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Advances in Applied Clifford Algebras最新文献

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On the Geometry of Quantum Spheres and Hyperboloids 论量子球和超球的几何学
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-07-13 DOI: 10.1007/s00006-024-01339-6
Giovanni Landi, Chiara Pagani
{"title":"On the Geometry of Quantum Spheres and Hyperboloids","authors":"Giovanni Landi,&nbsp;Chiara Pagani","doi":"10.1007/s00006-024-01339-6","DOIUrl":"10.1007/s00006-024-01339-6","url":null,"abstract":"<div><p>We study two classes of quantum spheres and hyperboloids, one class consisting of homogeneous spaces, which are <span>(*)</span>-quantum spaces for the quantum orthogonal group <span>(mathcal {O}(SO_q(3)))</span>. We construct line bundles over the quantum homogeneous space associated with the quantum subgroup <i>SO</i>(2) of <span>(SO_q(3))</span>. The line bundles are associated to the quantum principal bundle via representations of <i>SO</i>(2) and are described dually by finitely-generated projective modules <span>(mathcal {E}_n)</span> of rank 1 and of degree computed to be an even integer <span>(-2n)</span>. The corresponding idempotents, that represent classes in the K-theory of the base space, are explicitly worked out and are paired with two suitable Fredhom modules that compute the rank and the degree of the bundles. For <i>q</i> real, we show how to diagonalise the action (on the base space algebra) of the Casimir operator of the Hopf algebra <span>({mathcal {U}_{q^{1/2}}(sl_2)})</span> which is dual to <span>(mathcal {O}(SO_q(3)))</span>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01339-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141602718","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
Models of CR Manifolds and Their Symmetry Algebras CR 曼olds 的模型及其对称性代数
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-07-05 DOI: 10.1007/s00006-024-01341-y
Jan Gregorovič, Martin Kolář, Francine Meylan, David Sykes
{"title":"Models of CR Manifolds and Their Symmetry Algebras","authors":"Jan Gregorovič,&nbsp;Martin Kolář,&nbsp;Francine Meylan,&nbsp;David Sykes","doi":"10.1007/s00006-024-01341-y","DOIUrl":"10.1007/s00006-024-01341-y","url":null,"abstract":"<div><p>In this paper we give an exposition of several recent results on local symmetries of real submanifolds in complex space, featuring new examples and important corollaries. Departing from Levi non-degenerate hypersurfaces, treated in the classical Chern–Moser theory, we explore three important classes of manifolds, which naturally extend the classical case. We start with quadratic models for real submanifolds of higher codimension and review some recent striking results, which demonstrate that such higher codimension models may possess symmetries of arbitrarily high jet degree. This disproves the long held belief that the fundamental 2-jet determination results from Chern–Moser theory extend to this case. As a second case, we consider hypersurfaces with singular Levi form at a point, which are of finite multitype. This leads to the study of holomorphically nondegenerate polynomial models. We outline several results on their symmetry algebras including a characterization of models admitting nonlinear symmetries. In the third part we consider the class of structures with everywhere singular Levi forms that has received the most attention recently, namely everywhere 2-nondegenerate structures. We present a computation of their Catlin multitype and results on symmetry algebras of their weighted homogeneous (w.r.t. multitype) models.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141545937","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
A Multi-dimensional Unified Concavity and Convexity Detection Method Based on Geometric Algebra 基于几何代数的多维统一凹凸检测方法
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-07-02 DOI: 10.1007/s00006-024-01332-z
Jiyi Zhang, Huanhuan Liu, Tianzi Wei, Ruitong Liu, Chunwang Jia, Fan Yang
{"title":"A Multi-dimensional Unified Concavity and Convexity Detection Method Based on Geometric Algebra","authors":"Jiyi Zhang,&nbsp;Huanhuan Liu,&nbsp;Tianzi Wei,&nbsp;Ruitong Liu,&nbsp;Chunwang Jia,&nbsp;Fan Yang","doi":"10.1007/s00006-024-01332-z","DOIUrl":"10.1007/s00006-024-01332-z","url":null,"abstract":"<div><p>Detecting the concavity and convexity of three-dimensional (3D) geometric objects is a well-established challenge in the realm of computer graphics. Serving as the cornerstone for various related graphics algorithms and operations, researchers have put forth numerous algorithms for discerning the concavity and convexity of such objects. The majority of existing methods primarily rely on Euclidean geometry, determining concavity and convexity by calculating the vertices