Advances in Applied Clifford Algebras最新文献

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.

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.