Advances in Applied Clifford Algebras最新文献

To address the challenges of high specialization and fragmented learning resources in Geometric Algebra (GA), this paper introduces a multi-task Geometric Algebraic Large Language Model (GAGPT), which is built upon a GA vector base, a GA knowledge graph, and a GA multi-tasking agent. Additionally, to facilitate interactive GA teaching, the paper proposes the development of two specialized agents: a GA knowledge Q&A agent and a GA interactive exercises agent. The GAGPT is equipped with comprehensive GA contextual background information by constructing a GA vector base from an extensively curated GA corpus. A GA Knowledge Graph is developed from the selected corpus to provide the model with the necessary GA rules. In the GA knowledge Q&A experiment, the accuracy of both formula-based and concept-based quizzes was improved by 46% and 42%, respectively, when compared to GPT-4o. Moreover, in the experiment involving the gradual generation of GA exercises, GAGPT demonstrated superior performance, while GPT-4o, despite utilizing the appropriate GA calculation formulas, made computational errors that led to incorrect results.

This paper introduces and studies generalized degenerate Clifford and Lipschitz groups in geometric (Clifford) algebras. These Lie groups preserve the direct sums of the subspaces determined by the grade involution and reversion under the adjoint and twisted adjoint representations in degenerate geometric algebras. We prove that the generalized degenerate Clifford and Lipschitz groups can be defined using centralizers and twisted centralizers of fixed grades subspaces and the norm functions that are widely used in the theory of spin groups. We study the relations between these groups and consider them in the particular cases of plane-based geometric algebras and Grassmann algebras. The corresponding Lie algebras are studied. The presented groups are interesting for the study of generalized degenerate spin groups and applications in computer science, physics, and engineering.

We study the quaternionic Carleson measure, which provides an embedding of the slice regular Hardy space ({mathcal {H}}^p({mathbb {B}})) into (L^s({mathbb {B}}, text {d}mu )) with (s>p.) A new criterion is needed for a finite positive Borel measure to be an (({mathcal {H}}^p({mathbb {B}}),s))-Carleson measure, given by the uniform integrability of slice Cauchy kernels. It turns out that the symmetric box and the symmetric pseudo-hyperbolic disc are equivalent in the characterization of (({mathcal {H}}^p({mathbb {B}}),s))-Carleson measures, while they are not when (s=p.) We further study the s-Carleson measure for slice regular Bergman spaces ({{mathcal {A}}}^p({mathbb {B}})) for all indices s, p. When (sge p,) our characterization relies on a close relation between the Carleson measure for Hardy and Bergman spaces and is primarily based on the slice Cauchy kernel, rather than the slice Bergman kernel. The advantage of the slice Cauchy kernel over the slice Bergman kernel is that, when restricted to any slice plane, the former, as a sum of two terms, transforms into a fractional linear transform, whereas the latter does not. This enables a locally uniform lower bound estimate for the slice Cauchy kernel, which is crucial in applications. In the case where (s<p,) we need to apply Khinchine’s inequality and a point-wise estimate for atoms in slice Bergman spaces based on the convex combination identity.

We describe Rota-Baxter operators on split octonions. It turns out that up to some transformations there exists exactly one such non-splitting operator over any field. We also obtain a description of all decompositions of split octonions over a quadratically closed field of characteristic different from 2 into a sum of two subalgebras, which describes the splitting Rota-Baxter operators. It completes the classification of Rota-Baxter operators on composition algebras of any weight.

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.