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Advances in Applied Clifford Algebras最新文献

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Geometric Algebras of Light Cone Projective Graph Geometries 光锥投影图几何的几何代数
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-01-17 DOI: 10.1007/s00006-023-01307-6
Garret Sobczyk
{"title":"Geometric Algebras of Light Cone Projective Graph Geometries","authors":"Garret Sobczyk","doi":"10.1007/s00006-023-01307-6","DOIUrl":"10.1007/s00006-023-01307-6","url":null,"abstract":"<div><p>A null vector is an algebraic quantity with the property that its square is zero. I denote the universal algebra generated by taking all sums and products of null vectors over the real or complex numbers by <span>({{mathcal {N}}})</span>. The rules of addition and multiplication in <span>({{mathcal {N}}})</span> are taken to be the same as those for real and complex square matrices. A distinct pair of null vectors is <i>positively</i> or <i>negatively</i> correlated if their inner product is <i>positive</i> or <i>negative</i>, respectively. A <i>basis</i> of <span>(n+1)</span> null vectors, with pairwise inner products equal to plus or minus one half, defines the Clifford geometric algebras <span>({mathbb {G}}_{1,n})</span>, or <span>({mathbb {G}}_{n,1})</span>, respectively, and provides a foundation for a new Cayley–Grassman linear algebra, a theory of complete graphs, and other applications in pure and applied areas of science.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139489203","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
Algorithms for Conic Fitting Through Given Proper and Improper Waypoints in Geometric Algebra for Conics 圆锥曲线几何代数中通过给定适当和不适当航点进行圆锥拟合的算法
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-01-09 DOI: 10.1007/s00006-023-01308-5
Pavel Loučka, Petr Vašík
{"title":"Algorithms for Conic Fitting Through Given Proper and Improper Waypoints in Geometric Algebra for Conics","authors":"Pavel Loučka,&nbsp;Petr Vašík","doi":"10.1007/s00006-023-01308-5","DOIUrl":"10.1007/s00006-023-01308-5","url":null,"abstract":"<div><p>As an addition to proper points of the real plane, we introduce a representation of improper points, i.e. points at infinity, in terms of Geometric Algebra for Conics (GAC) and offer possible use of both types of points. More precisely, we present two algorithms fitting a conic to a dataset with a certain number of points lying on the conic precisely, referred to as the waypoints. Furthermore, we consider inclusion of one or two improper waypoints, which leads to the asymptotic directions of the fitted conic. The number of used waypoints may be up to four and we classify all the cases.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139400393","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
An Extension of Slice Regular Functions in Terms of Fiber Bundle Theory 从纤维束理论看片正则函数的扩展
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-01-08 DOI: 10.1007/s00006-023-01309-4
J. Oscar González-Cervantes
{"title":"An Extension of Slice Regular Functions in Terms of Fiber Bundle Theory","authors":"J. Oscar González-Cervantes","doi":"10.1007/s00006-023-01309-4","DOIUrl":"10.1007/s00006-023-01309-4","url":null,"abstract":"<div><p>This work presents an extension, called coordinate slice extension, of the union of a finite number of axially symmetric s domains according to the fiber bundle theory and a kind of slice regular functions are defined on this coordinate slice extension.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2024-01-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139400249","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
Concept of s-Numbers in Quaternionic Analysis and Schatten Classes 四元分析中的 s 数概念和沙腾类
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-12-30 DOI: 10.1007/s00006-023-01311-w
João Costa
{"title":"Concept of s-Numbers in Quaternionic Analysis and Schatten Classes","authors":"João Costa","doi":"10.1007/s00006-023-01311-w","DOIUrl":"10.1007/s00006-023-01311-w","url":null,"abstract":"<div><p>In this paper we introduce an axiomatic approach to the theory of s-numbers in quaternionic analysis. To this end, Pietsch’s approach to s-number theory is adapted to the quaternionic framework, following the works of Colombo and Sabadini on quaternionic spectral analysis. One of the main results of this paper is the uniqueness of s-numbers over quaternionic Hilbert spaces. Moreover, examples are given in the quaternionic framework together with the introduction of nuclear numbers. A consequence of the presented theory is a basis independent definition of the Schatten classes over quaternionic Hilbert and Banach spaces.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01311-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139060443","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
New Versions of the Plemelj–Sochocki Formula in Clifford Analysis 克利福德分析中普莱梅利-索霍基公式的新版本
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-12-26 DOI: 10.1007/s00006-023-01310-x
Yufeng Wang, Zhongxiang Zhang
{"title":"New Versions of the Plemelj–Sochocki Formula in Clifford Analysis","authors":"Yufeng Wang,&nbsp;Zhongxiang Zhang","doi":"10.1007/s00006-023-01310-x","DOIUrl":"10.1007/s00006-023-01310-x","url":null,"abstract":"<div><p>In this paper, we give some new versions of the Plemelj–Sochocki formula under weaker condition in real Clifford Analysis which are different from the result in Luo and Du (Adv Appl Clifford Algebras 27:2531-2583, 2017). By the new versions of the Plemelj–Sochocki formula, we can give a different proof of the generalized Plemelj–Sochocki formula for the symmetric difference of boundary values, which is obtained in Luo and Du (2017), the classical Plemelj–Sochocki formula can be also derived.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-12-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139041333","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
Fractional Powers of the Quaternionic d-Bar Derivative 四元数d-Bar导数的分数次幂
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-11-28 DOI: 10.1007/s00006-023-01306-7
Arran Fernandez, Cihan Güder, Walaa Yasin
