Pub Date : 2020-01-20DOI: 10.7546/JGSP-55-2020-75-103
S. E. Samokhvalov
In this work we use generalized deformed gauge groups for investigation of symmetry of general relativity (GR). GR is formulated in generalized reference frames, which are represented by (anholonomic in general case) affine frame fields. The general principle of relativity is extended to the requirement of invariance of the theory with respect to transitions between generalized reference frames, that is, with respect to the group $GL^g$ of local linear transformations of affine frame fields. GR is interpreted as the gauge theory of the gauge group of translations $T^g_M$, and therefore is invariant under the space-time diffeomorphisms. The groups $GL^g$ and $T^g_M$ are united into group $S^g_M$, which is their semidirect product and is the complete symmetry group of the general relativity in an affine frame (GRAF). The consequence of $GL^g$-invariance of GRAF is the Palatini equation, which in the absence of torsion goes into the metricity condition, and vice versa, that is, is fulfilled identically in the Riemannian space. The consequence of the $T^g_M$-invariance of GRAF is representation of the Einstein equation in the superpotential form, that is, in the form of dynamic Maxwell equations (or Young-Mills equations). Deformation of the group $S^g_M$ leads to renormalisation of energy-momentum of the gravitation field. At the end we show that by limiting admissible reference frames (by $GL^g$-gauge fixing) from GRAF, in addition to Einstein gravity, one can obtain other local equivalent formulations of GR: general relativity in an orthonormal frame or teleparallel equivalent of general relativity, dilaton gravity, unimodular gravity, etc.
{"title":"About the Symmetry of General Relativity","authors":"S. E. Samokhvalov","doi":"10.7546/JGSP-55-2020-75-103","DOIUrl":"https://doi.org/10.7546/JGSP-55-2020-75-103","url":null,"abstract":"In this work we use generalized deformed gauge groups for investigation of symmetry of general relativity (GR). GR is formulated in generalized reference frames, which are represented by (anholonomic in general case) affine frame fields. The general principle of relativity is extended to the requirement of invariance of the theory with respect to transitions between generalized reference frames, that is, with respect to the group $GL^g$ of local linear transformations of affine frame fields. GR is interpreted as the gauge theory of the gauge group of translations $T^g_M$, and therefore is invariant under the space-time diffeomorphisms. The groups $GL^g$ and $T^g_M$ are united into group $S^g_M$, which is their semidirect product and is the complete symmetry group of the general relativity in an affine frame (GRAF). \u0000The consequence of $GL^g$-invariance of GRAF is the Palatini equation, which in the absence of torsion goes into the metricity condition, and vice versa, that is, is fulfilled identically in the Riemannian space. The consequence of the $T^g_M$-invariance of GRAF is representation of the Einstein equation in the superpotential form, that is, in the form of dynamic Maxwell equations (or Young-Mills equations). Deformation of the group $S^g_M$ leads to renormalisation of energy-momentum of the gravitation field. At the end we show that by limiting admissible reference frames (by $GL^g$-gauge fixing) from GRAF, in addition to Einstein gravity, one can obtain other local equivalent formulations of GR: general relativity in an orthonormal frame or teleparallel equivalent of general relativity, dilaton gravity, unimodular gravity, etc.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47887209","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 : 2020-01-01DOI: 10.7546/JGSP-58-2020-81-97
I. Mladenov, Marin Drinov Academic Publishng House
Here we derive explicit formulas that parameterize the Cassinian ovals based on their recognition as the so called spiric sections of the standard tori in the three-dimensional Euclidean space which was suggested in the ancient time by Perseus. These formulas derived originally in terms of the toric parameters are expressed through the usual geometrical parameters that enter in the present day definition of the Cassinian curves. All three types of morphologically different curves are illustrated graphically using the corresponding sets of parameters and respective formulas. The geometry of the ovals can be studied in full details and this is done here to some extent. As examples explicit formulas for the embraced volume and the surface area of the dumbbell like surface generated by the oval are presented. Last, but not least, new alternative explicit parameterizations of the Cassinian ovals are derived in polar, and even in non-canonical Monge forms.