of these objects. However, within the realm of Euclidean geometric space, there exists a lack of uniformity in the expression and calculation rules for geometric objects of differing dimensions. Consequently, distinct concavity and convexity detection algorithms must be tailored for geometric objects with varying dimensions. This approach inevitably results in heightened complexity and instability within the algorithmic structure. To address these aforementioned issues, this paper introduces geometric algebra theory into the domain of concavity and convexity detection within 3D spatial objects. With the algorithms devised in this study, it becomes feasible to detect concavity and convexity for geometric objects of varying dimensions, all based on a uniform set of criteria. In comparison to concavity-convexity detection algorithms grounded in Euclidean geometry, this research effectively streamlines the algorithmic structure.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489601","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
The Clifford Algebra of the Density Matrix: An Elementary Approach 密度矩阵的克利福德代数：初级方法
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-29 DOI: 10.1007/s00006-024-01337-8
Pedro Amao, Hernan Castillo
{"title":"The Clifford Algebra of the Density Matrix: An Elementary Approach","authors":"Pedro Amao,&nbsp;Hernan Castillo","doi":"10.1007/s00006-024-01337-8","DOIUrl":"10.1007/s00006-024-01337-8","url":null,"abstract":"<div><p>This work studies the Clifford algebra approach to the density matrix. We discuss elementary examples of pure and mixed states by writing the density matrix as an element of the Clifford algebra of the three-dimensional space <span>(Cl_3)</span>. We also revisit the phenomenon of Larmor precession within the framework of Clifford algebra. Additionally, we discuss the geometrical interpretation of the so-called Clifford Density Element (CDE) for pure states in analogy to the Bloch sphere of conventional quantum theory. Finally, we discuss the dynamics of the CDE, which obeys an algebraic form of the Liouville von–Neumann equation.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141489606","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
Convex Characteristics of Quaternionic Positive Definite Functions on Abelian Groups 阿贝尔群上四元正定函数的凸特性
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-25 DOI: 10.1007/s00006-024-01336-9
Jingning Liu, Zeping Zhu
{"title":"Convex Characteristics of Quaternionic Positive Definite Functions on Abelian Groups","authors":"Jingning Liu,&nbsp;Zeping Zhu","doi":"10.1007/s00006-024-01336-9","DOIUrl":"10.1007/s00006-024-01336-9","url":null,"abstract":"<div><p>This paper is concerned with the topological space of normalized quaternion-valued positive definite functions on an arbitrary abelian group <i>G</i>, especially its convex characteristics. There are two main results. Firstly, we prove that the extreme elements in the family of such functions are exactly the homomorphisms from <i>G</i> to the sphere group <span>({mathbb {S}})</span>, i.e., the unit 3-sphere in the quaternion algebra. Secondly, we reveal a new phenomenon: The compact convex set of such functions is not a Bauer simplex except when <i>G</i> is of exponent <span>(le 2)</span>. In contrast, its complex counterpart is always a Bauer simplex, as is well known. We also present an integral representation for such functions as an application and some other minor results.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141448094","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
More About Bicomplex Möbius Transformations: Geometric, Algebraic and Analitical Aspects 关于双复莫比乌斯变换的更多信息：几何、代数与分析方面
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-24 DOI: 10.1007/s00006-024-01323-0
M. Elena Luna–Elizarrarás, Anatoly Golberg
{"title":"More About Bicomplex Möbius Transformations: Geometric, Algebraic and Analitical Aspects","authors":"M. Elena Luna–Elizarrarás,&nbsp;Anatoly Golberg","doi":"10.1007/s00006-024-01323-0","DOIUrl":"10.1007/s00006-024-01323-0","url":null,"abstract":"<div><p>The aim of this paper is to analyze and prove different facts related with bicomplex Möbius transformations. Various algebraic and geometric results were obtained, using the decomposition of the bicomplex set as: <span>({{mathbb {B}}}{{mathbb {C}}}= {{mathbb {D}}}+ textbf{i}{{mathbb {D}}})</span>, and there were used actively both, hyperbolic and bicomplex, geometric objects. The basics of bicomplex Lobachevsky’s geometry are given.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141444862","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
Integral Formulas for Slice Cauchy–Riemann Operator and Applications 片状考奇-黎曼算子的积分公式及其应用
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-24 DOI: 10.1007/s00006-024-01338-7
