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Advances in Applied Clifford Algebras最新文献

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Quaternion Quantum Neural Network for Classification 用于分类的四元数量子神经网络
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-06-21 DOI: 10.1007/s00006-023-01280-0
Guillermo Altamirano-Escobedo, Eduardo Bayro-Corrochano
{"title":"Quaternion Quantum Neural Network for Classification","authors":"Guillermo Altamirano-Escobedo,&nbsp;Eduardo Bayro-Corrochano","doi":"10.1007/s00006-023-01280-0","DOIUrl":"10.1007/s00006-023-01280-0","url":null,"abstract":"<div><p>We propose the quaternionic quantum neural network (QQNN) for pattern recognition based on the formulation of quaternionic qubits and the construction of activation operators. In this model, the inputs and targets are represented by quaternionic qubits. The proposed neural network is evaluated through a series of experiments using different benchmark datasets, where the results show its superiority as a classifier in terms of accuracy when it is compared to conventional (real-valued) neural networks.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48983285","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: 1
Propagators Beyond The Standard Model 超越标准模型的传播媒介
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-06-17 DOI: 10.1007/s00006-023-01287-7
Rodolfo José Bueno Rogerio, Luca Fabbri
{"title":"Propagators Beyond The Standard Model","authors":"Rodolfo José Bueno Rogerio,&nbsp;Luca Fabbri","doi":"10.1007/s00006-023-01287-7","DOIUrl":"10.1007/s00006-023-01287-7","url":null,"abstract":"<div><p>In this paper, we explore the field propagator with a structure that is general enough to encompas both the case of newly-defined mass-dimension 1 fermions and spin-1/2 bosons. The method we employ is to define a map between spinors of different Lounesto classes, and then write the propagator in terms of the corresponding dual structures.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 4","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-06-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01287-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46541270","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: 2
Clifford Algebras, Quantum Neural Networks and Generalized Quantum Fourier Transform Clifford代数，量子神经网络和广义量子傅里叶变换
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-06-13 DOI: 10.1007/s00006-023-01279-7
Marco A. S. Trindade, Vinícius N. A. Lula-Rocha, S. Floquet
{"title":"Clifford Algebras, Quantum Neural Networks and Generalized Quantum Fourier Transform","authors":"Marco A. S. Trindade,&nbsp;Vinícius N. A. Lula-Rocha,&nbsp;S. Floquet","doi":"10.1007/s00006-023-01279-7","DOIUrl":"10.1007/s00006-023-01279-7","url":null,"abstract":"<div><p>We propose models of quantum perceptrons and quantum neural networks based on Clifford algebras. These models are capable to capture geometric features of classical and quantum data as well as producing data entanglement. Due to their representations in terms of Pauli matrices, the Clifford algebras seem to be a natural framework for multidimensional data analysis in a quantum setting. In this context, the implementation of activation functions, and unitary learning rules are discussed. In this scheme, we also provide an algebraic generalization of the quantum Fourier transform containing additional parameters that allow performing quantum machine learning based on variational algorithms. Furthermore, some interesting properties of the generalized quantum Fourier transform have been proved.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01279-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47016234","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: 3
Beurling’s Theorem in the Clifford Algebras Clifford代数中的Beurling定理
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-06-03 DOI: 10.1007/s00006-023-01284-w
Othman Tyr, Radouan Daher
{"title":"Beurling’s Theorem in the Clifford Algebras","authors":"Othman Tyr,&nbsp;Radouan Daher","doi":"10.1007/s00006-023-01284-w","DOIUrl":"10.1007/s00006-023-01284-w","url":null,"abstract":"<div><p>In this research, the Clifford–Fourier transform introduced by E. Hitzer, satisfies some uncertainty principles similar to the Euclidean Fourier transform. An analog of the Beurling–Hörmander’s theorem for the Clifford–Fourier transform is obtained. As a straightforward consequence of Beurling’s theorem, other versions of the uncertainty principle, such as the Hardy, Gelfand–Shilov and Cowling–Price theorems are also deduced.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-06-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01284-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42980305","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: 2
