Advances in Applied Clifford Algebras最新文献_第8页

Boundedness of Multiparameter Forelli–Rudin Type Operators on Product (L^p) Spaces over Tubular Domains 管状域上积(L^p)空间上多参数Forelli-Rudin型算子的有界性

IF 1.2 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-06-25 DOI: 10.1007/s00006-025-01395-6

Lvchang Li, Yuheng Liang, Haichou Li

引用次数: 0

Scaled-Hyperbolic Clifford Algebras 标度双曲Clifford代数

IF 1.2 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-06-24 DOI: 10.1007/s00006-025-01393-8

Ilwoo Cho

{"title":"Scaled-Hyperbolic Clifford Algebras","authors":"Ilwoo Cho","doi":"10.1007/s00006-025-01393-8","DOIUrl":"10.1007/s00006-025-01393-8","url":null,"abstract":"<div>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}.)</div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 3","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145145164","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Exploiting Degeneracy in Projective Geometric Algebra 利用投影几何代数中的简并性

IF 1.2 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-06-24 DOI: 10.1007/s00006-025-01392-9

John Bamberg, Jeff Saunders

{"title":"Exploiting Degeneracy in Projective Geometric Algebra","authors":"John Bamberg, Jeff Saunders","doi":"10.1007/s00006-025-01392-9","DOIUrl":"10.1007/s00006-025-01392-9","url":null,"abstract":"<div>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.</div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 3","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-025-01392-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145145163","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Probabilities with Values in Scaled Hyperbolic Numbers 具有缩放双曲数值的概率

IF 1.2 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-06-15 DOI: 10.1007/s00006-025-01394-7

Daniel Alpay, Ilwoo Cho

引用次数: 0

The Symmetry of Hilbert Transformation in (mathbb {R}^3) 中的希尔伯特变换的对称性 (mathbb {R}^3)

IF 1.2 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-06-01 DOI: 10.1007/s00006-025-01387-6

Pei Dang, Hua Liu, Tao Qian

引用次数: 0

GAGPT and Its Application to the Interactive Learning of Geometric Algebra GAGPT及其在几何代数交互式学习中的应用

IF 1.2 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-05-30 DOI: 10.1007/s00006-025-01385-8

Jian Wang, Pei Du, Zhuo Zhao, Wen Luo, Zhaoyuan Yu, Linwang Yuan

{"title":"GAGPT and Its Application to the Interactive Learning of Geometric Algebra","authors":"Jian Wang, Pei Du, Zhuo Zhao, Wen Luo, Zhaoyuan Yu, Linwang Yuan","doi":"10.1007/s00006-025-01385-8","DOIUrl":"10.1007/s00006-025-01385-8","url":null,"abstract":"<div>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.</div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 3","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145145660","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Generalized Degenerate Clifford and Lipschitz Groups in Geometric Algebras 几何代数中的广义简并Clifford群和Lipschitz群

IF 1.2 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-05-28 DOI: 10.1007/s00006-025-01390-x

Ekaterina Filimoshina, Dmitry Shirokov

引用次数: 0

Carleson Measures for Slice Regular Hardy and Bergman Spaces in Quaternions 四元数中片正则Hardy和Bergman空间的Carleson测度

IF 1.1 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-05-26 DOI: 10.1007/s00006-025-01391-w

Wenwan Yang, Cheng Yuan

{"title":"Carleson Measures for Slice Regular Hardy and Bergman Spaces in Quaternions","authors":"Wenwan Yang, Cheng Yuan","doi":"10.1007/s00006-025-01391-w","DOIUrl":"10.1007/s00006-025-01391-w","url":null,"abstract":"<div>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.</div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144140198","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Rota-Baxter Operators of Nonzero Weight on the Split Octonions 分割八元上非零权的Rota-Baxter算子

IF 1.1 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-05-24 DOI: 10.1007/s00006-025-01389-4

A. S. Panasenko

引用次数: 0

Linear Canonical Space-Time Transform and Convolution Theorems 线性正则时空变换与卷积定理

IF 1.1 2区数学

Advances in Applied Clifford Algebras Pub Date : 2025-05-21 DOI: 10.1007/s00006-025-01386-7

Yi-Qiao Xu, Bing-Zhao Li

引用次数: 0