Chao Ding, Xiaoqian Cheng
{"title":"Integral Formulas for Slice Cauchy–Riemann Operator and Applications","authors":"Chao Ding,&nbsp;Xiaoqian Cheng","doi":"10.1007/s00006-024-01338-7","DOIUrl":"10.1007/s00006-024-01338-7","url":null,"abstract":"<div><p>The theory of slice regular functions has been developed rapidly in the past few years, and most properties are given in slices at the early stage. In 2013, Colombo et al. introduced a non-constant coefficients differential operator to describe slice regular functions globally, and this brought the study of slice regular functions in a global sense. In this article, we introduce a slice Cauchy–Riemann operator, which is motivated by the non-constant coefficients differential operator mentioned above. Then, A Borel–Pompeiu formula for this slice Cauchy–Riemann operator is discovered, which leads to a Cauchy integral formula for slice regular functions. A Plemelj integral formula for the slice Cauchy–Riemann operator is introduced, which gives rise to results on slice regular extension at the end.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141444809","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 Symmetries of Geometric Algebra Cl(3, 1) for Space-Time 论时空几何代数 Cl(3, 1) 的对称性
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-20 DOI: 10.1007/s00006-024-01331-0
Eckhard Hitzer
{"title":"On Symmetries of Geometric Algebra Cl(3, 1) for Space-Time","authors":"Eckhard Hitzer","doi":"10.1007/s00006-024-01331-0","DOIUrl":"10.1007/s00006-024-01331-0","url":null,"abstract":"<div><p>From viewpoints of crystallography and of elementary particles, we explore symmetries of multivectors in the geometric algebra <i>Cl</i>(3, 1) that can be used to describe space-time.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141430412","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
Harmonic Analysis on Exceptional Domain (E_{6(-14)}/U(1)Spin(10)) 例外域上的谐波分析 $$E_{6(-14)}/U(1)Spin(10)$$
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-13 DOI: 10.1007/s00006-024-01335-w
Fouzia El Wassouli, Daoud Oukacha
{"title":"Harmonic Analysis on Exceptional Domain (E_{6(-14)}/U(1)Spin(10))","authors":"Fouzia El Wassouli,&nbsp;Daoud Oukacha","doi":"10.1007/s00006-024-01335-w","DOIUrl":"10.1007/s00006-024-01335-w","url":null,"abstract":"<div><p>Let </p><div><div><span>$$begin{aligned} mathcal {D}_{16}=left{ Zin mathcal {M}_{1,2}(mathfrak {C}^{c}):;begin{array}{lll} 1-leftlangle Z,Z rightrangle +leftlangle Z^{sharp },Z^{sharp }rightrangle&gt;0, 2-leftlangle Z,Z rightrangle ; &gt;0end{array}right} end{aligned}$$</span></div></div><p>be an exceptional domain of non-tube type and let <span>(mathcal {U}_{nu })</span> and <span>(mathcal {W}_{nu })</span> the associated generalized Hua operators. In this paper, we determine the explicit formula of the action of the group <span>( E_{6(-14)})</span> on <span>(mathcal {D}_{16})</span>. We characterized the <span>(L^{p})</span>-range, <span>(1le p &lt; infty )</span> of the generalized Poisson transform on the Shilov boundary of the domain <span>(mathcal {D}_{16})</span>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141315762","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
Short Time Quaternion Quadratic Phase Fourier Transform and Its Uncertainty Principles 短时四元数二次相傅里叶变换及其不确定性原理
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-11 DOI: 10.1007/s00006-024-01334-x
Bivek Gupta, Amit K. Verma
{"title":"Short Time Quaternion Quadratic Phase Fourier Transform and Its Uncertainty Principles","authors":"Bivek Gupta,&nbsp;Amit K. Verma","doi":"10.1007/s00006-024-01334-x","DOIUrl":"10.1007/s00006-024-01334-x","url":null,"abstract":"<div><p>In this paper, we extend the quadratic phase Fourier transform of a complex valued functions to that of the quaternion-valued functions of two variables. We call it the quaternion quadratic phase Fourier transform (QQPFT). Based on the relation between the QQPFT and the quaternion Fourier transform (QFT) we obtain the sharp Hausdorff–Young inequality for QQPFT, which in particular sharpens the constant in the inequality for the quaternion offset linear canonical transform (QOLCT). We define the short time quaternion quadratic phase Fourier transform (STQQPFT) and explore some of its properties including inner product relation and inversion formula. We find its relation with that of the 2<i>D</i> quaternion ambiguity function and the quaternion Wigner–Ville distribution associated with QQPFT and obtain the Lieb’s uncertainty and entropy uncertainty principles for these three transforms.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141309093","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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