{"title":"The Spiric Sections of Perseus and the Uniform Parameterizations of the Cassinian Ovals","authors":"I. Mladenov, Marin Drinov Academic Publishng House","doi":"10.7546/JGSP-58-2020-81-97","DOIUrl":"https://doi.org/10.7546/JGSP-58-2020-81-97","url":null,"abstract":"Here we derive explicit formulas that parameterize the Cassinian ovals based on their recognition as the so called spiric sections of the standard tori in the three-dimensional Euclidean space which was suggested in the ancient time by Perseus. These formulas derived originally in terms of the toric parameters are expressed through the usual geometrical parameters that enter in the present day definition of the Cassinian curves. All three types of morphologically different curves are illustrated graphically using the corresponding sets of parameters and respective formulas. The geometry of the ovals can be studied in full details and this is done here to some extent. As examples explicit formulas for the embraced volume and the surface area of the dumbbell like surface generated by the oval are presented. Last, but not least, new alternative explicit parameterizations of the Cassinian ovals are derived in polar, and even in non-canonical Monge forms.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"8 Suppl 4 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197077","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 : 2020-01-01DOI: 10.7546/jgsp-56-2020-31-43
S. Aktay
In this work we investigate the possible classes of seven-dimensional almost paracontact metric structures induced by the three-forms of $G_2^*$ structures. We write the projections that determine to which class the almost paracontact structure belongs, by using the properties of the $G_2^*$ structures. Then we study the properties that the characteristic vector field of the almost paracontact metric structure should have such that the structure belongs to a specific subclass of almost paracontact metric structures.
{"title":"An Overview of the History of Projective Representations of Groups","authors":"S. Aktay","doi":"10.7546/jgsp-56-2020-31-43","DOIUrl":"https://doi.org/10.7546/jgsp-56-2020-31-43","url":null,"abstract":"In this work we investigate the possible classes of seven-dimensional almost paracontact metric structures induced by the three-forms of $G_2^*$ structures. We write the projections that determine to which class the almost paracontact structure belongs, by using the properties of the $G_2^*$ structures. Then we study the properties that the characteristic vector field of the almost paracontact metric structure should have such that the structure belongs to a specific subclass of almost paracontact metric structures.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71196777","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 : 2020-01-01DOI: 10.7546/jgsp-56-2020-1-29
T. Hirai
An overview of the history of projective representations (= spin representations) of groups, preceded by the prehistory of studies on the theory of quaternion due to Rodrigues and Hamilton. Beginning with Schur, we cover many mathematicians until today, and also physicists Pauli and Dirac.
{"title":"An Overview of the History of Projective Representations of Groups","authors":"T. Hirai","doi":"10.7546/jgsp-56-2020-1-29","DOIUrl":"https://doi.org/10.7546/jgsp-56-2020-1-29","url":null,"abstract":"An overview of the history of projective representations (= spin representations) of groups, preceded by the prehistory of studies on the theory of quaternion due to Rodrigues and Hamilton. Beginning with Schur, we cover many mathematicians until today, and also physicists Pauli and Dirac.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71196653","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 : 2020-01-01DOI: 10.7546/jgsp-57-2020-99-109
Nicola Sottocornola
The problem of separating variables in integrable Hamiltonian systems has been extensively studied in the last decades. A recent approach is based on the so called Kowalewski's Conditions used to characterized a Control Matrix (M) whose eigenvalues give the desired coordinates. In this paper we calculate directly a second compatible Hamiltonian structure for the cubic Hénon-Heiles systems and in this way we obtain the separation variables as the eigenvalues of a recursion operator (N). Finally we re-obtain the Control Matrix (M) from (N).
