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Advances in Applied Clifford Algebras最新文献

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Multidimensional Generalized Fractional ({pmb {S}}) Transform 多维广义分式 $${pmb {S}}$ 变换
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-04-17 DOI: 10.1007/s00006-024-01317-y
Lakshmanan Subbiah, Roopkumar Rajakumar
{"title":"Multidimensional Generalized Fractional ({pmb {S}}) Transform","authors":"Lakshmanan Subbiah,&nbsp;Roopkumar Rajakumar","doi":"10.1007/s00006-024-01317-y","DOIUrl":"10.1007/s00006-024-01317-y","url":null,"abstract":"<div><p>In this paper, we introduce a new multidimensional fractional <i>S</i> transform <span>(S_{phi ,varvec{alpha },lambda })</span> using a generalized fractional convolution <span>(star _{varvec{alpha },lambda })</span> and a general window function <span>(phi )</span> satisfying some admissibility condition. The value of <span>(S_{phi ,varvec{alpha },lambda }f)</span> is also written in the form of inner product of the input function <i>f</i> with a suitable function <span>(phi _{textbf{t},textbf{u}}^{varvec{alpha }_{lambda }})</span>. The representation of <span>(S_{phi ,varvec{alpha },lambda }f)</span> in terms of the generalized fractional convolution helps us to obtain the Parseval’s formula for <span>(S_{phi ,varvec{alpha },lambda })</span> using the generalized fractional convolution theorem. Then, the inversion theorem is proved as a consequence of the Parseval’s identity. Using a generalized window function in the kernel of <span>(S_{phi ,varvec{alpha },lambda })</span> gives option to choose window function whose Fourier transform as a compactly supported smooth function or a rapidly decreasing function. We also discuss about the characterization of range of <span>(S_{phi ,varvec{alpha },lambda })</span> on <span>(L^2(mathbb {R}^N, mathbb {C}))</span>. Finally, we extend the transform to a class of quaternion valued functions consistently.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140603899","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
A Note on Cohomology of Clifford Algebras 关于克利福德代数同调的说明
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-04-09 DOI: 10.1007/s00006-024-01324-z
Bikram Banerjee, Goutam Mukherjee
{"title":"A Note on Cohomology of Clifford Algebras","authors":"Bikram Banerjee,&nbsp;Goutam Mukherjee","doi":"10.1007/s00006-024-01324-z","DOIUrl":"10.1007/s00006-024-01324-z","url":null,"abstract":"<div><p>In this article we construct a cochain complex of a complex Clifford algebra with coefficients in itself in a combinatorial fashion and we call the corresponding cohomology by <i>Clifford cohomology.</i> We show that <i>Clifford cohomology</i> controls the deformation of a complex Clifford algebra and can classify them up to Morita equivalence. We also study Hochschild cohomology groups and formal deformations of the algebra of smooth sections of a complex Clifford algebra bundle over an even dimensional orientable Riemannian manifold <i>M</i> which admits a <span>(Spin^{c})</span> structure.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140538358","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
Inequalities Pertaining to Quaternion Ambiguity Function 与四元数模糊函数有关的不等式
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-04-08 DOI: 10.1007/s00006-024-01320-3
Imanuel Agung Sembe, Mawardi Bahri, Nasrullah Bachtiar, Muhammad Zakir
{"title":"Inequalities Pertaining to Quaternion Ambiguity Function","authors":"Imanuel Agung Sembe,&nbsp;Mawardi Bahri,&nbsp;Nasrullah Bachtiar,&nbsp;Muhammad Zakir","doi":"10.1007/s00006-024-01320-3","DOIUrl":"10.1007/s00006-024-01320-3","url":null,"abstract":"<div><p>The quaternion ambiguity function is an expansion of the standard ambiguity function using quaternion algebra. Various properties such as linearity, translation, modulation, Moyal’s formula and inversion identity are studied in detail. In addition, an interesting interaction between the quaternion ambiguity function and the quaternion Fourier transform is demonstrated. Based on these facts, we seek for several versions of the uncertainty inequalities associated with the proposed quaternion ambiguity function.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140538521","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
Lipschitz Norm Estimate for a Higher Order Singular Integral Operator 高阶奇异积分算子的 Lipschitz Norm 估计数
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-04-08 DOI: 10.1007/s00006-024-01321-2
Tania Rosa Gómez Santiesteban, Ricardo Abreu Blaya, Juan Carlos Hernández Gómez, José Luis Sánchez Santiesteban
