Pub Date : 2022-01-01DOI: 10.7546/giq-23-2022-1-38
R. G. Calvet
The dynamic equations of the $n$-body problem are solved in relative coordinates and applied to the solar system, whence the mean variation rates of the longitudes of the ascending nodes and of the inclinations of the planetary orbits at J2000 have been calculated with respect to the ecliptic and to the Laplace invariable plane under the approximation of circular orbits. The theory so obtained supersedes the Lagrange-Laplace secular evolution theory. Formulas for the change from the equatorial and ecliptic coordinates to those of the Laplace invariable plane are also provided.
{"title":"On the Dynamics of the Solar System I: Orbital Inclination and Nodal Precession","authors":"R. G. Calvet","doi":"10.7546/giq-23-2022-1-38","DOIUrl":"https://doi.org/10.7546/giq-23-2022-1-38","url":null,"abstract":"The dynamic equations of the $n$-body problem are solved in relative coordinates and applied to the solar system, whence the mean variation rates of the longitudes of the ascending nodes and of the inclinations of the planetary orbits at J2000 have been calculated with respect to the ecliptic and to the Laplace invariable plane under the approximation of circular orbits. The theory so obtained supersedes the Lagrange-Laplace secular evolution theory. Formulas for the change from the equatorial and ecliptic coordinates to those of the Laplace invariable plane are also provided.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79184740","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 : 2022-01-01DOI: 10.7546/giq-24-2022-65-83
C. Mladenova, I. Mladenov
Here we demonstrate how the very definition of the numerical range leads to its direct geometrical identification. The procedure which we follow can be even slightly refined by making use of the famous Jacobi's method for diagonalization in reverse direction. From mathematical point of view, the Jacobi's idea here is used to reduce the number of the independent parameters from three to two which simplifies significantly the problem. As a surplus we have found an explicit recipe how to associate a Cassinian oval with the numerical range of any real $2times 2$ matrix. Last, but not least, we have derived their explicit parameterizations.
{"title":"Numerical Ranges of the Real 2×2 Matrices Derived by First Principles","authors":"C. Mladenova, I. Mladenov","doi":"10.7546/giq-24-2022-65-83","DOIUrl":"https://doi.org/10.7546/giq-24-2022-65-83","url":null,"abstract":"Here we demonstrate how the very definition of the numerical range leads to its direct geometrical identification. The procedure which we follow can be even slightly refined by making use of the famous Jacobi's method for diagonalization in reverse direction. From mathematical point of view, the Jacobi's idea here is used to reduce the number of the independent parameters from three to two which simplifies significantly the problem. As a surplus we have found an explicit recipe how to associate a Cassinian oval with the numerical range of any real $2times 2$ matrix. Last, but not least, we have derived their explicit parameterizations.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"83646515","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 : 2022-01-01DOI: 10.7546/giq-24-2022-39-64
R. G. Calvet
The evolution of the orientations of the orbital planes of the planets is calculated under the approximation of circular orbits. The inclination and the longitude of the ascending node of each orbital plane are then described by means of a linear combination of complex exponentials of time with periods of several thousand years. The evolution of these orbital elements for Mercury, Jupiter and Saturn is displayed as well as that of the ecliptic. Finally, the obliquity of the ecliptic is computed from $-2,000,000$ to $+2,000,000$ years since J2000. It ranges from $10^circ$ to $35^circ$ in this time interval.
