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Advances in Applied Clifford Algebras最新文献

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Hyperbolic Spinor Representations of Non-Null Framed Curves 非零框架曲线的双曲旋量表示
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-21 DOI: 10.1007/s00006-025-01425-3
Zehra İşbilir, Bahar Doğan Yazıcı, Mehmet Güner
{"title":"Hyperbolic Spinor Representations of Non-Null Framed Curves","authors":"Zehra İşbilir,&nbsp;Bahar Doğan Yazıcı,&nbsp;Mehmet Güner","doi":"10.1007/s00006-025-01425-3","DOIUrl":"10.1007/s00006-025-01425-3","url":null,"abstract":"<div><p>In this paper, we intend to bring together the hyperbolic spinors, which are useful frameworks from mathematics to physics, and non-null framed curves in Minkowski 3-space <span>(mathbb {R}_1^3)</span>, which are new type attractive frames and a very crucial issue for singularity theory especially. First, we obtain new adapted frames for framed curves in <span>(mathbb {R}_1^3)</span>. Then, we investigate the hyperbolic spinor representations of non-null framed curves of the general and adapted frames. Also, we find some geometric results and interpretations with respect to them, and we obtain illustrative and numerical examples with figures in order to support the given theorems and results.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145796146","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
Algebraic Properties of the Primitive Idempotent in Clifford Analysis Clifford分析中原始幂等的代数性质
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-20 DOI: 10.1007/s00006-025-01416-4
Hilde De Ridder, Hennie De Schepper, Alí Guzmán Adán, Srđan Lazendić
{"title":"Algebraic Properties of the Primitive Idempotent in Clifford Analysis","authors":"Hilde De Ridder,&nbsp;Hennie De Schepper,&nbsp;Alí Guzmán Adán,&nbsp;Srđan Lazendić","doi":"10.1007/s00006-025-01416-4","DOIUrl":"10.1007/s00006-025-01416-4","url":null,"abstract":"<div><p>This work provides an overview of the algebraic properties of primitive idempotents, which are fundamental in defining spinor spaces within the Clifford algebra framework. In addition to the key concepts, we also present novel results. In particular, we show that the primitive idempotent can be expressed as a polynomial in a specific <i>special bivector</i>. More generally, we demonstrate that every endomorphism on the spinor space can be represented as a polynomial in this special bivector. We also establish that the primitive idempotent, interpreted as a zero projection, represents a special case of this broader polynomial framework. By combining established insights with new contributions, this article offers a fresh perspective on these fundamental structures.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145779327","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
On Commutative Analogues of Clifford Algebras and Their Decompositions Clifford代数的交换类似物及其分解
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-19 DOI: 10.1007/s00006-025-01422-6
Heerak Sharma, Dmitry Shirokov
{"title":"On Commutative Analogues of Clifford Algebras and Their Decompositions","authors":"Heerak Sharma,&nbsp;Dmitry Shirokov","doi":"10.1007/s00006-025-01422-6","DOIUrl":"10.1007/s00006-025-01422-6","url":null,"abstract":"<div><p>We investigate commutative analogues of Clifford algebras—algebras whose generators square to <span>(pm {1})</span> but commute, instead of anti-commuting as they do in Clifford algebras. We observe that commutativity allows for elegant results. We note that these algebras generalise multicomplex spaces—we show that a commutative analogue of Clifford algebra is either isomorphic to a multicomplex space or to ‘multi split-complex space’ (space defined just like multicomplex numbers but uses split-complex numbers instead of complex numbers). We do a general study of commutative analogues of Clifford algebras and use tools like operations of conjugation and idempotents to give a tensor product decomposition and a direct sum decomposition for them. Tensor product decomposition follows relatively easily from the definition. For the direct sum decomposition, we give explicit basis using new techniques.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145779255","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
Quadratic Programming Problem in Power System Engineering Based on Projective Geometric Algebra 基于投影几何代数的电力系统工程二次规划问题
