发布求助
文献互助 智能选刊 最新文献

Advances in Applied Clifford Algebras最新文献

筛选
英文 中文
Parametrizing Clifford Algebras’ Matrix Generators with Euler Angles 用欧拉角范化克利福德代数的矩阵生成器
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-09-02 DOI: 10.1007/s00006-024-01353-8
Manuel Beato Vásquez, Melvin Arias Polanco
{"title":"Parametrizing Clifford Algebras’ Matrix Generators with Euler Angles","authors":"Manuel Beato Vásquez,&nbsp;Melvin Arias Polanco","doi":"10.1007/s00006-024-01353-8","DOIUrl":"10.1007/s00006-024-01353-8","url":null,"abstract":"<div><p>A parametrization, given by the Euler angles, of Hermitian matrix generators of even and odd non-degenerate Clifford algebras is constructed by means of the Kronecker product of a parametrized version of Pauli matrices and by the identification of all possible anticommutation sets. The internal parametrization of the matrix generators allows a straightforward interpretation in terms of rotations, and in the absence of a similarity transformation can be reduced to the canonical representations by an appropriate choice of parameters. The parametric matrix generators of second and fourth-order are linearly decomposed in terms of Pauli, Dirac, and fourth-order Gell–Mann matrices establishing a direct correspondence between the different representations and matrix algebra bases. In addition, and with the expectation for further applications in group theory, a linear decomposition of GL(4) matrices on the basis of the parametric fourth-order matrix generators and in terms of four-vector parameters is explored. By establishing unitary conditions, a parametrization of two subgroups of SU(4) is achieved.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 5","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142123680","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
Higher Order Geometric Algebras and Their Implementations Using Bott Periodicity 高阶几何代数及其利用底周期性的实现
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-08-31 DOI: 10.1007/s00006-024-01346-7
Marek Stodola, Jaroslav Hrdina
{"title":"Higher Order Geometric Algebras and Their Implementations Using Bott Periodicity","authors":"Marek Stodola,&nbsp;Jaroslav Hrdina","doi":"10.1007/s00006-024-01346-7","DOIUrl":"10.1007/s00006-024-01346-7","url":null,"abstract":"<div><p>Using the classification of Clifford algebras and Bott periodicity, we show how higher geometric algebras can be realized as matrices over classical low dimensional geometric algebras. This matrix representation allows us to use standard geometric algebra software packages more easily. As an example, we express the geometric algebra for conics (GAC) as a matrix over the Compass ruler algebra (CRA).</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01346-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142101019","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
Quaternion Convolutional Neural Networks: Current Advances and Future Directions 四元卷积神经网络：当前进展与未来方向
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-08-28 DOI: 10.1007/s00006-024-01350-x
Gerardo Altamirano-Gomez, Carlos Gershenson
{"title":"Quaternion Convolutional Neural Networks: Current Advances and Future Directions","authors":"Gerardo Altamirano-Gomez,&nbsp;Carlos Gershenson","doi":"10.1007/s00006-024-01350-x","DOIUrl":"10.1007/s00006-024-01350-x","url":null,"abstract":"<div><p>Since their first applications, Convolutional Neural Networks (CNNs) have solved problems that have advanced the state-of-the-art in several domains. CNNs represent information using real numbers. Despite encouraging results, theoretical analysis shows that representations such as hyper-complex numbers can achieve richer representational capacities than real numbers, and that Hamilton products can capture intrinsic interchannel relationships. Moreover, in the last few years, experimental research has shown that Quaternion-valued CNNs (QCNNs) can achieve similar performance with fewer parameters than their real-valued counterparts. This paper condenses research in the development of QCNNs from its very beginnings. We propose a conceptual organization of current trends and analyze the main building blocks used in the design of QCNN models. Based on this conceptual organization, we propose future directions of research.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01350-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090002","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
Hypercomplex Representation of Finite-Dimensional Unital Archimedean f-Algebras 有限维单元阿基米德 f 结构的超复数表示
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-08-28 DOI: 10.1007/s00006-024-01352-9
Sayed Kossentini
{"title":"Hypercomplex Representation of Finite-Dimensional Unital Archimedean f-Algebras","authors":"Sayed Kossentini","doi":"10.1007/s00006-024-01352-9","DOIUrl":"10.1007/s00006-024-01352-9","url":null,"abstract":"<div><p>In this paper, we characterize all <i>N</i>-dimensional hypercomplex numbers having unital Archimedean <i>f</i>-algebra structure. We use matrix representation of hypercomplex numbers to define an order structure on the matrix spectra. We prove that the unique (up to isomorphism) unital Archimedean <i>f</i>-algebra of hypercomplex numbers of dimension <span>(N ge 1)</span> is that with real and simple spectrum. We also show that these number systems can be made into unital Banach lattice algebras and we establish some of their properties. Furthermore, we prove that every 2<i>N</i>-dimensional unital Archimedean <i>f</i>-algebra is the <i>hyperbolization</i> of that of dimension <i>N</i>. Finally, we consider hypercomplex number systems of dimension <span>(N=2,3,4,6)</span> and give their explicit matrix representation and eigenvalue operators. This work is a multidimensional generalization of the results obtained in Gargoubi and Kossentini (Adv Appl Clifford Algebras 26(4):1211–1233, 2016) and Bilgin and Ersoy S (Adv Appl Clifford Algebras 30:13, 2020) for, respectively, the two and four-dimensional systems.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01352-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142090003","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
