clifford Analysis lets us extend many results from Complex Analysis to higher dimensions. In this paper, I decompose the Cauchy-Riemann equations for a general Clifford algebra into grades using the Geometric Algebra formalism, and show that for the Spacetime Algebra Cl(3, 1) these equations are the equations for a self-dual source free Maxwell field, and for a massless uncharged Spinor. This shows a deep link between fundamental physics and the Clifford geometry of Spacetime.