Advances in Applied Clifford Algebras最新文献

This paper focuses on finding solutions of generalized Sylvester-type matrix equations over the quaternion skew-field. We express the general least–squares solutions, and perhermitian, skew-perhermitian least-squares solutions of (AXB+CYD=E) and (AXB+CXD=E) over the quaternion skew-field in terms of a vec operator (defined specifically for matrices over the quaternion skew-field) and the Moore–Penrose pseudoinverse. In addition, characterizations that facilitate the computation of the least-squares solutions closest to prescribed quaternion matrices are deduced. We illustrate our theoretical findings on several numerical examples, most of which originate from color image restoration via Tikhonov regularization.

The Atiyah-Singer index formula for Dirac operators acting on the space of spinors put across a kind of topological invariant ((hat{{A}}) genus) of a closed spin manifold ({{mathcal {M}}}), hence offering a bridge between geometric and analytical aspects of the original spin manifold. In this study, we prove the index theorem for a family of fractional Dirac operators in particular for complex analytic coordinates.

The polynomial null solutions of the k-hyperbolic Dirac operator are investigated by the (mathfrak {osp}(1|2)) approach. These solutions are then utilized to construct the (fractional) Fourier transform associated to the k-hyperbolic Dirac operator. The resulting integral kernels are found to be a specific kind of Dunkl kernels. Additionally, we give tight uncertainty inequalities for three distinct fractional Fourier transforms that we have defined. These inequalities are new even for the ordinary fractional Hankel and Weinstein transforms.

In this paper, we study the properties of the S-spectrum of right linear bounded operators on a right quaternionic Banach space. We prove some relations between the S-spectrum and most of its important parts; the approximate S-spectrum, the compression S-spectrum and the surjective S-spectrum. Among other results, we provide some properties of duality and orthogonality on a right quaternionic Banach space.

The “10-fold way” refers to the combined classification of the 3 associative division algebras (of real, complex and quaternionic numbers) and of the 7, ({mathbb Z}_2)-graded, superdivision algebras (in a superdivision algebra each homogeneous element is invertible). The connection of the 10-fold way with the periodic table of topological insulators and superconductors is well known. Motivated by the recent interest in ({mathbb Z}_2times {mathbb Z}_2)-graded physics (classical and quantum invariant models, parastatistics) we classify the associative ({mathbb Z}_2times {mathbb Z}_2)-graded superdivision algebras and show that 13 inequivalent cases have to be added to the 10-fold way. Our scheme is based on the “alphabetic presentation of Clifford algebras”, here extended to graded superdivision algebras. The generators are expressed as equal-length words in a 4-letter alphabet (the letters encode a basis of invertible (2times 2) real matrices and in each word the symbol of tensor product is skipped). The 13 inequivalent ({mathbb Z}_2times {mathbb Z}_2)-graded superdivision algebras are split into real series (4 subcases with 4 generators each), complex series (5 subcases with 8 generators) and quaternionic series (4 subcases with 16 generators). As an application, the connection of ({mathbb Z}_2times {mathbb Z}_2)-graded superdivision algebras with a parafermionic Hamiltonian possessing time-reversal and particle-hole symmetries is presented.

We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems (big ()for short RHBVPs(big )) for axially monogenic functions on Euclidean space ({mathbb {R}}^{n+1},nin {mathbb {N}}). With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to (big ({mathcal {D}}-alpha big )phi =0,alpha in {mathbb {R}}).

The k-Cauchy–Fueter operator and the tangential k-Cauchy–Fueter operator are the quaternionic counterpart of Cauchy–Riemann operator and the tangential Cauchy–Riemann operator in the theory of several complex variables, respectively. In Wang (On the boundary complex of the k-Cauchy–Fueter complex, arXiv:2210.13656), Wang introduced the notion of right-type groups, which have the structure of nilpotent Lie groups of step-two, and many aspects of quaternionic analysis can be generalized to this kind of group. In this paper we generalize the right-type group to any step-two case, and introduce the generalization of Cauchy–Fueter operator on ({mathbb {H}}^ntimes {mathbb {R}}^r.) Then we establish the Bochner–Martinelli type formula for tangential k-Cauchy–Fueter operator on stratified right-type groups.

We study the notions of centrally-extended higher (*)-derivations and centrally-extended generalized higher (*)-derivations. Both are shown to be additive in a (*)-ring without nonzero central ideals. Also, we prove that in semiprime (*)-rings with no nonzero central ideals, every centrally-extended (generalized) higher (*)-derivation is a (generalized) higher (*)-derivation.

Bicomplex convolutional neural networks (BCCNN) are a natural extension of the quaternion convolutional neural networks for the bicomplex case. As it happens with the quaternionic case, BCCNN has the capability of learning and modelling external dependencies that exist between neighbour features of an input vector and internal latent dependencies within the feature. This property arises from the fact that, under certain circumstances, it is possible to deal with the bicomplex number in a component-wise way. In this paper, we present a BCCNN, and we apply it to a classification task involving the colourized version of the well-known dataset MNIST. Besides the novelty of considering bicomplex numbers, our CNN considers an activation function a Bessel-type function. As we see, our results present better results compared with the one where the classical ReLU activation function is considered.