Advances in Applied Clifford Algebras最新文献

In this paper, we establish a novel dual matrix representation for dual quaternion matrices, which forms the foundation for a fast and innovative dual structure-preserving algorithm for dual quaternion singular value decomposition (DQSVD). By leveraging the dual quaternion Householder transformation and exploiting the existing properties of dual quaternions, we design a structure-preserving algorithm. This algorithm has a remarkable advantage in that it can convert quaternion operations in the process of bidiagonalizing the dual quaternion matrix into a dual matrix during DQSVD into real operations. As a result, computational efficiency is significantly enhanced. To verify the effectiveness of our proposed algorithm, we present a series of numerical examples. In these examples, we construct the dual complex matrix representation of color images and apply the concept of the structure-preserving algorithm to the dual complex singular value decomposition (DCSVD). This has been successfully employed in the watermark design of color images.

In this paper, we introduce and study two classes of multiparameter Forelli–Rudin type operators from (L^{vec {p}}left( {mathcal {D}}right) ) to (L^{vec {q}}left( {mathcal {D}}right) ), especially on their boundedness, where (L^{vec {p}}left( {mathcal {D}}right) ) and (L^{vec {q}}left( {mathcal {D}}right) ) are both weighted Lebesgue spaces over the Cartesian product of two tubular domains (T_B), with mixed-norm and appropriate weights. We completely characterize the boundedness of these two operators when (1le vec {p}le vec {q}<infty ). Moreover, we provide the necessary and sufficient condition of the case that (vec {q}=(infty ,infty )). As an application, we obtain the boundedness of three common classes of integral operators, including the weighted multiparameter Bergman-type projection and the weighted multiparameter Berezin-type transform.

In this paper, starting from recently known scaled hypercomplexes ({mathbb {H}}_{t}), we define scaled hyperbolics ({mathbb {D}}_{t}) for scales (tin {mathbb {R}}). In particular, the (left( -1right) )-scaled hyperbolics ({mathbb {D}}_{-1}) is isomorphic to the complex field ({mathbb {C}}), the 0-scaled hyperbolics ({mathbb {D}}_{0}) is isomorphic to the dual numbers ({textbf{D}}), and the 1-scaled hyperbolics ({mathbb {D}}_{1}) is isomorphic to the classical hyperbolic numbers ({mathcal {D}}). For any fixed (tin {mathbb {R}}), initiated from the t-scaled hyperbolics ({mathbb {D}}_{t}), we construct the t-scaled-hyperbolic Clifford algebra ({mathscr {C}}_{t}=underrightarrow{textrm{lim}}{mathscr {C}}_{t,n}), where ({mathscr {C}}_{t,n}) are the n-th t-scaled-hyperbolic Clifford algebras for all (nin {mathbb {N}}cup left{ 0right} ), with ({mathscr {C}}_{t,0}={mathbb {R}}) and ({mathscr {C}}_{t,1}={mathbb {D}}_{t}), just like the classical Clifford algebra ({mathscr {C}}={mathscr {C}}_{-1}). To analyze this ({mathbb {R}})-algebra ({mathscr {C}}_{t}), we establish an operator algebra ({mathscr {M}}_{t}) (over ({mathbb {C}}), as usual), containing ({mathscr {C}}_{t}), and then construct a free-probabilistic structure (left( {mathscr {M}}_{t},tau _{t}right) ). From the operator theory, operator algebra and free probability on ({mathscr {M}}_{t}), we apply these analysis for studying ({mathscr {C}}_{t}.)

The last two decades, since the seminal work of Selig [18], has seen projective geometric algebra (PGA) gain popularity as a modern coordinate-free framework for doing classical Euclidean geometry and other Cayley-Klein geometries. This framework is based upon a degenerate Clifford algebra, and it is the purpose of this paper to delve deeper into its internal algebraic structure and extract meaningful information for the purposes of PGA. This includes exploiting the split extension structure to realise the natural decomposition of elements of this Clifford algebra into Euclidean and ideal parts. This leads to a beautiful demonstration of how Playfair’s axiom for affine geometry arises from the ambient degenerate quadratic space. The highlighted split extension property of the Clifford algebra also corresponds to a splitting of the group of units and the Lie algebra of bivectors. Central to these results is that the degenerate Clifford algebra ({{,textrm{Cl},}}(V)) is isomorphic to the twisted trivial extension ({{,textrm{Cl},}}(V/mathbb {F}{e_{0}})ltimes _alpha {{,textrm{Cl},}}(V/mathbb {F}{e_{0}})), where ({e_{0}}) is a degenerate vector and (alpha ) is the grade-involution.

In this paper, we introduce a notion of a probabilistic measure which takes values in t-scaled hyperbolic numbers for (tin mathbb {R}), with a system of axioms generalizing directly Kolmogorov’s axioms. i.e., we establish a suitable measure theory in the set (mathbb {D}_{t}) of all t-scaled hyperbolic numbers for arbitrarily fixed (tin mathbb {R}).

In this paper we study symmetry properties of the Hilbert transformation of the three real variables in the quaternion setting. In order to describe the symmetry properties we introduce the group (rtextrm{Spin}(3)+mathbb {R}^3) which is essentially an extension of the ax+b group. The study concludes that the Hilbert transformation has certain characteristic symmetry properties in terms of (rtextrm{Spin}(3)+mathbb {R}^3.) We first obtain the spinor representation of the group induced by one of (textrm{Spin}(2)) in (mathbb {H}). Then we decompose the natural representation of (rtextrm{Spin}(3)+mathbb {R}^3) into the direct sum of some two irreducible spinor representations, by which we characterize the Hilbert transformation in (mathbb {R}^3). Precisely, we show that a nontrivial operator is essentially the Hilbert transformation if and only if it is invariant under the action of the (rtextrm{Spin}(3)+mathbb {R}^3) group.

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.