Pub Date : 2019-01-01DOI: 10.7546/giq-giq-20-2019-122-130
A. Dobrogowska, G. Jakimowicz, M. Szajewska, Karolina Wojciechowicz
The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras. MSC : 53D17, 37K10
{"title":"Deformation of the Poisson Structure Related to Algebroid Bracket of Differential Forms and Application to Real Low Dimentional Lie Algebras","authors":"A. Dobrogowska, G. Jakimowicz, M. Szajewska, Karolina Wojciechowicz","doi":"10.7546/giq-giq-20-2019-122-130","DOIUrl":"https://doi.org/10.7546/giq-giq-20-2019-122-130","url":null,"abstract":"The main goal of this paper is to present the possibility of application of some well known tools of Poisson geometry to classification of real low dimensional Lie algebras. MSC : 53D17, 37K10","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"53 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75347252","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 : 2019-01-01DOI: 10.7546/giq-20-2019-131-160
D. Ghisa
The concept of fundamental domain, as defined by Ahlfors, plays an important role in the study of different classes of analytic functions. For more than a century the Dirichlet functions have been intensely studied by mathematicians working in the field of number theory as well as by those interested in their analytic properties. The fundamental domains pertain to the last field, yet we found a lot of theoretic aspects which can be dealt with by knowing in detail those domains. We gathered together in this survey paper some recent advances in this field. Proofs are provided for some of the theorems, so that the reader can navigate easily through it. MSC : 30C35, 11M26
{"title":"Fundamental Domains of Dirichlet Functions","authors":"D. Ghisa","doi":"10.7546/giq-20-2019-131-160","DOIUrl":"https://doi.org/10.7546/giq-20-2019-131-160","url":null,"abstract":"The concept of fundamental domain, as defined by Ahlfors, plays an important role in the study of different classes of analytic functions. For more than a century the Dirichlet functions have been intensely studied by mathematicians working in the field of number theory as well as by those interested in their analytic properties. The fundamental domains pertain to the last field, yet we found a lot of theoretic aspects which can be dealt with by knowing in detail those domains. We gathered together in this survey paper some recent advances in this field. Proofs are provided for some of the theorems, so that the reader can navigate easily through it. MSC : 30C35, 11M26","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80818621","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 : 2019-01-01DOI: 10.7546/GIQ-20-2019-99-114
R. Campoamor-Stursberg
Some aspects of the simple Lie algebra sl(2,R) realized as a subalgebra of Noether point symmetries of Lagrangian systems and related inverse problems are discussed, specially in connection to Lagrangians of kinetic type and some geometric properties like sectional curvatures. MSC : 70F17, 70H03, 70H33
{"title":"The Lie Algebra sl(2,R) and Noether Point Symmetries of Lagrangian Systems","authors":"R. Campoamor-Stursberg","doi":"10.7546/GIQ-20-2019-99-114","DOIUrl":"https://doi.org/10.7546/GIQ-20-2019-99-114","url":null,"abstract":"Some aspects of the simple Lie algebra sl(2,R) realized as a subalgebra of Noether point symmetries of Lagrangian systems and related inverse problems are discussed, specially in connection to Lagrangians of kinetic type and some geometric properties like sectional curvatures. MSC : 70F17, 70H03, 70H33","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80257205","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 : 2019-01-01DOI: 10.7546/GIQ-20-2019-255-265
N. Turgay, Alev Kelleci, R. Şen, E. O. Canfes
In this paper, first we gave a summary of recent results on biconservative immersions. Then, we obtained a necessary and sufficient condition for the existence of biconservative quasi-minimal immersions into a four dimensional semi-Riemannian space form of index two with non-zero sectional curvatures. We also constructed an explicit example of biconservative quasiminimal surface. MSC : 53D12, 53C40, 53C42
{"title":"On Quasi-Minimal Isometric Immersions into Non-Flat Semi Riemannian Space Forms of Index Two","authors":"N. Turgay, Alev Kelleci, R. Şen, E. O. Canfes","doi":"10.7546/GIQ-20-2019-255-265","DOIUrl":"https://doi.org/10.7546/GIQ-20-2019-255-265","url":null,"abstract":"In this paper, first we gave a summary of recent results on biconservative immersions. Then, we obtained a necessary and sufficient condition for the existence of biconservative quasi-minimal immersions into a four dimensional semi-Riemannian space form of index two with non-zero sectional curvatures. We also constructed an explicit example of biconservative quasiminimal surface. MSC : 53D12, 53C40, 53C42","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"211 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80711232","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 : 2019-01-01DOI: 10.7546/GIQ-20-2019-184-207
T. Kanazawa, A. Yoshioka
{"title":"Quasi-Classical Calculation of Eigenvalues by Maslov Quantization Condition","authors":"T. Kanazawa, A. Yoshioka","doi":"10.7546/GIQ-20-2019-184-207","DOIUrl":"https://doi.org/10.7546/GIQ-20-2019-184-207","url":null,"abstract":"","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80927354","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 : 2019-01-01DOI: 10.7546/giq-20-2019-297-310
A. Yanovski
We consider the theory of the hierarchies of nonlinear evolution equations associated with two gauge-equivalent systems we denote by L± and GMV± (GMVsystem). They are obtained from the Generalized Zakharov-Shabat system on sl(3,C) in general position making a Z2 × Z2 reductions of Mikhailov type in canonical and in pole gauge respectively. Using the Recursion Operators approach and expansions over the adjoint solutions we study the symplectic structures of the hierarchies of the nonlinear evolution equations associated with L± and GMV± and calculate the relation between them.
