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Advances in Applied Clifford Algebras最新文献

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A Note on Centralizers and Twisted Centralizers in Clifford Algebras 关于克利福德代数中的中心子和扭曲中心子的说明
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-09-17 DOI: 10.1007/s00006-024-01345-8
Ekaterina Filimoshina, Dmitry Shirokov
{"title":"A Note on Centralizers and Twisted Centralizers in Clifford Algebras","authors":"Ekaterina Filimoshina,&nbsp;Dmitry Shirokov","doi":"10.1007/s00006-024-01345-8","DOIUrl":"10.1007/s00006-024-01345-8","url":null,"abstract":"<div><p>This paper investigates centralizers and twisted centralizers in degenerate and non-degenerate Clifford (geometric) algebras. We provide an explicit form of the centralizers and twisted centralizers of the subspaces of fixed grades, subspaces determined by the grade involution and the reversion, and their direct sums. The results can be useful for applications of Clifford algebras in computer science, physics, and engineering.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 5","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142236236","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 Discovers Invariants of Braids and Flat Braids 机器学习发现辫子和扁平辫子的不变量
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-09-11 DOI: 10.1007/s00006-024-01349-4
Alexei Lisitsa, Mateo Salles, Alexei Vernitski
{"title":"Machine Learning Discovers Invariants of Braids and Flat Braids","authors":"Alexei Lisitsa,&nbsp;Mateo Salles,&nbsp;Alexei Vernitski","doi":"10.1007/s00006-024-01349-4","DOIUrl":"10.1007/s00006-024-01349-4","url":null,"abstract":"<div><p>We use machine learning to classify examples of braids (or flat braids) as trivial or non-trivial. Our machine learning takes the form of supervised learning, specifically multilayer perceptron neural networks. When they achieve good results in classification, we are able to interpret their structure as mathematical conjectures and then prove these conjectures as theorems. As a result, we find new invariants of braids and prove several theorems related to them. This work evolves from our experiments exploring how different types of AI cope with untangling braids with 3 strands, this is why we concentrate mostly on braids with 3 strands.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 5","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01349-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142166289","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
Recent Advances for Meson Algebras and their Lipschitz Monoids 介子代数及其 Lipschitz Monoids 的最新进展
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-09-10 DOI: 10.1007/s00006-024-01354-7
Jacques Helmstetter
{"title":"Recent Advances for Meson Algebras and their Lipschitz Monoids","authors":"Jacques Helmstetter","doi":"10.1007/s00006-024-01354-7","DOIUrl":"10.1007/s00006-024-01354-7","url":null,"abstract":"<div><p>This article has two purposes. After a short reminder of classical properties of meson algebras (also called Duffin-Kemmer algebras), Sects. 4 to 7 present recent advances in the study of their algebraic structure. Then Sects. 8 to 11 explain that each meson algebra contains a Lipschitz monoid with properties quite similar to those of Lipschitz monoids in Clifford algebras.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 5","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01354-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142160799","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 Octonionic Submodules Generated by One Element 关于由一个元素生成的八离子子模块
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-09-09 DOI: 10.1007/s00006-024-01355-6
Qinghai Huo, Guangbin Ren
{"title":"On Octonionic Submodules Generated by One Element","authors":"Qinghai Huo,&nbsp;Guangbin Ren","doi":"10.1007/s00006-024-01355-6","DOIUrl":"10.1007/s00006-024-01355-6","url":null,"abstract":"<div><p>The aim of this article is to characterize the octonionic submodules generated by one element, which is very complicated compared with other normed division algebras. To this end, we introduce a novel identity that elucidates the relationship between the commutator and associator within an octonionic bimodule. Remarkably, the commutator can be expressed in terms of the linear combination of associators. This phenomenon starkly contrasts with the quaternionic case, which leads to a unique right octonionic scalar multiplication compatible with the original left octonionic module structure in the sense of forming an octonionic bimodule. With the help of this identity, we get a new expression of the real part and imaginary part of an element in an octonionic bimodule. Ultimately, we obtain that the submodule generated by one element <i>x</i> is <span>({mathbb {O}}^5x)</span> instead of <span>({mathbb {O}}x)</span>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 5","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142158719","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 Bessel–Clifford Function Associated to the Cayley–Laplace Operator 与卡莱-拉普拉斯算子相关的贝塞尔-克利福德函数
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-09-09 DOI: 10.1007/s00006-024-01351-w
David Eelbode
{"title":"The Bessel–Clifford Function Associated to the Cayley–Laplace Operator","authors":"David Eelbode","doi":"10.1007/s00006-024-01351-w","DOIUrl":"10.1007/s00006-024-01351-w","url":null,"abstract":"<div><p>In this paper the Cayley–Laplace operator <span>(Delta _{xu})</span> is considered, a rotationally invariant differential operator which can be seen as a generalisation of the classical Laplace operator for functions depending on wedge variables <span>(X_{ab})</span> (the minors of a matrix variable). We will show that the Bessel–Clifford function appears naturally in the framework of two-wedge variables, and explain how this function somehow plays the role of the exponential function in the framework of Grassmannians. This will be used to obtain a generalisation of the series expansion for the Newtonian potential, and to investigate a new kind of binomial polynomials related to Nayarana numbers.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 5","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142160467","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
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
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