{"title":"Direct Construction of a Bi-Hamiltonian Structure for Cubic Hénon-Heiles Systems","authors":"Nicola Sottocornola","doi":"10.7546/jgsp-57-2020-99-109","DOIUrl":"https://doi.org/10.7546/jgsp-57-2020-99-109","url":null,"abstract":"The problem of separating variables in integrable Hamiltonian systems has been extensively studied in the last decades. A recent approach is based on the so called Kowalewski's Conditions used to characterized a Control Matrix (M) whose eigenvalues give the desired coordinates. In this paper we calculate directly a second compatible Hamiltonian structure for the cubic Hénon-Heiles systems and in this way we obtain the separation variables as the eigenvalues of a recursion operator (N). Finally we re-obtain the Control Matrix (M) from (N).","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71197014","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 : 2019-12-07DOI: 10.7546/jgsp-62-2021-1-28
L. Georgiev
Using the decomposition of rational conformal filed theory characters for the $Z_k$ parafermion quantum Hall droplets for general $k=2,3,dots$, we derive analytically the full modular $S$ matrix for these states, including the $uu$ parts corresponding to the charged sector of the full conformal field theory and the neutral parafermion contributions corresponding to the diagonal affine coset models. This precise neutral-part parafermion $S$ matrix is derived from the explicit relations between the coset matrix and those for the numerator and denominator of the coset and the latter is expressed in compact form due to the level-rank duality between the affine Lie algebras $widehat{frak{su}(k)_2}$ and $widehat{frak{su}(2)_k}$. The exact results obtained for the $S$ matrix elements are expected to play an important role for identifying interference patterns of fractional quantum Hall states in Fabry-P'erot interferometers which can be used to distinguish between Abelian and non-Abelian statistics of quasiparticles localized in the bulk of fractional quantum Hall droplets as well as for nondestructive interference measurement of Fibonacci anyons which can be used for universal topological quantum computation
{"title":"Exact Modular $S$ Matrix for ${mathbb Z}_{k}$ Parafermion Quantum Hall Islands and Measurement of Non-Abelian Anyons","authors":"L. Georgiev","doi":"10.7546/jgsp-62-2021-1-28","DOIUrl":"https://doi.org/10.7546/jgsp-62-2021-1-28","url":null,"abstract":"Using the decomposition of rational conformal filed theory characters for the $Z_k$ parafermion quantum Hall droplets for general $k=2,3,dots$, we derive analytically the full modular $S$ matrix for these states, including the $uu$ parts corresponding to the charged sector of the full conformal field theory and the neutral parafermion contributions corresponding to the diagonal affine coset models. This precise neutral-part parafermion $S$ matrix is derived from the explicit relations between the coset matrix and those for the numerator and denominator of the coset and the latter is expressed in compact form due to the level-rank duality between the affine Lie algebras $widehat{frak{su}(k)_2}$ and $widehat{frak{su}(2)_k}$. The exact results obtained for the $S$ matrix elements are expected to play an important role for identifying interference patterns of fractional quantum Hall states in Fabry-P'erot interferometers which can be used to distinguish between Abelian and non-Abelian statistics of quasiparticles localized in the bulk of fractional quantum Hall droplets as well as for nondestructive interference measurement of Fibonacci anyons which can be used for universal topological quantum computation","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-12-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44581811","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 : 2019-10-14DOI: 10.7546/jgsp-65-2023-1-39
J. Escudero
We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile rearrangements in the substitution rules associated to edge flips. Special types of random tilings for Euclidean, spherical and hyperbolic three-manifolds are also considered.
{"title":"Deltoid Tangents with Evenly Distributed Orientations and Random Tilings","authors":"J. Escudero","doi":"10.7546/jgsp-65-2023-1-39","DOIUrl":"https://doi.org/10.7546/jgsp-65-2023-1-39","url":null,"abstract":"We study the construction of substitution tilings of the plane based on certain simplicial configurations of tangents of the deltoid with evenly distributed orientations. The random tiling ensembles are obtained as a result of tile rearrangements in the substitution rules associated to edge flips. Special types of random tilings for Euclidean, spherical and hyperbolic three-manifolds are also considered.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-10-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43620437","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 : 2019-10-09DOI: 10.7546/jgsp-54-2019-79-110
Maria Robaszewska
We recall the well-known Chern-Terng theorem concerning affine minimal surfaces. Next we formulate some complementary (with transversal fields necessarily not parallel) affine Backlund theorem. We describe some geometrical conditions which imply the local symmetry of both induced connections. We give also some necessary and sufficient conditions under which the affine fundamental forms are proportional.