{"title":"On the Dynamics of the Solar System II: Evolution of the Orbital Planes","authors":"R. G. Calvet","doi":"10.7546/giq-24-2022-39-64","DOIUrl":"https://doi.org/10.7546/giq-24-2022-39-64","url":null,"abstract":"The evolution of the orientations of the orbital planes of the planets is calculated under the approximation of circular orbits. The inclination and the longitude of the ascending node of each orbital plane are then described by means of a linear combination of complex exponentials of time with periods of several thousand years. The evolution of these orbital elements for Mercury, Jupiter and Saturn is displayed as well as that of the ecliptic. Finally, the obliquity of the ecliptic is computed from $-2,000,000$ to $+2,000,000$ years since J2000. It ranges from $10^circ$ to $35^circ$ in this time interval.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71196236","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 : 2021-12-28DOI: 10.7546/giq-24-2022-1-37
T. Hirai
From the standpoint of the History of Mathematics, beginning with pioneering work of Hurwitz on invariant integrals (or invariant measures) on Lie groups, we pick up epoch-making works successively and draw the main stream among so many contributions to the study of invariant integrals on groups, due to Hurwitz, Schur, Weyl, Haar, Neumann, Kakutani, Weil, and Kakutani-Kodaira, and explain their contents and give the relationships among them.
{"title":"Lineage of the Theory of Invariant Integrals on Groups","authors":"T. Hirai","doi":"10.7546/giq-24-2022-1-37","DOIUrl":"https://doi.org/10.7546/giq-24-2022-1-37","url":null,"abstract":"From the standpoint of the History of Mathematics, beginning with pioneering work of Hurwitz on invariant integrals (or invariant measures) on Lie groups, we pick up epoch-making works successively and draw the main stream among so many contributions to the study of invariant integrals on groups, due to Hurwitz, Schur, Weyl, Haar, Neumann, Kakutani, Weil, and Kakutani-Kodaira, and explain their contents and give the relationships among them.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-12-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43119184","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/GIQ-22-2021-107-120
Adina Crișan, I. Vancea
In this paper, we review recent results on the interaction of the topological electromagnetic fields with matter, in particular with spinless and spin half charged particles obtained earlier. The problems discussed here are the generalized Finsler geometries and their dualities in the Trautman-Ra~{n}ada backgrounds, the classical dynamics of the charged particles in the single non-null knot mode background and the quantization in the same background in the strong field approximation.
{"title":"Geometric and Quantum Properties of Charged Particles in Monochromatic Electromagnetic Knot Background","authors":"Adina Crișan, I. Vancea","doi":"10.7546/GIQ-22-2021-107-120","DOIUrl":"https://doi.org/10.7546/GIQ-22-2021-107-120","url":null,"abstract":"In this paper, we review recent results on the interaction of the topological electromagnetic fields with matter, in particular with spinless and spin half charged particles obtained earlier. The problems discussed here are the generalized Finsler geometries and their dualities in the Trautman-Ra~{n}ada backgrounds, the classical dynamics of the charged particles in the single non-null knot mode background and the quantization in the same background in the strong field approximation.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80640765","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/GIQ-22-2021-64-77
P. Aniello
A quantum stochastic product is a binary operation on the space of quantum states preserving the convex structure. We describe a class of associative stochastic products, the twirled products, that have interesting connections with quantum measurement theory. Constructing such a product involves a square integrable group representation, a probability measure and a fiducial state. By extending a twirled product to the full space of trace class operators, one obtains a Banach algebra. This algebra is commutative if the underlying group is abelian. In the case of the group of translations on phase space, one gets a quantum convolution algebra, a quantum counterpart of the classical phase-space convolution algebra. The peculiar role of the fiducial state characterizing each quantum convolution product is highlighted.
{"title":"Quantum Stochastic Products and the Quantum Convolution","authors":"P. Aniello","doi":"10.7546/GIQ-22-2021-64-77","DOIUrl":"https://doi.org/10.7546/GIQ-22-2021-64-77","url":null,"abstract":"A quantum stochastic product is a binary operation on the space of quantum states preserving the convex structure. We describe a class of associative stochastic products, the twirled products, that have interesting connections with quantum measurement theory. Constructing such a product involves a square integrable group representation, a probability measure and a fiducial state. By extending a twirled product to the full space of trace class operators, one obtains a Banach algebra. This algebra is commutative if the underlying group is abelian. In the case of the group of translations on phase space, one gets a quantum convolution algebra, a quantum counterpart of the classical phase-space convolution algebra. The peculiar role of the fiducial state characterizing each quantum convolution product is highlighted.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88445314","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/giq-22-2021-13-32
D. Djordjevic
In this survey paper we present some aspects of generalized inverses, which are related to inner and outer invertibility, Moore-Penrose inverse, the appropriate reverse order law, and Drazin inverse.