Functional Calculus for Dual Quaternions 对偶四元数的泛函微积分
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-05-24 DOI: 10.1007/s00006-023-01282-y
Stephen Montgomery-Smith
{"title":"Functional Calculus for Dual Quaternions","authors":"Stephen Montgomery-Smith","doi":"10.1007/s00006-023-01282-y","DOIUrl":"10.1007/s00006-023-01282-y","url":null,"abstract":"<div><p>We give a formula for <span>(f(eta ),)</span> where <span>(f:{mathbb {C}} rightarrow {mathbb {C}})</span> is a continuously differentiable function satisfying <span>(f(bar{z}) = overline{f(z)},)</span> and <span>(eta )</span> is a dual quaternion. Note this formula is straightforward or well known if <span>(eta )</span> is merely a dual number or a quaternion. If one is willing to prove the result only when <i>f</i> is a polynomial, then the methods of this paper are elementary.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-05-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47595304","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: 2
A Geometric Algebra Approach to Invariance Control in Sliding Regimes for Switched Systems 切换系统滑动域不变控制的几何代数方法
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-05-19 DOI: 10.1007/s00006-023-01281-z
H. Sira-Ramírez, B. C. Gómez-León, M. A. Aguilar-Orduña
{"title":"A Geometric Algebra Approach to Invariance Control in Sliding Regimes for Switched Systems","authors":"H. Sira-Ramírez,&nbsp;B. C. Gómez-León,&nbsp;M. A. Aguilar-Orduña","doi":"10.1007/s00006-023-01281-z","DOIUrl":"10.1007/s00006-023-01281-z","url":null,"abstract":"<div><p>Within a Geometric Algebra (GA) framework, this article presents a general method for synthesis of sliding mode (SM) controllers in Single Input Single Output (SISO) switched nonlinear systems. The method, addressed as the <i>invariance control method</i>, rests on a reinterpretation of the necessary and sufficient conditions for the local existence of a sliding regime on a given smooth manifold. This consideration leads to a natural decomposition of the SM control scheme resulting in an invariance state feedback controller feeding a Delta–Sigma modulator that, ultimately, provides the required binary-valued switched input to the plant. As application examples, the obtained results are used to illustrate the design of an invariance controller for a switched power converter system. Using the invariance control design procedure, it is shown how well-known second order sliding regime algorithms can be obtained, via a limiting process, from traditional sliding regimes induced on linear sliding manifolds for certain nonlinear switched systems.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47969828","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
Right-Covariant Differential Calculus on Hopf Superalgebra ({{mathbb {F}}}({mathbb {C}}_q^{2|1})) Hopf超代数上的右协变微分学
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-05-18 DOI: 10.1007/s00006-023-01273-z
Salih Celik
{"title":"Right-Covariant Differential Calculus on Hopf Superalgebra ({{mathbb {F}}}({mathbb {C}}_q^{2|1}))","authors":"Salih Celik","doi":"10.1007/s00006-023-01273-z","DOIUrl":"10.1007/s00006-023-01273-z","url":null,"abstract":"<div><p>We define a new <span>({{mathbb {Z}}}_2)</span>-graded quantum (2+1)-space and show that the extended <span>({{mathbb {Z}}}_2)</span>-graded algebra of polynomials on this <span>({{mathbb {Z}}}_2)</span>-graded quantum space, denoted by <span>({mathbb F}({{mathbb {C}}}_q^{2vert 1 }))</span>, is a <span>({{mathbb {Z}}}_2)</span>-graded Hopf algebra. We construct a right-covariant differential calculus on <span>({{mathbb {F}}}({{mathbb {C}}}_q^{2vert 1 }))</span> and define a <span>({mathbb Z}_2)</span>-graded quantum Weyl algebra and mention a few algebraic properties of this algebra. Finally, we explicitly construct the dual <span>({{mathbb {Z}}}_2)</span>-graded Hopf algebra of <span>({{mathbb {F}}}({mathbb C}_q^{2vert 1 }))</span>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50494444","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
Dynamical Systems of Operators Induced by Scaled Hypercomplex Rings 标度超复环诱导的算子动力学系统
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-05-17 DOI: 10.1007/s00006-023-01272-0
Daniel Alpay, Ilwoo Cho
