Advances in Applied Clifford Algebras最新文献

Following the idea of the fractional space-time Fourier transform, a linear canonical space-time transform for 16-dimensional space-time (Cell _{3,1})-valued signals is investigated in this paper. First, the definition of the proposed linear canonical space-time transform is given, and some related properties of this transform are obtained. Second, the convolution operator and the corresponding convolution theorem are proposed. Third, the convolution theorem associated with the two-sided linear canonical space-time transform is derived.

We discuss a generalization of Clifford algebras known as generalized Clifford algebras (in particular, ternary Clifford algebras). In these objects, we have a fixed higher-degree form (in particular, a ternary form) instead of a quadratic form in ordinary Clifford algebras. We present a natural realization of unitary Lie groups, which are important in physics and other applications, using only operations in generalized Clifford algebras and without using the corresponding matrix representations. Basis-free definitions of the determinant, trace, and characteristic polynomial in generalized Clifford algebras are introduced. Explicit formulas for all coefficients of the characteristic polynomial and inverse in generalized Clifford algebras are presented. The operation of Hermitian conjugation (or Hermitian transpose) in generalized Clifford algebras is introduced without using the corresponding matrix representations.

It is easier to investigate phenomena in particle physics geometrically by exploring a real solution to the Dirac–Hestenes equation instead of a complex solution to the Dirac equation. The current research presents a formulation of the multidimensional Dirac–Hestenes equation. Since the matrix representation of the complexified (Clifford) geometric algebra (mathbb {C}otimes C hspace{-1.00006pt}ell _{1,n}) depends on the parity of n, we examine even and odd cases separately. In the geometric algebra (C hspace{-1.00006pt}ell _{1,3}), there is a lemma on a unique decomposition of an element of the minimal left ideal into the product of the idempotent and an element of the real even subalgebra. The lemma is used to construct the four-dimensional Dirac–Hestenes equation. The analogous lemma is not valid in the multidimensional case, since the dimension of the real even subalgebra of (C hspace{-1.00006pt}ell _{1,n}) is bigger than the dimension of the minimal left ideal for (n>4). Hence, we consider the auxiliary real subalgebra of (C hspace{-1.00006pt}ell _{1,n}) to prove a similar statement. We present the multidimensional Dirac–Hestenes equation in (C hspace{-1.00006pt}ell _{1,n}). We prove that one might obtain a solution to the multidimensional Dirac–Hestenes equation using a solution to the multidimensional Dirac equation and vice versa. We also show that the multidimensional Dirac–Hestenes equation has gauge invariance.

This work presents a coordinate sphere bundle defined from theory of slice regular functions whose bundle projection and some real Banach spaces induce coordinate sphere bundles in which the quaternionic Banach modules of the slice regular functions of Bloch, Besov and Dirichlet are the base spaces. Finally, this work shows that Möbius invariant property of these quaternionic Banach modules defines pullback bundles or automorphisms on sphere bundles.

Holomorphic Cliffordian functions of order k are functions in the kernel of the differential operator (overline{partial }Delta ^k). When (overline{partial }Delta ^k) is applied to functions defined in the paravector space of some Clifford Algebra (mathbb {R}_m) with an odd number of imaginary units, the Fueter–Sce construction establishes a critical index (k=frac{m-1}{2}) (sometimes called Sce exponent) for which the class of slice regular functions is contained in the one of holomorphic Cliffordian functions of order (frac{m-1}{2}). In this paper, we analyze the case (k<frac{m-1}{2}) and find that the polynomials of degree at most 2k are the only slice regular holomorphic Cliffordian functions of order k.

This paper completely describes the form of all unital additive surjective maps, on the algebra of all bounded right linear operators acting on a two-sided quaternionic Banach space, that preserve any one of (left, right) invertibility, (left, right) zero divisors and (left, right) topological divisors of zero in both directions.

A monogenic function of two vector variables is a function annihilated by two Dirac operators. We give the explicit form of differential operators in the Dirac complex resolving two Dirac operators and prove its ellipticity directly. This opens the door to apply the method of several complex variables to investigate this kind of monogenic functions. We prove the Poincaré lemma for this complex, i.e. the non-homogeneous equations are solvable under the compatibility condition, by solving the associated Hodge Laplacian equations of fourth order. As corollaries, we establish the Bochner–Martinelli integral representation formula for two Dirac operators and the Hartogs’ extension phenomenon for monogenic functions. We also apply abstract duality theorem to the Dirac complex to obtain the generalization of Malgrange’s vanishing theorem and establish the Hartogs–Bochner extension phenomenon for monogenic functions under the moment condition.

The paper is concerned with the definition, properties and uncertainty principles for the multi-dimensional quaternionic offset linear canonical transform. First, we define the multi-dimensional offset linear canonical transform based on matrices with symplectic structure. Then, we focus on the definition of the multi-dimensional quaternionic offset linear canonical transform and the corresponding convolution theorem. Finally, some uncertainty principles are established for the proposed multi-dimensional (quaternionic) offset linear canonical transform.

In this article, a Geometric Algebra (GA) and Geometric Calculus (GC) based exposition is carried out dealing with the formal characterization of sliding regimes for general Single-Input-Single-Output (SISO) nonlinear switched controlled Hamiltonian systems. Necessary and sufficient conditions for the local existence of a sliding regime on a given vector manifold are presented. Feedback controller design strategies for achieving local sliding regimes on a given smooth vector manifold—defined in the phase space of the system—are also derived using the GA-GC framework. One such controller design method, which is mathematically justified, is based on the invariance property of the leaves of the foliation induced by the sliding surface coordinate function level sets. The idealized average smooth sliding motions are shown to arise from an extrinsic projection operator whose geometric properties are exploited for characterizing robustness with respect to unknown exogenous perturbation vector fields. An application example is provided from the power electronics area.

The paper deals with two second order elliptic systems of partial differential equations in Clifford analysis. They are of the form ({^phi !underline{partial }}f{^psi !underline{partial }}=0) and (f{^phi !underline{partial }}{^psi !underline{partial }}=0), where ({^phi !underline{partial }}) stands for the Dirac operator related to a structural set (phi ). Their solutions, known as ((phi ,psi ))-inframonogenic and ((phi ,psi ))-harmonic functions, not every enjoy the nice properties and usual structure of the harmonic ones. We describe the precise relation between these two classes of functions and show their strong link to the Laplace operator. Finally, we apply a multi-dimensional Ahlfors-Beurling transform, to prove that some relative function spaces are indeed isomorphic.