{"title":"Hierarchies of Symplectic Structures for sl(3,C) Zakharov-Shabat Systems in Canonical and Pole Gauge with Z2×Z2 Reduction of Mikhailov Type","authors":"A. Yanovski","doi":"10.7546/giq-20-2019-297-310","DOIUrl":"https://doi.org/10.7546/giq-20-2019-297-310","url":null,"abstract":"We consider the theory of the hierarchies of nonlinear evolution equations associated with two gauge-equivalent systems we denote by L± and GMV± (GMVsystem). They are obtained from the Generalized Zakharov-Shabat system on sl(3,C) in general position making a Z2 × Z2 reductions of Mikhailov type in canonical and in pole gauge respectively. Using the Recursion Operators approach and expansions over the adjoint solutions we study the symplectic structures of the hierarchies of the nonlinear evolution equations associated with L± and GMV± and calculate the relation between them.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"44 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79952093","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 : 2019-01-01DOI: 10.7546/GIQ-20-2019-285-296
T. Valchev, A. Yanovski
In this report we consider a vector generalization of Heisenberg ferromagnet equation. That completely integrable system is related to a spectral problem in pole gauge for the Lie algebra sl(n + 1,C). We construct special solutions over constant background using dressing technique. MSC : 35C05, 35C08, 35G50, 37K15, 37K35
{"title":"Solutions to a Vector Heisenberg Ferromagnet Equation Related to Symmetric Spaces","authors":"T. Valchev, A. Yanovski","doi":"10.7546/GIQ-20-2019-285-296","DOIUrl":"https://doi.org/10.7546/GIQ-20-2019-285-296","url":null,"abstract":"In this report we consider a vector generalization of Heisenberg ferromagnet equation. That completely integrable system is related to a spectral problem in pole gauge for the Lie algebra sl(n + 1,C). We construct special solutions over constant background using dressing technique. MSC : 35C05, 35C08, 35G50, 37K15, 37K35","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72441742","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 : 2019-01-01DOI: 10.7546/giq-20-2019-239-245
M. Petrov, T. Todorov
{"title":"Refinement Strategies for 4D Regular Domains","authors":"M. Petrov, T. Todorov","doi":"10.7546/giq-20-2019-239-245","DOIUrl":"https://doi.org/10.7546/giq-20-2019-239-245","url":null,"abstract":"","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84626988","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 : 2019-01-01DOI: 10.7546/giq-20-2019-88-98
M. Arminjon
In the electrodynamics of special relativity (SR) or general relativity (GR), the energy tensors of the charged medium and its EM field add to give the total energy tensor that obeys the dynamical equation without external force. In the investigated scalar theory of gravitation ("SET"), this assumption leads to charge non-conservation, hence an additional, "interaction" energy tensor T inter has to be postulated. The present work aims at constraining this tensor. First we study the independent equations of electrodynamics and their number, beginning with SR and GR. As in SR and GR, the system of electrodynamics of SET is closed in the absence of T inter. Hence, with T inter , at least one additional equation must be provided. This is done by assuming that T inter is Lorentz-invariant in the situation of SR. We derive equations allowing one in principle to compute T inter in a given gravitational plus EM field. T inter may contribute to the dark matter.
{"title":"Interaction Energy of a Charged Medium and its EM Field in a Curved Spacetime","authors":"M. Arminjon","doi":"10.7546/giq-20-2019-88-98","DOIUrl":"https://doi.org/10.7546/giq-20-2019-88-98","url":null,"abstract":"In the electrodynamics of special relativity (SR) or general relativity (GR), the energy tensors of the charged medium and its EM field add to give the total energy tensor that obeys the dynamical equation without external force. In the investigated scalar theory of gravitation (\"SET\"), this assumption leads to charge non-conservation, hence an additional, \"interaction\" energy tensor T inter has to be postulated. The present work aims at constraining this tensor. First we study the independent equations of electrodynamics and their number, beginning with SR and GR. As in SR and GR, the system of electrodynamics of SET is closed in the absence of T inter. Hence, with T inter , at least one additional equation must be provided. This is done by assuming that T inter is Lorentz-invariant in the situation of SR. We derive equations allowing one in principle to compute T inter in a given gravitational plus EM field. T inter may contribute to the dark matter.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"2 1-2","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72472356","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 : 2019-01-01DOI: 10.7546/GIQ-20-2019-79-87
Kamel Al-khaled
This paper aims to introduce a comparison of variation iteration method and wavelet basis method for the numerical solution of the Kawahara equation. Test problem is used to compare between the two methods. The comparison shows that variation iteration method is efficient and easy to use. On the other hand, the wavelet method is more stable as time increases. MSC : 35Q53, 65M60, 65R20
{"title":"Wavelet-Based Numerical Scheme Compared with VIM for Solving Kawahara Equation","authors":"Kamel Al-khaled","doi":"10.7546/GIQ-20-2019-79-87","DOIUrl":"https://doi.org/10.7546/GIQ-20-2019-79-87","url":null,"abstract":"This paper aims to introduce a comparison of variation iteration method and wavelet basis method for the numerical solution of the Kawahara equation. Test problem is used to compare between the two methods. The comparison shows that variation iteration method is efficient and easy to use. On the other hand, the wavelet method is more stable as time increases. MSC : 35Q53, 65M60, 65R20","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82445443","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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