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Advances in Applied Clifford Algebras最新文献

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MiTopos MiTopos
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-12-19 DOI: 10.1007/s00006-024-01362-7
Bernd Schmeikal
{"title":"MiTopos","authors":"Bernd Schmeikal","doi":"10.1007/s00006-024-01362-7","DOIUrl":"10.1007/s00006-024-01362-7","url":null,"abstract":"<div><p>In the present article, the research work of many years is summarized in an interim report. This concerns the connection between logic, space, time and matter. The author always had in mind two things, namely 1. The discovery/construction of an interface between matter and mind, and 2. some entry points for the topos view that concern graphs, grade rotations and contravariant involutions in geometric Boolean lattices. In this part of the MiTopos theme I follow the historic approach to mathematical physics and remain with the Clifford algebra of the Minkowski space. It turns out that this interface is a basic morphogenetic structure inherent in both matter and thought. It resides in both oriented spaces and logic, and most surprisingly is closely linked to the symmetries of elementary particle physics.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142845127","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
Self-Dual Maxwell Fields from Clifford Analysis 自对偶麦克斯韦场从克利福德分析
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-12-11 DOI: 10.1007/s00006-024-01368-1
C. J. Robson
{"title":"Self-Dual Maxwell Fields from Clifford Analysis","authors":"C. J. Robson","doi":"10.1007/s00006-024-01368-1","DOIUrl":"10.1007/s00006-024-01368-1","url":null,"abstract":"<div><p>The study of complex functions is based around the study of holomorphic functions, satisfying the Cauchy-Riemann equations. The relatively recent field of Clifford Analysis lets us extend many results from Complex Analysis to higher dimensions. In this paper, I decompose the Cauchy-Riemann equations for a general Clifford algebra into grades using the Geometric Algebra formalism, and show that for the Spacetime Algebra <i>Cl</i>(3, 1) these equations are the equations for a self-dual source free Maxwell field, and for a massless uncharged Spinor. This shows a deep link between fundamental physics and the Clifford geometry of Spacetime.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01368-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142809682","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
STP Method for Solving the Least Squares Special Solutions of Quaternion Matrix Equations 求解四元数矩阵方程最小二乘特解的STP方法
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-12-02 DOI: 10.1007/s00006-024-01367-2
Weihua Chen, Caiqin Song
{"title":"STP Method for Solving the Least Squares Special Solutions of Quaternion Matrix Equations","authors":"Weihua Chen,&nbsp;Caiqin Song","doi":"10.1007/s00006-024-01367-2","DOIUrl":"10.1007/s00006-024-01367-2","url":null,"abstract":"<div><p>In this paper, we apply the semi-tensor product of matrices and the real vector representation of a quaternion matrix to find the least squares lower (upper) triangular Toeplitz solution of <span>(AX-XB=C)</span>, <span>(AXB-CX^{T}D=E)</span> and (anti)centrosymmetric solution of <span>(AXB-CYD=E)</span>. And the expressions of the least squares lower (upper) triangular Toeplitz and (anti)centrosymmetric solution for the studied equations are derived. Additionally, the necessary and sufficient conditions for the existence of solutions and general expression of the studied equations are given. Eventually, some numerical examples are provided for showing the validity and superiority of our method.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142757908","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
Construction of an Infinite-Dimensional Family of Exact Solutions of a Three-Dimensional Biharmonic Equation by the Hypercomplex Method 用超复杂法构建三维双谐方程的无限维精确解族
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-11-25 DOI: 10.1007/s00006-024-01365-4
Vitalii Shpakivskyi
{"title":"Construction of an Infinite-Dimensional Family of Exact Solutions of a Three-Dimensional Biharmonic Equation by the Hypercomplex Method","authors":"Vitalii Shpakivskyi","doi":"10.1007/s00006-024-01365-4","DOIUrl":"10.1007/s00006-024-01365-4","url":null,"abstract":"<div><p>An infinite-dimensional family of exact solutions of a three-dimensional biharmonic equation was constructed by the hypercomplex method.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142694783","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
Eigenvalues of Quaternion Tensors: Properties, Algorithms and Applications 四元张量的特征值：特性、算法和应用
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-11-22 DOI: 10.1007/s00006-024-01366-3
Zhuo-Heng He, Ting-Ting Liu, Xiang-Xiang Wang
{"title":"Eigenvalues of Quaternion Tensors: Properties, Algorithms and Applications","authors":"Zhuo-Heng He,&nbsp;Ting-Ting Liu,&nbsp;Xiang-Xiang Wang","doi":"10.1007/s00006-024-01366-3","DOIUrl":"10.1007/s00006-024-01366-3","url":null,"abstract":"<div><p>In this paper, we investigate the eigenvalues of quaternion tensors under Einstein Product and their applications in color video processing. We present the Ger<span>(check{s})</span>gorin theorem for quaternion tensors. Furthermore, we have executed some experiments to validate the efficacy of our proposed theoretical framework and algorithms. Finally, we contemplate the application of this methodology in color video compression, in which the reconstruction of an approximate original image is achieved by computing a limited number of the largest eigenvalues, yielding a favorable outcome. In summary, by utilizing block tensors in its iterations, this method converges more rapidly to the desired eigenvalues and eigentensors, which significantly reduces the time required for videos compression.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-11-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142690705","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