Geometric Structures on the Quaternionic Unit Ball and Slice Regular Möbius Transformations 四元单位球上的几何结构和切片正则莫比乌斯变换
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-08-17 DOI: 10.1007/s00006-024-01343-w
Raul Quiroga-Barranco
{"title":"Geometric Structures on the Quaternionic Unit Ball and Slice Regular Möbius Transformations","authors":"Raul Quiroga-Barranco","doi":"10.1007/s00006-024-01343-w","DOIUrl":"10.1007/s00006-024-01343-w","url":null,"abstract":"<div><p>Building from ideas of hypercomplex analysis on the quaternionic unit ball, we introduce Hermitian, Riemannian and Kähler-like structures on the latter. These are built from the so-called regular Möbius transformations. Such geometric structures are shown to be natural generalizations of those from the complex setup. Our structures can be considered as more natural, from the hypercomplex viewpoint, than the usual quaternionic hyperbolic geometry. Furthermore, our constructions provide solutions to problems not achieved by hyper-Kähler and quaternion-Kähler geometries when applied to the quaternionic unit ball. We prove that the Riemannian metric obtained in this work yields the same tensor previously computed by Arcozzi–Sarfatti. However, our approach is completely geometric as opposed to the function theoretic methods of Arcozzi–Sarfatti.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01343-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141994373","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
Bounds for the Zeros of a Quaternionic Polynomial with Restricted Coefficients 具有受限系数的四元多项式的零点界限
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-08-07 DOI: 10.1007/s00006-024-01344-9
Abdullah Mir, Abrar Ahmad
{"title":"Bounds for the Zeros of a Quaternionic Polynomial with Restricted Coefficients","authors":"Abdullah Mir,&nbsp;Abrar Ahmad","doi":"10.1007/s00006-024-01344-9","DOIUrl":"10.1007/s00006-024-01344-9","url":null,"abstract":"<div><p>In this paper, we are concerned with the problem of locating the zeros of polynomials and regular functions with quaternionic coefficients when their real and imaginary parts are restricted. The extended Schwarz’s lemma, the maximum modulus theorem, and the structure of the zero sets defined in the newly constructed theory of regular functions and polynomials of a quaternionic variable are used to deduce the bounds for the zeros of these polynomials and regular functions. Our findings generalise certain recently established results about the zero distribution for this subclass of regular functions.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01344-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141904567","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
On the Construction of Beltrami Fields and Associated Boundary Value Problems 论贝尔特拉米场的构造及相关的边值问题
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-08-01 DOI: 10.1007/s00006-024-01340-z
Pablo E. Moreira, Briceyda B. Delgado
{"title":"On the Construction of Beltrami Fields and Associated Boundary Value Problems","authors":"Pablo E. Moreira,&nbsp;Briceyda B. Delgado","doi":"10.1007/s00006-024-01340-z","DOIUrl":"10.1007/s00006-024-01340-z","url":null,"abstract":"<div><p>In this paper, we present two simple methods for constructing Beltrami fields. The first one consists of a composition of operators, including a quaternionic transmutation operator as well as the computation of formal powers for the function <span>(f(x)=e^{textbf{i}lambda x})</span>. For the second method, we generate Beltrami fields from harmonic functions, and using the intrinsic relation between the normal and tangential derivative, we solve an associated Neumann-type boundary value problem.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-08-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01340-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141862340","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
Quaternionic Subspace Gabor Frames and Their Duals 四元子空间 Gabor 帧及其对偶
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-07-14 DOI: 10.1007/s00006-024-01342-x
Yun-Zhang Li, Xiao-Li Zhang
{"title":"Quaternionic Subspace Gabor Frames and Their Duals","authors":"Yun-Zhang Li,&nbsp;Xiao-Li Zhang","doi":"10.1007/s00006-024-01342-x","DOIUrl":"10.1007/s00006-024-01342-x","url":null,"abstract":"<div><p>Due to its potential application in signal analysis and image processing, quaternionic Fourier analysis has received increasing attention. This paper addresses quaternionic subspace Gabor frames under the condition that the products of time-frequency shift parameters are rational numbers. We characterize subspace quaternionic Gabor frames in terms of quaternionic Zak transformation matrices. For an arbitrary subspace Gabor frame, we give a parametric expression of its Gabor duals of type I and type II, and characterize the uniqueness Gabor duals of type I and type II. And as an application, given a Gabor frame for the whole space <span>(L^{2}({mathbb {R}}^{2},,{mathbb {H}}))</span>, we give a parametric expression of its all Gabor duals, and derive its unique Gabor dual of type II. Some examples are also provided.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 4","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-07-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01342-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141618336","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
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
首页 上一页 下一页 尾页
0
×
引用
GB/T 7714-2015
复制
MLA
复制
APA
复制
导出至
BibTeX EndNote RefMan NoteFirst NoteExpress
×
提示
您的信息不完整，为了账户安全，请先补充。
现在去补充
×
提示
您因"违规操作"
具体请查看互助需知
我知道了
×
提示
现在去查看 取消
×
提示
确定
请完成安全验证×
相关产品
×
本文献相关产品
联系我们：info@booksci.cn Book学术提供免费学术资源搜索服务，方便国内外学者检索中英文文献。致力于提供最便捷和优质的服务体验。 Copyright © 2023 布克学术 All rights reserved.
京ICP备2023020795号-1
ghs 京公网安备 11010802042870号
Book学术文献互助
Book学术文献互助群
群 号：481959085
Book学术官方微信