{"title":"Fractional Powers of the Quaternionic d-Bar Derivative","authors":"Arran Fernandez,&nbsp;Cihan Güder,&nbsp;Walaa Yasin","doi":"10.1007/s00006-023-01306-7","DOIUrl":"10.1007/s00006-023-01306-7","url":null,"abstract":"<div><p>This work introduces fractional d-bar derivatives in the setting of quaternionic analysis, by giving meaning to fractional powers of the quaternionic d-bar derivative. The definition is motivated by starting from <i>n</i>th-order d-bar derivatives for <span>(nin {mathbb {N}})</span>, and further justified by various natural properties such as composition laws and its action on special functions such as Fueter polynomials.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138449100","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 Algebra Speaks Quantum Esperanto 几何代数讲量子世界语
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-11-11 DOI: 10.1007/s00006-023-01304-9
Sebastian Xambó-Descamps
{"title":"Geometric Algebra Speaks Quantum Esperanto","authors":"Sebastian Xambó-Descamps","doi":"10.1007/s00006-023-01304-9","DOIUrl":"10.1007/s00006-023-01304-9","url":null,"abstract":"<div><p>The goal of this paper is to elucidate the hermitian structure of the algebra of geometric quaternions <span>({textbf {H}})</span> (that is, the even algebra of the geometric algebra of the euclidean 3d space), use it to couch a geometric and dynamical presentation of the <i>q</i>-bit, and to see its bearing on the geometric structure of <i>q</i>-registers (arrangements of finite number of <i>q</i>-bits) and with that to pay a brief revisit to the formal structure of <i>q</i>-computations, with emphasis on the <i>algebra</i> structure of <span>(textbf{H}^{otimes n})</span>. The main underlying theme is the unraveling of the subtle geometric relations between <span>(textbf{H})</span> and the sphere <span>(S^2)</span> in the 3d euclidean space.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 1","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-11-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72365059","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
General Right-Sided Orthogonal 2D-Planes Split Quaternionic Wave-Packet Transform 一般右侧正交二维平面分割四元数波包变换
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-10-23 DOI: 10.1007/s00006-023-01303-w
Hakim Monaim, Said Fahlaoui
{"title":"General Right-Sided Orthogonal 2D-Planes Split Quaternionic Wave-Packet Transform","authors":"Hakim Monaim,&nbsp;Said Fahlaoui","doi":"10.1007/s00006-023-01303-w","DOIUrl":"10.1007/s00006-023-01303-w","url":null,"abstract":"<div><p>In this paper, we present the general right-sided quaternionic orthogonal 2D-planes split wave-packet transform that combines windowed and wavelet transforms. We derive fundamental properties: Plancherel–Parseval theorems, reconstruction formulas, and orthogonality relations, and we provide characterization range, convolutions, and some estimates. Additionally, we derive component-wise, directional and logarithmic uncertainty principles for the given transform and give a discrete formula on the square-integrable function space.\u0000</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 5","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-10-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50509026","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 (mathcal {L_C})-Structure-Preserving Algorithms of Quaternion (LDL^H) Decomposition and Cholesky Decomposition 四元数分解和Cholesky分解的保结构算法
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-10-16 DOI: 10.1007/s00006-023-01298-4
Mingcui Zhang, Ying Li, Jianhua Sun, Wenxv Ding
{"title":"The (mathcal {L_C})-Structure-Preserving Algorithms of Quaternion (LDL^H) Decomposition and Cholesky Decomposition","authors":"Mingcui Zhang,&nbsp;Ying Li,&nbsp;Jianhua Sun,&nbsp;Wenxv Ding","doi":"10.1007/s00006-023-01298-4","DOIUrl":"10.1007/s00006-023-01298-4","url":null,"abstract":"<div><p>In this paper, the <span>(mathcal {L_C})</span>-structure-preserving algorithms of <span>(LDL^H)</span> decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices based on the semi-tensor product of matrices are studied. We first propose <span>(mathcal {L_C})</span>-representation by using the semi-tensor product of matries and the structure matrix of the product of the quaternion. Then, <span>(mathcal {L_C})</span>-structure-preserving algorithms of <span>(LDL^H)</span> decomposition and Cholesky decomposition of quaternion Hermitian positive definite matrices are proposed by using <span>(mathcal {L_C})</span>-representation, and the advantages of our method are obtained by comparing the operation time and error with the real structure-preserving algorithms in Wei et al. (Quaternion matrix computations. Nova Science Publishers, Hauppauge, 2018). Finally, we apply the <span>(mathcal {L_C})</span>-structure-preserving algorithm of Cholesky decomposition to strict authentication of color images.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 5","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50487941","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
Dual Boas Type Results for the Quaternion Transform and Generalized Lipschitz Spaces 四元数变换与广义Lipschitz空间的对偶Boas型结果
IF 1.5 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2023-10-14 DOI: 10.1007/s00006-023-01301-y
Sergey Volosivets
{"title":"Dual Boas Type Results for the Quaternion Transform and Generalized Lipschitz Spaces","authors":"Sergey Volosivets","doi":"10.1007/s00006-023-01301-y","DOIUrl":"10.1007/s00006-023-01301-y","url":null,"abstract":"<div><p>For the quaternion algebra <span>({mathbb {H}})</span> and <span>(f:mathbb R^2rightarrow {mathbb {H}})</span>, we consider a two-sided quaternion Fourier transform <span>({widehat{f}})</span>. Necessary and sufficient conditions for <span>({widehat{f}})</span> to belong to generalized uniform Lipschitz spaces are given in terms of behavior of <i>f</i>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"33 5","pages":""},"PeriodicalIF":1.5,"publicationDate":"2023-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50482888","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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