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-19 DOI: 10.1007/s00006-025-01417-3
Johanka Brdečková
{"title":"Quadratic Programming Problem in Power System Engineering Based on Projective Geometric Algebra","authors":"Johanka Brdečková","doi":"10.1007/s00006-025-01417-3","DOIUrl":"10.1007/s00006-025-01417-3","url":null,"abstract":"<div><p>To find an optimal current in a three-phase four-wire power system we have to solve a quadratic programming problem with a positive definite quadratic form with an equality constraint. We offer an approach which solves this and similar problems using an apparatus of geometric algebras, namely Projective geometric algebra. We add dimensions to encode parts of a quadratic function and reformulate the problem to seeking an orthogonal projection of the origin to an intersection of hyperplanes.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-025-01417-3.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145779256","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
Free Probability Theory over the Scaled Hyperbolic Numbers 缩放双曲数的自由概率论
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-12 DOI: 10.1007/s00006-025-01427-1
Daniel Alpay, Ilwoo Cho
{"title":"Free Probability Theory over the Scaled Hyperbolic Numbers","authors":"Daniel Alpay,&nbsp;Ilwoo Cho","doi":"10.1007/s00006-025-01427-1","DOIUrl":"10.1007/s00006-025-01427-1","url":null,"abstract":"<div><p>In this paper, we introduce a notion of free probability over the scaled hyperbolic numbers. Scaled hypercomplex numbers <span>(left{ mathbb {D}_{t}right} _{tin mathbb {R}})</span> are constructed as sub-structures of scaled hypercomplex numbers <span>(left{ mathbb {H}_{t}right} _{tin mathbb {R}})</span> under the scales (or, the moments) of the set <span>(mathbb {R})</span> of real numbers. We show that if <span>(t&lt;0)</span>, then the classical free probability theory covers our free probability on <span>(left{ mathbb {D}_{t}right} _{t&lt;0})</span>; if <span>(t&gt;0)</span>, then our free probability on <span>(left{ mathbb {D}_{t}right} _{t&gt;0})</span> is represented by the free probability over the classical hyperbolic numbers <span>(mathcal {D}=mathbb {D}_{1})</span>; and if <span>(t=0)</span>, then the free probability on <span>(mathbb {D}_{0})</span> is actually over the dual numbers <span>(textbf{D}=mathbb {D}_{0})</span>. Since the usual free probability theory is over <span>(mathbb {C})</span>, we here concentrate on establishing our free probability theory on <span>(mathcal {D})</span>, or that on <span>(textbf{D})</span>. Our approaches are motivated by the Speicher’s combinatorial free probability. As applications, the <span>(mathbb {D}_{t})</span>-free-probabilistic versions of semicircular elements and circular elements are considered.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-025-01427-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145730279","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
Beurling’s Theorem for the Cayley Heisenberg Group 海森堡群的伯林定理
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-10 DOI: 10.1007/s00006-025-01430-6
Said Fahlaoui, Zakariyae Mouhcine
{"title":"Beurling’s Theorem for the Cayley Heisenberg Group","authors":"Said Fahlaoui,&nbsp;Zakariyae Mouhcine","doi":"10.1007/s00006-025-01430-6","DOIUrl":"10.1007/s00006-025-01430-6","url":null,"abstract":"<div><p>We formulate and prove an analogue of Beurling’s theorem for the Fourier transform on the Cayley Heisenberg group. As a consequence we deduce some qualitative uncertainty principles associated with this transform.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145711449","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
Lorentz Invariance of the Multidimensional Dirac–Hestenes Equation 多维Dirac-Hestenes方程的Lorentz不变性
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-05 DOI: 10.1007/s00006-025-01418-2
Sofia Rumyantseva, Dmitry Shirokov