{"title":"Dynamical Systems of Operators Induced by Scaled Hypercomplex Rings","authors":"Daniel Alpay,&nbsp;Ilwoo Cho","doi":"10.1007/s00006-023-01272-0","DOIUrl":"10.1007/s00006-023-01272-0","url":null,"abstract":"<div><p>In this paper, we consider a family of the hypercomplex rings <span>({mathscr {H}}=left{ {mathbb {H}}_{t}right} _{tin {mathbb {R}}})</span> scaled by <span>({mathbb {R}})</span>, and the dynamical system of <span>({mathbb {R}})</span> acting on <span>({mathscr {H}})</span> via a certain action <span>(theta )</span> of <span>({mathbb {R}})</span>. i.e., we study an analysis on dynamical system induced by <span>({mathscr {H}})</span>. In particular, we are interested in free-probabilistic information on the dynamical system dictated by our hypercomplex analysis.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41733949","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: 2
Left-Right Symmetric Fermions and Sterile Neutrinos from Complex Split Biquaternions and Bioctonions 来自复杂分裂双四元数和生物八元数的左右对称费米子和无菌中微子
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-05-17 DOI: 10.1007/s00006-023-01278-8
Vatsalya Vaibhav, Tejinder P. Singh
{"title":"Left-Right Symmetric Fermions and Sterile Neutrinos from Complex Split Biquaternions and Bioctonions","authors":"Vatsalya Vaibhav,&nbsp;Tejinder P. Singh","doi":"10.1007/s00006-023-01278-8","DOIUrl":"10.1007/s00006-023-01278-8","url":null,"abstract":"<div><p>In this article we investigate the application of complex split biquaternions and bioctonions to the standard model. We show that the Clifford algebras <i>Cl</i>(3) and <i>Cl</i>(7) can be used for making left-right symmetric fermions. Hence we incorporate right-handed neutrinos in the division algebras-based approach to the standard model. The right-handed neutrinos, or sterile neutrinos, are a potential dark-matter candidate. Using the division algebra approach, we discuss the left-right symmetric fermions and their phenomenology. We describe the gauge groups associated with the left-right symmetric model and prospects for unification through division algebras. We briefly discuss the possibility of obtaining three generations of fermions and charge/mass ratios through the exceptional Jordan algebra <span>(J_3(O))</span> and the exceptional groups <span>(F_4)</span> and <span>(E_6)</span>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45967798","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: 14
Graded Symmetry Groups: Plane and Simple 渐变对称群:平面与简单
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-05-13 DOI: 10.1007/s00006-023-01269-9
Martin Roelfs, Steven De Keninck
{"title":"Graded Symmetry Groups: Plane and Simple","authors":"Martin Roelfs,&nbsp;Steven De Keninck","doi":"10.1007/s00006-023-01269-9","DOIUrl":"10.1007/s00006-023-01269-9","url":null,"abstract":"<div><p>The symmetries described by Pin groups are the result of combining a finite number of discrete reflections in (hyper)planes. The current work shows how an analysis using geometric algebra provides a picture complementary to that of the classic matrix Lie algebra approach, while retaining information about the number of reflections in a given transformation. This imposes a type of graded structure on Lie groups, not evident in their matrix representation. Embracing this graded structure, we prove the invariant decomposition theorem: any composition of <i>k</i> linearly independent reflections can be decomposed into <span>(lceil {k/2}{rceil })</span> commuting factors, each of which is the product of at most two reflections. This generalizes a conjecture by M. Riesz, and has e.g. the Mozzi–Chasles’ theorem as its 3D Euclidean special case. To demonstrate its utility, we briefly discuss various examples such as Lorentz transformations, Wigner rotations, and screw transformations. The invariant decomposition also directly leads to closed form formulas for the exponential and logarithmic functions for all Spin groups, and identifies elements of geometry such as planes, lines, points, as the invariants of <i>k</i>-reflections. We conclude by presenting a novel algorithm for the construction of matrix/vector representations for geometric algebras <span>({mathbb {R}}^{{}}_{pqr})</span>, and use this in <span>(text {E}({3}))</span> to illustrate the relationship with the classic covariant, contravariant and adjoint representations for the transformation of points, planes and lines.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 3","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-05-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44738113","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: 9
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