Pub Date : 2021-11-30DOI: 10.7546/jgsp-61-2021-17-40
M. Kharinov
In this paper, aiming to develop the group and out-of-group formalization of the symmetry concept, the preservation of a matrix symmetry after row permutation is considered by the example of the maximally permutable emph{normalized} Hadamard matrices which row and column elements are either plus or minus one. These matrices are used to extend the additive decomposition of a linear operator into symmetric and skew-symmetric parts using several commuting operations of the Hermitian conjugation type, for the quaternionic generalization of a vector cross product, as well as for creating educational puzzles and other applications.
{"title":"Permutable Symmetric Hadamard Matrices in Quaternion Algebra and Engineering Applications","authors":"M. Kharinov","doi":"10.7546/jgsp-61-2021-17-40","DOIUrl":"https://doi.org/10.7546/jgsp-61-2021-17-40","url":null,"abstract":"In this paper, aiming to develop the group and out-of-group formalization of the symmetry concept, the preservation of a matrix symmetry after row permutation is considered by the example of the maximally permutable emph{normalized} Hadamard matrices which row and column elements are either plus or minus one. These matrices are used to extend the additive decomposition of a linear operator into symmetric and skew-symmetric parts using several commuting operations of the Hermitian conjugation type, for the quaternionic generalization of a vector cross product, as well as for creating educational puzzles and other applications.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-11-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44482826","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.7546/jgsp-60-2021-1-23
Christiam Figueroa
In this paper we study the Gauss map of surfaces in three-dimensional Heisenberg group using the Gans model of the hyperbolic plane. We establish a relationship between the tension fields of the Gauss maps and the mean curvatures of the surfaces in $mathcal{H}_{3}$.
{"title":"Gauss Maps of the Surfaces in the Three-Dimensional Heisenberg Group","authors":"Christiam Figueroa","doi":"10.7546/jgsp-60-2021-1-23","DOIUrl":"https://doi.org/10.7546/jgsp-60-2021-1-23","url":null,"abstract":"In this paper we study the Gauss map of surfaces in three-dimensional Heisenberg group using the Gans model of the hyperbolic plane. We establish a relationship between the tension fields of the Gauss maps and the mean curvatures of the surfaces in $mathcal{H}_{3}$.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"56 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77238897","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.7546/jgsp-60-2021-47-64
L. Silvestre, E. Miro, Job A. Nable
This paper characterizes primitive substitution tiling systems for Lie groups for which every element of the associated tiling space generates coherent frames for corresponding representation spaces.
本文刻画了李群的基元替换平铺系统，其相关平铺空间的每个元素都为相应的表示空间生成相干帧。
{"title":"Primitive Tilings and Coherent Frames","authors":"L. Silvestre, E. Miro, Job A. Nable","doi":"10.7546/jgsp-60-2021-47-64","DOIUrl":"https://doi.org/10.7546/jgsp-60-2021-47-64","url":null,"abstract":"This paper characterizes primitive substitution tiling systems for Lie groups for which every element of the associated tiling space generates coherent frames for corresponding representation spaces.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47705280","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.7546/jgsp-60-2021-65-81
T. Valchev, C. Mladenova, I. Mladenov
Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $mathrm{Sp}(2,bbr)congmathrm{SL}(2,bbr)$. Relying on the properties of the exponential map $mathfrak{sl}(2,bbr)tomathrm{SL}(2,bbr)$, we have found a few explicit formulas for the logarithm of the matrices in these groups. Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.
{"title":"New Parameterizations of (mathrm{SL}(2,mathbb{R})) and Some Explicit Formulas for Its Logarithm","authors":"T. Valchev, C. Mladenova, I. Mladenov","doi":"10.7546/jgsp-60-2021-65-81","DOIUrl":"https://doi.org/10.7546/jgsp-60-2021-65-81","url":null,"abstract":"Here we demonstrate some of the benefits of a novel parameterization of the Lie groups $mathrm{Sp}(2,bbr)congmathrm{SL}(2,bbr)$. Relying on the properties of the exponential map $mathfrak{sl}(2,bbr)tomathrm{SL}(2,bbr)$, we have found a few explicit formulas for the logarithm of the matrices in these groups. Additionally, the explicit analytic description of the ellipse representing their field of values is derived and this allows a direct graphical identification of various types.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47424255","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-07-01DOI: 10.7546/jgsp-60-2021-83-94
H. Yoldaş
The purpose of present paper is to study cosymplectic manifolds admitting certain special vector fields such as holomorphically planar conformal (in short HPC) vector field. First, we prove that an HPC vector field on a cosymplectic manifold is also a Jacobi-type vector field. Then, we obtain the necessary conditions for such a vector field to be Killing. Finally, we give an important characterization for a torse-forming vector field on such a manifold given as to be recurrent.
{"title":"Some Results on Cosymplectic Manifolds Admitting Certain Vector Fields","authors":"H. Yoldaş","doi":"10.7546/jgsp-60-2021-83-94","DOIUrl":"https://doi.org/10.7546/jgsp-60-2021-83-94","url":null,"abstract":"The purpose of present paper is to study cosymplectic manifolds admitting certain special vector fields such as holomorphically planar conformal (in short HPC) vector field. First, we prove that an HPC vector field on a cosymplectic manifold is also a Jacobi-type vector field. Then, we obtain the necessary conditions for such a vector field to be Killing. Finally, we give an important characterization for a torse-forming vector field on such a manifold given as to be recurrent.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47769893","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7546/JGSP-59-2021-47-65
P. Bracken
General classes of non-linear sigma models originating from a specified action are developed and studied. Models can be grouped and considered within a single mathematical structure this way. The geometrical properties of these models and the theories they describe are developed in detail. The zero curvature representation of the equations of motion are found. Those representations which have a spectral parameter are of importance here. Some new models with Lax pairs which depend on a spectral parameter are found. Some particular classes of solutions are worked out and discussed.