{"title":"Lorentz Invariance of the Multidimensional Dirac–Hestenes Equation","authors":"Sofia Rumyantseva,&nbsp;Dmitry Shirokov","doi":"10.1007/s00006-025-01418-2","DOIUrl":"10.1007/s00006-025-01418-2","url":null,"abstract":"<div><p>This paper investigates the Lorentz invariance of the multidimensional Dirac–Hestenes equation, that is, whether the equation remains form-invariant under pseudo-orthogonal transformations of the coordinates. We examine two distinct approaches: the tensor formulation and the spinor formulation. We first present a detailed examination of the four-dimensional Dirac–Hestenes equation, comparing both transformation approaches. These results are subsequently generalized to the multidimensional case with (1, <i>n</i>) signature. The tensor approach requires explicit invariants, while the spinor formulation naturally maintains Lorentz covariance through spin group action.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145675545","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
On Generalized Right Eigenvalues of Split Quaternion Matrix Pencil 关于分裂四元数矩阵铅笔的广义右特征值
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-02 DOI: 10.1007/s00006-025-01412-8
Shan-Qi Duan, Qing-Wen Wang
{"title":"On Generalized Right Eigenvalues of Split Quaternion Matrix Pencil","authors":"Shan-Qi Duan,&nbsp;Qing-Wen Wang","doi":"10.1007/s00006-025-01412-8","DOIUrl":"10.1007/s00006-025-01412-8","url":null,"abstract":"<div><p>In this paper, by utilizing the complex adjoint matrices, we transform the generalized right eigenvalue problem of the split quaternion matrix pencil into an equivalent generalized complex eigenvalue problem. This transformation enables us to propose an effective algebraic method for solving generalized eigenvalues and their corresponding eigenvectors. Additionally, we investigate the corresponding generalized right least squares eigenvalue problem for the split quaternion matrix pencil, providing a comprehensive framework for these types of problems. Secondly, we define the standard generalized right eigenvalues for the split quaternion matrix pencil. We rigorously prove that a split quaternion matrix pencil of order <i>n</i> has exactly <i>n</i> standard generalized right eigenvalues, all of which are complex numbers. Thirdly, we introduce the Rayleigh quotient for the split quaternion matrix pencil and study its fundamental properties. The definition and analysis of the Rayleigh quotient contribute to the theoretical understanding and potential applications of generalized split quaternion eigenvalue problems.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145657538","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
q-Numerical Range of Quaternionic Right Linear Bounded Operators 四元数右线性有界算子的q-数值范围
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-12-02 DOI: 10.1007/s00006-025-01428-0
Somayya Moulaharabbi, Mohamed Barraa
{"title":"q-Numerical Range of Quaternionic Right Linear Bounded Operators","authors":"Somayya Moulaharabbi,&nbsp;Mohamed Barraa","doi":"10.1007/s00006-025-01428-0","DOIUrl":"10.1007/s00006-025-01428-0","url":null,"abstract":"<div><p>In this paper, we establish and study various properties of the q-numerical range and the q-numerical radius for right linear bounded operators on a right quaternionic Hilbert space.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145657537","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
Dirac Operators on Conformal Manifolds 共形流形上的狄拉克算子
IF 1.2 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2025-11-14 DOI: 10.1007/s00006-025-01421-7
H. Orelma, N. Vieira
{"title":"Dirac Operators on Conformal Manifolds","authors":"H. Orelma,&nbsp;N. Vieira","doi":"10.1007/s00006-025-01421-7","DOIUrl":"10.1007/s00006-025-01421-7","url":null,"abstract":"<div><p>Conformal manifolds <span>(M_lambda )</span> are open subsets of <span>(mathbb {R}^n)</span> endowed with the metric </p><div><div><span>$$begin{aligned} g_lambda =frac{dx_1^2+ldots +dx_n^2}{lambda ^2} end{aligned}$$</span></div></div><p>where <span>(lambda )</span> is called the conformal function. We show that there exists the <span>(alpha )</span>-Dirac operator <span>(D_alpha )</span>, with <span>(alpha in mathbb {R})</span>, acting on functions valued by the Clifford algebra on <span>(M_lambda )</span>. The operator behaves similarly to the usual Euclidean Dirac operator. We develop <span>(alpha )</span>-dependent potential theory for <span>(Delta _alpha )</span> on conformal manifolds, prove refined Poincaré lemmata, and establish Helmholtz-type decompositions for multivector fields.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"36 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2025-11-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"145510859","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
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