{"title":"Lipschitz Norm Estimate for a Higher Order Singular Integral Operator","authors":"Tania Rosa Gómez Santiesteban,&nbsp;Ricardo Abreu Blaya,&nbsp;Juan Carlos Hernández Gómez,&nbsp;José Luis Sánchez Santiesteban","doi":"10.1007/s00006-024-01321-2","DOIUrl":"10.1007/s00006-024-01321-2","url":null,"abstract":"<div><p>Let <span>(Gamma )</span> be a <i>d</i>-summable surface in <span>(mathbb {R}^m)</span>, i.e., the boundary of a Jordan domain in <span>( mathbb {R}^m)</span>, such that <span>(int nolimits _{0}^{1}N_{Gamma }(tau )tau ^{d-1}textrm{d}tau &lt;+infty )</span>, where <span>(N_{Gamma }(tau ))</span> is the number of balls of radius <span>(tau )</span> needed to cover <span>(Gamma )</span> and <span>(m-1&lt;d&lt;m)</span>. In this paper, we consider a singular integral operator <span>(S_Gamma ^*)</span> associated with the iterated equation <span>({mathcal {D}}_{underline{x}}^k f=0)</span>, where <span>({mathcal {D}}_{underline{x}})</span> stands for the Dirac operator constructed with the orthonormal basis of <span>( mathbb {R}^m)</span>. The fundamental result obtained establishes that if <span>(alpha &gt;frac{d}{m})</span>, the operator <span>(S_Gamma ^*)</span> transforms functions of the higher order Lipschitz class <span>(text{ Lip }(Gamma , k +alpha ))</span> into functions of the class <span>(text{ Lip }(Gamma , k +beta ))</span>, for <span>(beta =frac{malpha -d}{m-d})</span>. In addition, an estimate for its norm is obtained.\u0000</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140538363","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
Some Uncertainty Principles for the Right-Sided Multivariate Continuous Quaternion Wavelet Transform 右侧多变量连续四元数小波变换的一些不确定性原理
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-04-08 DOI: 10.1007/s00006-024-01319-w
Manel Hleili
{"title":"Some Uncertainty Principles for the Right-Sided Multivariate Continuous Quaternion Wavelet Transform","authors":"Manel Hleili","doi":"10.1007/s00006-024-01319-w","DOIUrl":"10.1007/s00006-024-01319-w","url":null,"abstract":"<div><p>For the right-sided multivariate continuous quaternion wavelet transform (CQWT), we analyse the concentration of this transform on sets of finite measure. We also establish an analogue of Heisenberg’s inequality for the quaternion wavelet transform. Finally, we extend local uncertainty principle for a set of finite measure to CQWT.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140538505","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
A Real Method for Solving Octonion Matrix Equation (AXB=C) Based on Semi-tensor Product of Matrices 基于矩阵半张量积的求解八音矩阵方程 $$AXB=C$$ 的实数法
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-03-23 DOI: 10.1007/s00006-024-01316-z
Xiaochen Liu, Ying Li, Wenxv Ding, Ruyu Tao
{"title":"A Real Method for Solving Octonion Matrix Equation (AXB=C) Based on Semi-tensor Product of Matrices","authors":"Xiaochen Liu,&nbsp;Ying Li,&nbsp;Wenxv Ding,&nbsp;Ruyu Tao","doi":"10.1007/s00006-024-01316-z","DOIUrl":"10.1007/s00006-024-01316-z","url":null,"abstract":"<div><p>In this paper, the octonion matrix equation <span>(AXB=C)</span> is studied based on semi-tensor product of matrices. Firstly, we propose the left real element representation and the right real element representation of octonion. Then we obtain the expression of the least squares Hermitian solution to the octonion matrix equation <span>(AXB=C)</span> by combining these representations with <span>(mathcal {H})</span>-representation of the special matrices. In addition, we also put forward the equivalent condition of existence and general expression of the Hermitian solution to the octonion matrix equation <span>(AXB=C.)</span> Finally, the validity and stability of our method is verified by numerical experiments.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140192587","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
Common Spectral Properties of Bounded Right Linear Operators AC and BA in the Quaternionic Setting 四元数背景下有界右线性算子 AC 和 BA 的共谱特性
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-03-18 DOI: 10.1007/s00006-024-01315-0
Rachid Arzini, Ali Jaatit