{"title":"On Analogues of B\" acklund Theorem in Affine Differential Geometry of Surfaces","authors":"Maria Robaszewska","doi":"10.7546/jgsp-54-2019-79-110","DOIUrl":"https://doi.org/10.7546/jgsp-54-2019-79-110","url":null,"abstract":"We recall the well-known Chern-Terng theorem concerning affine minimal surfaces. Next we formulate some complementary (with transversal fields necessarily not parallel) affine Backlund theorem. We describe some geometrical conditions which imply the local symmetry of both induced connections. We give also some necessary and sufficient conditions under which the affine fundamental forms are proportional.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46473509","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 : 2019-09-04DOI: 10.7546/jgsp-54-2019-13-35
S. Matsutani, Hiroshi Nishiguchi, K. Higashida, A. Nakatani, Hiroyasu Hamada
In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $phi_0$; we refer the piece-wised elastica $Lambda_{phi_0}$-elastica or $Lambda$-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly due the rupture, the shapes of $Lambda$-elastica appear. We construct a mathematical theory to describe the phenomena and represent the $Lambda$-elastica in terms of the elliptic $zeta$-function completely. Using the mathematical theory, we discuss the experimental results from an energetic viewpoint and numerically show the explicit shape of $Lambda$-elastica. It means that this paper provides a novel investigation on elastica theory with rupture.
{"title":"On (Lambda)-Elastica","authors":"S. Matsutani, Hiroshi Nishiguchi, K. Higashida, A. Nakatani, Hiroyasu Hamada","doi":"10.7546/jgsp-54-2019-13-35","DOIUrl":"https://doi.org/10.7546/jgsp-54-2019-13-35","url":null,"abstract":"In this paper, we investigate a transition from an elastica to a piece-wised elastica whose connected point defines the hinge angle $phi_0$; we refer the piece-wised elastica $Lambda_{phi_0}$-elastica or $Lambda$-elastica. The transition appears in the bending beam experiment; we compress elastic beams gradually and then suddenly due the rupture, the shapes of $Lambda$-elastica appear. We construct a mathematical theory to describe the phenomena and represent the $Lambda$-elastica in terms of the elliptic $zeta$-function completely. Using the mathematical theory, we discuss the experimental results from an energetic viewpoint and numerically show the explicit shape of $Lambda$-elastica. It means that this paper provides a novel investigation on elastica theory with rupture.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":"1 1","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42583011","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 : 2019-08-23DOI: 10.7546/jgsp-53-2019-21-53
J. Bernal, J. Lawrence
The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two corresponding sets of points in d-dimensional Euclidean space. This problem generalizes to the so-called Wahba's problem which is the same problem with nonnegative weights. Given a dxd matrix M, solutions to these problems are intimately related to the problem of finding a dxd rotation matrix U that maximizes the trace of UM, i.e., that makes UM a matrix of maximal trace over rotations, and it is well known this can be achieved with a method based on the computation of the singular value decomposition (SVD) of M. As the main goal of this paper, we characterize dxd matrices of maximal trace over rotation matrices in terms of their eigenvalues, and for d = 2, 3, we show how this characterization can be used to determine whether a matrix is of maximal trace over rotation matrices. Finally, although depending only slightly on the characterization, as a secondary goal of the paper, for d = 2, 3, we identify alternative ways, other than the SVD, of obtaining solutions to the aforementioned problems.
{"title":"Characterization and Computation of Matrices of Maximal Trace Over Rotations","authors":"J. Bernal, J. Lawrence","doi":"10.7546/jgsp-53-2019-21-53","DOIUrl":"https://doi.org/10.7546/jgsp-53-2019-21-53","url":null,"abstract":"The constrained orthogonal Procrustes problem is the least-squares problem that calls for a rotation matrix that optimally aligns two corresponding sets of points in d-dimensional Euclidean space. This problem generalizes to the so-called Wahba's problem which is the same problem with nonnegative weights. Given a dxd matrix M, solutions to these problems are intimately related to the problem of finding a dxd rotation matrix U that maximizes the trace of UM, i.e., that makes UM a matrix of maximal trace over rotations, and it is well known this can be achieved with a method based on the computation of the singular value decomposition (SVD) of M. As the main goal of this paper, we characterize dxd matrices of maximal trace over rotation matrices in terms of their eigenvalues, and for d = 2, 3, we show how this characterization can be used to determine whether a matrix is of maximal trace over rotation matrices. Finally, although depending only slightly on the characterization, as a secondary goal of the paper, for d = 2, 3, we identify alternative ways, other than the SVD, of obtaining solutions to the aforementioned problems.","PeriodicalId":43078,"journal":{"name":"Journal of Geometry and Symmetry in Physics","volume":" ","pages":""},"PeriodicalIF":0.4,"publicationDate":"2019-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46698282","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}
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