Advances in Applied Clifford Algebras最新文献_第6页

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, Huanhuan Liu, Tianzi Wei, Ruitong Liu, Chunwang Jia, Fan Yang","doi":"10.1007/s00006-024-01332-z","DOIUrl":"10.1007/s00006-024-01332-z","url":null,"abstract":"<div>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.</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

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

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

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

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

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

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

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, Haiyan Wang","doi":"10.1007/s00006-024-01333-y","DOIUrl":"10.1007/s00006-024-01333-y","url":null,"abstract":"<div>The aim of this paper is to study the properties of the Möbius addition (oplus ) under the action of the gyration operator gyr[a, b], and the relation between ((sigma ,t))-translation defined by the Möbius addition and the generalized Laplace–Beltrami operator (Delta _{sigma ,t} ) 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 (aoplus x) and (xoplus a ) is a specific orthogonal matrix factor. Finally, we demonstrate that the ((sigma ,t))-translation is a unitary operator in (L^2 left( {mathbb {B}^8_t,dtau _{sigma ,t} } right) ) and it commutes with the generalized Laplace–Beltrami operator (Delta _{sigma ,t} ) in the octonionic space.</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>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-s 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-s Position Algebras to form the foundation for a generalisation to arbitrary spin of the Clifford and Duffin–Kemmer–Petiau algebras.</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