{"title":"Common Spectral Properties of Bounded Right Linear Operators AC and BA in the Quaternionic Setting","authors":"Rachid Arzini,&nbsp;Ali Jaatit","doi":"10.1007/s00006-024-01315-0","DOIUrl":"10.1007/s00006-024-01315-0","url":null,"abstract":"<div><p>Let <i>X</i> be a two-sided quaternionic Banach space and let <span>(A, B, C: X longrightarrow X)</span> be bounded right linear quaternionic operators such that <span>(ACA=ABA)</span>. Let <i>q</i> be a non-zero quaternion. In this paper, we investigate the common properties of <span>((AC)^{2}-2Re(q)AC+|q|^2I)</span> and <span>((BA)^{2}-2Re(q)BA+|q|^2I)</span> where <i>I</i> stands for the identity operator on <i>X</i>. In particular, we show that </p><div><div><span>$$begin{aligned} sigma ^{S}_{{mathcal {F}}}(AC)backslash {0} = sigma ^{S}_{{mathcal {F}}}(BA)backslash {0} end{aligned}$$</span></div></div><p>where <span>(sigma ^{S}_{{mathcal {F}}}(.))</span> is a distinguished part of the spherical spectrum.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140161940","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
Generalized Partial-Slice Monogenic Functions: A Synthesis of Two Function Theories 广义部分片单原函数：两种函数理论的合成
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-03-09 DOI: 10.1007/s00006-024-01314-1
Zhenghua Xu, Irene Sabadini
{"title":"Generalized Partial-Slice Monogenic Functions: A Synthesis of Two Function Theories","authors":"Zhenghua Xu,&nbsp;Irene Sabadini","doi":"10.1007/s00006-024-01314-1","DOIUrl":"10.1007/s00006-024-01314-1","url":null,"abstract":"<div><p>In this paper, we review the notion of generalized partial-slice monogenic functions that was introduced by the authors in Xu and Sabadini (Generalized partial-slice monogenic functions, arXiv:2309.03698, 2023). The class of these functions includes both the theory of monogenic functions and of slice monogenic functions over Clifford algebras and it is obtained via a synthesis operator which combines a generalized Cauchy–Riemann operator with an operator acting on slices. Besides recalling the fundamental features, we provide a notion of <span>(*)</span>-product based on the CK-extension and discuss the smoothness of generalized partial-slice functions.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-03-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-024-01314-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140067802","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
Heron’s Formula in Higher Dimensions 高维赫伦公式
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-02-17 DOI: 10.1007/s00006-023-01305-8
Timothy F. Havel
{"title":"Heron’s Formula in Higher Dimensions","authors":"Timothy F. Havel","doi":"10.1007/s00006-023-01305-8","DOIUrl":"10.1007/s00006-023-01305-8","url":null,"abstract":"<div><p>This paper shows how geometric algebra can be used to derive a novel generalization of Heron’s classical formula for the area of a triangle in the plane to higher dimensions. It begins by illustrating some of the many ways in which the conformal model of three-dimensional Euclidean space yields provocative insights into some of our most basic intuitive notions of solid geometry. It then uses this conceptual framework to elucidate the geometric meaning of Heron’s formula in the plane, and explains in detail how it extends naturally to the volumes of tetrahedra in space. The paper closes by outlining a proof of a previously conjectured extension of the formula to the hyper-volumes of simplices in all dimensions.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 2","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-02-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-023-01305-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139898782","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 Optimal Inequalities for Anti-invariant Riemannian Submersions from Conformal Sasakian Space Forms 论来自共形萨萨基空间形式的反不变黎曼潜影的最优不等式
IF 1.1 2区 数学
Advances in Applied Clifford Algebras Pub Date : 2024-01-21 DOI: 10.1007/s00006-023-01312-9
Mehraj Ahmad Lone, Towseef Ali Wani
{"title":"On Optimal Inequalities for Anti-invariant Riemannian Submersions from Conformal Sasakian Space Forms","authors":"Mehraj Ahmad Lone,&nbsp;Towseef Ali Wani","doi":"10.1007/s00006-023-01312-9","DOIUrl":"10.1007/s00006-023-01312-9","url":null,"abstract":"<div><p>The aim of this paper is two-fold. First, we obtain various inequalities which involve the Ricci and scalar curvatures of horizontal and vertical distributions of anti-invariant Riemannian submersion defined from conformal Sasakian space form onto a Riemannian manifold. Second, we obtain the Chen–Ricci inequality for the said Riemannian submersion.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"34 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2024-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139510875","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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