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

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

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

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

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

引用次数: 0

Distribution Function and Nonincreasing Rearrangement of ({mathbb {B}}{mathbb {C}})-Valued Functions with ({mathbb {B}} {mathbb {C}})-Measure 有$${mathbb {B}}{mathbb {C}}$ 值函数的分布函数和非递增重排{{mathbb {C}}$ -度量

IF 1.1 2区数学

Advances in Applied Clifford Algebras Pub Date : 2024-05-18 DOI: 10.1007/s00006-024-01327-w

İlker Eryılmaz

引用次数: 0

Hausdorff–Young Inequalities for Fourier Transforms over Cayley–Dickson Algebras Cayley-Dickson 代数上傅立叶变换的 Hausdorff-Young 不等式

IF 1.1 2区数学

Advances in Applied Clifford Algebras Pub Date : 2024-05-10 DOI: 10.1007/s00006-024-01326-x

Shihao Fan, Guangbin Ren

{"title":"Hausdorff–Young Inequalities for Fourier Transforms over Cayley–Dickson Algebras","authors":"Shihao Fan, Guangbin Ren","doi":"10.1007/s00006-024-01326-x","DOIUrl":"10.1007/s00006-024-01326-x","url":null,"abstract":"<div>In this study, we extend Beckner’s seminal work on the Fourier transform to the domain of Cayley–Dickson algebras, establishing a precise form of the Hausdorff–Young inequality for functions that take values in these algebras. Our extension faces significant hurdles due to the unique characteristics of the Cayley–Dickson Fourier transform. This transformation diverges from the classical Fourier transform in several key aspects: it does not conform to the Plancherel theorem, alters the interplay between derivatives and multiplication, and the product of algebra elements does not necessarily maintain the magnitude relationships found in classical settings. To overcome these challenges, our approach involves constructing the Cayley–Dickson Fourier transform by sequentially applying classical Fourier transforms. A pivotal part of our strategy is the utilization of a theorem that facilitates the norm-preserving extension of linear operators between spaces (L^p) and (L^q.) Furthermore, our investigation brings new insights into the complexities surrounding the Beckner–Hirschman Entropic inequality in the context of non-associative algebras.</div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140903199","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

Machine Learning Clifford Invariants of ADE Coxeter Elements ADE Coxeter 元素的机器学习克利福德不变式

IF 1.1 2区数学

Advances in Applied Clifford Algebras Pub Date : 2024-05-04 DOI: 10.1007/s00006-024-01325-y

Siqi Chen, Pierre-Philippe Dechant, Yang-Hui He, Elli Heyes, Edward Hirst, Dmitrii Riabchenko

{"title":"Machine Learning Clifford Invariants of ADE Coxeter Elements","authors":"Siqi Chen, Pierre-Philippe Dechant, Yang-Hui He, Elli Heyes, Edward Hirst, Dmitrii Riabchenko","doi":"10.1007/s00006-024-01325-y","DOIUrl":"10.1007/s00006-024-01325-y","url":null,"abstract":"<div>There has been recent interest in novel Clifford geometric invariants of linear transformations. This motivates the investigation of such invariants for a certain type of geometric transformation of interest in the context of root systems, reflection groups, Lie groups and Lie algebras: the Coxeter transformations. We perform exhaustive calculations of all Coxeter transformations for (A_8), (D_8) and (E_8) for a choice of basis of simple roots and compute their invariants, using high-performance computing. This computational algebra paradigm generates a dataset that can then be mined using techniques from data science such as supervised and unsupervised machine learning. In this paper we focus on neural network classification and principal component analysis. Since the output—the invariants—is fully determined by the choice of simple roots and the permutation order of the corresponding reflections in the Coxeter element, we expect huge degeneracy in the mapping. This provides the perfect setup for machine learning, and indeed we see that the datasets can be machine learned to very high accuracy. This paper is a pump-priming study in experimental mathematics using Clifford algebras, showing that such Clifford algebraic datasets are amenable to machine learning, and shedding light on relationships between these novel and other well-known geometric invariants and also giving rise to analytic results.</div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-05-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01325-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140845242","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

Exploring Quaternion Neural Network Loss Surfaces 探索四元数神经网络损失曲面

IF 1.1 2区数学

Advances in Applied Clifford Algebras Pub Date : 2024-04-24 DOI: 10.1007/s00006-024-01313-2

Jeremiah Bill, Bruce Cox

引用次数: 0

Fractional Elliptic Operators with Multiple Poles on Riemannian Manifold with Clifford Bundle 带克里福德束的黎曼曼体上具有多个极点的分数椭圆算子

IF 1.1 2区数学

Advances in Applied Clifford Algebras Pub Date : 2024-04-23 DOI: 10.1007/s00006-024-01318-x

Rami Ahmad El-Nabulsi, Waranont Anukool

引用次数: 0