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Advances in Applied Clifford Algebras最新文献

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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
The Möbius Addition and Generalized Laplace–Beltrami Operator in Octonionic Space 八音空间中的莫比乌斯加法和广义拉普拉斯-贝尔特拉米算子
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-08 DOI: 10.1007/s00006-024-01333-y
Wei Xia, Haiyan Wang
{"title":"The Möbius Addition and Generalized Laplace–Beltrami Operator in Octonionic Space","authors":"Wei Xia,&nbsp;Haiyan Wang","doi":"10.1007/s00006-024-01333-y","DOIUrl":"10.1007/s00006-024-01333-y","url":null,"abstract":"<div><p>The aim of this paper is to study the properties of the Möbius addition <span>(oplus )</span> under the action of the gyration operator <i>gyr</i>[<i>a</i>, <i>b</i>], and the relation between <span>((sigma ,t))</span>-translation defined by the Möbius addition and the generalized Laplace–Beltrami operator <span>(Delta _{sigma ,t} )</span> in the octonionic space. Despite the challenges posed by the non-associativity and non-commutativity of octonions, Möbius addition still exhibits many significant properties in the octonionic space, such as the left cancellation law and the gyrocommutative law. We introduce a novel approach to computing the Jacobian determinant of Möbius addition. Then, we discover that the gyration operator is closely related to the Jacobian matrix of Möbius addition. Importantly, we determine that the distinction between <span>(aoplus x)</span> and <span>(xoplus a )</span> is a specific orthogonal matrix factor. Finally, we demonstrate that the <span>((sigma ,t))</span>-translation is a unitary operator in <span>(L^2 left( {mathbb {B}^8_t,dtau _{sigma ,t} } right) )</span> and it commutes with the generalized Laplace–Beltrami operator <span>(Delta _{sigma ,t} )</span> in the octonionic space.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141295005","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 Relationship Between Spin and Geometry 自旋与几何之间的关系
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-06-03 DOI: 10.1007/s00006-024-01322-1
Peter T. J. Bradshaw
{"title":"A Relationship Between Spin and Geometry","authors":"Peter T. J. Bradshaw","doi":"10.1007/s00006-024-01322-1","DOIUrl":"10.1007/s00006-024-01322-1","url":null,"abstract":"<div><p>In physics, spin is often seen exclusively through the lens of its phenomenological character: as an intrinsic form of angular momentum. However, there is mounting evidence that spin fundamentally originates as a quality of geometry, not of dynamics, and recent work further suggests that the structure of non-relativistic Euclidean three-space is sufficient to define it. In this paper, we directly explicate this fundamentally non-relativistic, geometric nature of spin by constructing non-commutative algebras of position operators which subsume the structure of an arbitrary spin system. These “Spin-<i>s</i> Position Algebras” are defined by elementary means and from the properties of Euclidean three-space alone, and constitute a fundamentally new model for quantum mechanical systems with non-zero spin, within which neither position and spin degrees of freedom, nor position degrees of freedom within themselves, commute. This reveals that the observables of a system with spin can be described completely geometrically as tensors of oriented planar elements, and that the presence of non-zero spin in a system naturally generates a non-commutative geometry within it. We will also discuss the potential for the Spin-<i>s</i> Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01322-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141235940","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
Fourier-Poisson Transforms Associated with the Principal Series Representations of Sp(1, n) 与 Sp(1, n) 主数列表示相关的傅立叶-泊松变换
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-05-28 DOI: 10.1007/s00006-024-01330-1
Xingya Fan, Jianxun He, Xiaoke Jia
{"title":"Fourier-Poisson Transforms Associated with the Principal Series Representations of Sp(1, n)","authors":"Xingya Fan,&nbsp;Jianxun He,&nbsp;Xiaoke Jia","doi":"10.1007/s00006-024-01330-1","DOIUrl":"10.1007/s00006-024-01330-1","url":null,"abstract":"<div><p>Let <span>(X=Sp(1,n)/Sp(n))</span> be the quaternion hyperbolic space with a left invariant Haar measure, unique up to scalars, where <i>n</i> is greater than or equal to 1. The Fürstenberg boundary of <i>X</i> is denoted as <span>(Sigma )</span>. In this paper, we focus on the Plancherel formula on <i>X</i> associated with the Poisson transform of vector-valued <span>(L^2)</span>-space on <span>(Sigma )</span>. Through the Fourier-Jacobi transform and the Fourier-Poisson transform, we derive the Plancherel decomposition of the unitary representation of <i>Sp</i>(1, <i>n</i>) on <span>(L^2(X))</span>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141165253","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
Mobility Analysis of Multi-loop Coupling Mechanisms Using Geometric Algebra 利用几何代数分析多环耦合机制的流动性
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-05-27 DOI: 10.1007/s00006-024-01329-8
Jinqun Guo, Yu Xiao, Qinchuan Li, Lingmin Xu, Xinxue Chai
{"title":"Mobility Analysis of Multi-loop Coupling Mechanisms Using Geometric Algebra","authors":"Jinqun Guo,&nbsp;Yu Xiao,&nbsp;Qinchuan Li,&nbsp;Lingmin Xu,&nbsp;Xinxue Chai","doi":"10.1007/s00006-024-01329-8","DOIUrl":"10.1007/s00006-024-01329-8","url":null,"abstract":"<div><p>Multi-loop coupling mechanisms (MCMs) have been widely used in spacedeployable antennas. However, the mobility of MCMs is difficult to analyze due to their complicated structure and coupled limbs. This paper proposes a general method for calculating the mobility of MCMs using geometric algebra (GA). For the independent limbs in the MCM, the twist spaces are constructed by the join operator. For coupled limbs coupled with closed loops in the MCM, the equivalent limbs can be found by solving the analytical expressions of the twist space on each closed loop’s output link. Then, the twist spaces of the coupled limbs can be easily obtained. The twist space of the MCM’s output link is the intersection of all the limb twist spaces, which can be calculated by the meet operator. The proposed method provides a simplified way of analyzing the mobility of MCMs, and three typical MCMs are chosen to validate this method. The analytical mobility of the MCM’s output link can be obtained, and it naturally indicates both the number and the property of the degrees of freedom (DOFs).</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141159673","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 SVD and Polar Decomposition in Real and Complexified Clifford Algebras 论实数和复数克利福德代数中的 SVD 和极性分解
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-05-27 DOI: 10.1007/s00006-024-01328-9
Dmitry Shirokov
{"title":"On SVD and Polar Decomposition in Real and Complexified Clifford Algebras","authors":"Dmitry Shirokov","doi":"10.1007/s00006-024-01328-9","DOIUrl":"10.1007/s00006-024-01328-9","url":null,"abstract":"<div><p>In this paper, we present a natural implementation of singular value decomposition (SVD) and polar decomposition of an arbitrary multivector in nondegenerate real and complexified Clifford geometric algebras of arbitrary dimension and signature. The new theorems involve only operations in geometric algebras and do not involve matrix operations. We naturally define these and other related structures such as Hermitian conjugation, Euclidean space, and Lie groups in geometric algebras. The results can be used in various applications of geometric algebras in computer science, engineering, and physics.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141156686","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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