Geometric Product of Two Oriented Points in Conformal Geometric Algebra 共形几何代数中两个定向点的几何积
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-11-15 DOI: 10.1007/s00006-024-01363-6
Eckhard Hitzer
{"title":"Geometric Product of Two Oriented Points in Conformal Geometric Algebra","authors":"Eckhard Hitzer","doi":"10.1007/s00006-024-01363-6","DOIUrl":"10.1007/s00006-024-01363-6","url":null,"abstract":"<div><p>We compute and explore the full geometric product of two oriented points in conformal geometric algebra <i>Cl</i>(4, 1) of three-dimensional Euclidean space. We comment on the symmetry of the various components, and state for all expressions also a representation in terms of point pair center and radius vectors.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-11-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142636951","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
Riemann–Hilbert Problems for Biaxially Symmetric Monogenic Functions in (mathbb {R}^{n}) Riemann-Hilbert Problems for Biaxially Symmetric Monogenic Functions in (mathbb {R}^{n})
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-11-13 DOI: 10.1007/s00006-024-01364-5
Dian Zuo, Min Ku, Fuli He
{"title":"Riemann–Hilbert Problems for Biaxially Symmetric Monogenic Functions in (mathbb {R}^{n})","authors":"Dian Zuo,&nbsp;Min Ku,&nbsp;Fuli He","doi":"10.1007/s00006-024-01364-5","DOIUrl":"10.1007/s00006-024-01364-5","url":null,"abstract":"<div><p>We are dedicated to addressing Riemann–Hilbert boundary value problems (RHBVPs) with variable coefficients, where the solutions are valued in the Clifford algebra of <span>(mathbb {R}_{0,n})</span>, for biaxially monogenic functions defined in the biaxially symmetric domains of the Euclidean space <span>(mathbb {R}^{n})</span>. Our research establishes the equivalence between RHBVPs for biaxially monogenic functions defined in biaxially domains and RHBVPs for generalized analytic functions on the complex plane. We derive explicit solutions and conditions for solvability of RHBVPs for biaxially monogenic functions. Additionally, we explore related Schwarz problems and RHBVPs for biaxially meta-monogenic functions.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-11-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142600706","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
Conics, Their Pencils and Intersections in Geometric Algebra 几何代数中的圆锥曲线、其铅笔和交点
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-11-05 DOI: 10.1007/s00006-024-01356-5
Clément Chomicki, Stéphane Breuils, Venceslas Biri, Vincent Nozick
{"title":"Conics, Their Pencils and Intersections in Geometric Algebra","authors":"Clément Chomicki,&nbsp;Stéphane Breuils,&nbsp;Venceslas Biri,&nbsp;Vincent Nozick","doi":"10.1007/s00006-024-01356-5","DOIUrl":"10.1007/s00006-024-01356-5","url":null,"abstract":"<div><p>This paper presents an approach for extracting points from conic intersections by using the concept of pencils. This method is based on QC2GA—the two-dimensional version of QCGA (Quadric Conformal Geometric Algebra)—that is demonstrated to be equivalent to GAC (Geometric Algebra for Conics). A new interpretation of QC2GA and its objects based on pencils of conics and point space elements is presented, enabling the creation, constraining, and exploitation of pencils of conics. A Geometric Algebra method for computing the discriminants and center point of a conic will also be presented, enabling the proposition of an algorithm for extracting points from a conic intersection object.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-11-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142579494","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
Fueter’s Theorem for One Class of Pseudoanalytic Functions 一类伪解析函数的 Fueter 定理
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-10-26 DOI: 10.1007/s00006-024-01361-8
Yuanyuan Han, Pan Lian
{"title":"Fueter’s Theorem for One Class of Pseudoanalytic Functions","authors":"Yuanyuan Han,&nbsp;Pan Lian","doi":"10.1007/s00006-024-01361-8","DOIUrl":"10.1007/s00006-024-01361-8","url":null,"abstract":"<div><p>In this paper, we extend Fueter’s theorem in hypercomplex function theory to encompass a class of pseudoanalytic functions associated with the main Vekua equation. This class includes Duffin’s <span>(mu )</span>-regular functions as special cases, which correspond to the Yukawa equation. As the parameter <span>(mu rightarrow 0)</span>, we recover the classical Fueter’s theorem.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 5","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-10-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142490663","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
Multivector Contractions Revisited, Part I 重温多向量收缩，第一部分
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-10-24 DOI: 10.1007/s00006-024-01357-4
André L. G. Mandolesi
{"title":"Multivector Contractions Revisited, Part I","authors":"André L. G. Mandolesi","doi":"10.1007/s00006-024-01357-4","DOIUrl":"10.1007/s00006-024-01357-4","url":null,"abstract":"<div><p>We reorganize, simplify and expand the theory of contractions or interior products of multivectors, and related topics like Hodge star duality. Many results are generalized and new ones are given, like: geometric characterizations of blade contractions and regressive products, higher-order graded Leibniz rules, determinant formulas, improved complex star operators, etc. Different contractions found in the literature are discussed and compared, in special those of Clifford Geometric Algebra. Applications of the theory are developed in a follow-up paper.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 5","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142488420","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
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