{"title":"Classically Integrable Non-Linear Sigma Models and their Geometric Properties","authors":"P. Bracken","doi":"10.7546/JGSP-59-2021-47-65","DOIUrl":"https://doi.org/10.7546/JGSP-59-2021-47-65","url":null,"abstract":"General classes of non-linear sigma models originating from a specified action are developed and studied. Models can be grouped and considered within a single mathematical structure this way. The geometrical properties of these models and the theories they describe are developed in detail. The zero curvature representation of the equations of motion are found. Those representations which have a spectral parameter are of importance here. Some new models with Lax pairs which depend on a spectral parameter are found. Some particular classes of solutions are worked out and discussed.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"37 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82238336","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7546/jgsp-62-2021-67-84
L. B. Natividad, Job A. Nable
The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $star$-product of phase space functions. In this article, the $star$-product of functions on the Euclidean motion group of rank three, $mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $star_s$ on $mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.
{"title":"Symbol Correspondence for Euclidean Systems","authors":"L. B. Natividad, Job A. Nable","doi":"10.7546/jgsp-62-2021-67-84","DOIUrl":"https://doi.org/10.7546/jgsp-62-2021-67-84","url":null,"abstract":"The three main objects that serve as the foundation of quantum mechanics on phase space are the Weyl transform, the Wigner distribution function, and the $star$-product of phase space functions. In this article, the $star$-product of functions on the Euclidean motion group of rank three, $mathrm{E}(3)$, is constructed. $C^*$-algebra properties of $star_s$ on $mathrm{E}(3)$ are presented, establishing a phase space symbol calculus for functions whose parameters are translations and rotations. The key ingredients in the construction are the unitary irreducible representations of the group.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71196975","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7546/jgsp-62-2021-29-52
Ying-Qiu Gu
In this note we construct explicit complex and real faithful matrix representations of the Clifford algebras $Cl_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. In the cases $p+q=4m$, the representation is unique in equivalent sense, and the $1+3$ dimensional space-time corresponds to the simplest and best case. Besides, the relation between the curvilinear coordinate frame and the local orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent space is calculated. These results are helpful for both theoretical analysis and practical calculation. The basis matrices are the faithful representation of Clifford algebras in any $p+q$ dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the complicated relations in geometry and physics into simple and concise algebraic operations. Clifford numbers over any number field $mathbb{F}$ expressed by this matrix basis form a well-defined $2^n$ dimensional hypercomplex number system. Therefore, we can expect that Clifford algebras will complete a large synthesis in science.
{"title":"A Note on the Representation of Clifford Algebras","authors":"Ying-Qiu Gu","doi":"10.7546/jgsp-62-2021-29-52","DOIUrl":"https://doi.org/10.7546/jgsp-62-2021-29-52","url":null,"abstract":"In this note we construct explicit complex and real faithful matrix representations of the Clifford algebras $Cl_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. In the cases $p+q=4m$, the representation is unique in equivalent sense, and the $1+3$ dimensional space-time corresponds to the simplest and best case. Besides, the relation between the curvilinear coordinate frame and the local orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent space is calculated. These results are helpful for both theoretical analysis and practical calculation. The basis matrices are the faithful representation of Clifford algebras in any $p+q$ dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the complicated relations in geometry and physics into simple and concise algebraic operations. Clifford numbers over any number field $mathbb{F}$ expressed by this matrix basis form a well-defined $2^n$ dimensional hypercomplex number system. Therefore, we can expect that Clifford algebras will complete a large synthesis in science.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197101","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.7546/jgsp-62-2021-53-66
F. Latti, H. Elhendi, L. Belarbi
In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.
本文在黎曼流形$(M,g)$的切束$TM$上引入了一类新的自然度量，表示为$ g ^{f,h}$，命名为扭曲Sasakian度量。建立了向量场相对于扭曲Sasakian度规是调和的充分必要条件。文中还给出了一些谐波矢量场的例子。
{"title":"Twisted Sasakian Metric on the Tangent Bundle and Harmonicity","authors":"F. Latti, H. Elhendi, L. Belarbi","doi":"10.7546/jgsp-62-2021-53-66","DOIUrl":"https://doi.org/10.7546/jgsp-62-2021-53-66","url":null,"abstract":"In the present paper, we introduce a new class of natural metrics on the tangent bundle $TM$ of the Riemannian manifold $(M,g)$ denoted by $G^{f,h}$ which is named a twisted Sasakian metric. A necessary and sufficient conditions under which a vector field is harmonic with respect to the twisted Sasakian metric are established. Some examples of harmonic vector fields are presented as well.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-09-05DOI: 10.7546/JGSP-58-2020-1-12
A. Blaga, D. Laţcu
We study almost Riemann solitons and almost Ricci solitons in an $(alpha,beta)$-contact metric manifold satisfying some Ricci symmetry conditions, treating the case when the potential vector field of the soliton is pointwise collinear with the structure vector field.
{"title":"Remarks on Riemann and Ricci Solitons in $(alpha,beta)$-Contact Metric Manifolds","authors":"A. Blaga, D. Laţcu","doi":"10.7546/JGSP-58-2020-1-12","DOIUrl":"https://doi.org/10.7546/JGSP-58-2020-1-12","url":null,"abstract":"We study almost Riemann solitons and almost Ricci solitons in an $(alpha,beta)$-contact metric manifold satisfying some Ricci symmetry conditions, treating the case when the potential vector field of the soliton is pointwise collinear with the structure vector field.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-09-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42151573","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
京公网安备 11010802042870号
京ICP备2023020795号-1
扫码关注我们
求助内容：
应助结果提醒方式：