{"title":"Generalized Inverses","authors":"D. Djordjevic","doi":"10.7546/giq-22-2021-13-32","DOIUrl":"https://doi.org/10.7546/giq-22-2021-13-32","url":null,"abstract":"In this survey paper we present some aspects of generalized inverses, which are related to inner and outer invertibility, Moore-Penrose inverse, the appropriate reverse order law, and Drazin inverse.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"13 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76334755","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/giq-21-2020-232-241
Á. Pámpano
. We study invariant Willmore-like tori in total spaces of Killing submersions. In particular, using a relation with elastic curves with potentials in the base surfaces, we analyze Willmore tori in total spaces of Killing submersions. Finally, we apply our findings to construct foliations of these total spaces by constant mean curvature Willmore tori.
{"title":"Willmore-Like Energies and Elastic Curves with Potential","authors":"Á. Pámpano","doi":"10.7546/giq-21-2020-232-241","DOIUrl":"https://doi.org/10.7546/giq-21-2020-232-241","url":null,"abstract":". We study invariant Willmore-like tori in total spaces of Killing submersions. In particular, using a relation with elastic curves with potentials in the base surfaces, we analyze Willmore tori in total spaces of Killing submersions. Finally, we apply our findings to construct foliations of these total spaces by constant mean curvature Willmore tori.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2020-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71196357","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-20-2019-246-254
V. Pulov, M. Hadzhilazova, I. Mladenov
Here we consider an interesting class of free of bending deformations of thin axial symmetric shells subjected to uniform normal pressure. The meridional kμ and the parallel kπ principal curvatures of the middle surfaces of such shells obey the non-linear relationship kμ = 2ak π + 3kπ , a = const. These non-bending shells depend on two arbitrary parameters, which are the principal radii rμ and rπ of some fixed parallel of the shell. Besides, these surfaces have no closed form description in elementary functions. Our principle aim here is to present such a parameterization of the whole class of non-bending closed surfaces by making use of the canonical forms of the elliptic integrals. The obtained explicit formulas are then applied for the derivation of the basic geometrical characteristics of these surfaces. MSC : 74K25, 74A10, 53A04, 53A05, 33E05
{"title":"Deformations Without Bending: Explicit Examples","authors":"V. Pulov, M. Hadzhilazova, I. Mladenov","doi":"10.7546/giq-20-2019-246-254","DOIUrl":"https://doi.org/10.7546/giq-20-2019-246-254","url":null,"abstract":"Here we consider an interesting class of free of bending deformations of thin axial symmetric shells subjected to uniform normal pressure. The meridional kμ and the parallel kπ principal curvatures of the middle surfaces of such shells obey the non-linear relationship kμ = 2ak π + 3kπ , a = const. These non-bending shells depend on two arbitrary parameters, which are the principal radii rμ and rπ of some fixed parallel of the shell. Besides, these surfaces have no closed form description in elementary functions. Our principle aim here is to present such a parameterization of the whole class of non-bending closed surfaces by making use of the canonical forms of the elliptic integrals. The obtained explicit formulas are then applied for the derivation of the basic geometrical characteristics of these surfaces. MSC : 74K25, 74A10, 53A04, 53A05, 33E05","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"84 1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76113834","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-20-2019-227-238
Á. Pámpano
. We variationally characterize the profile curves of rotational linear Weingarten surfaces as planar p-elastic curves. Moreover, by evolving these planar p-elasticae under the binormal flow with prescribed velocity, we describe a procedure to construct all rotational linear Weingarten surfaces, locally. Finally, we apply our findings to two well-known family of surfaces.
{"title":"Planar p-Elasticae and Rotational Linear Weingarten Surfaces","authors":"Á. Pámpano","doi":"10.7546/giq-20-2019-227-238","DOIUrl":"https://doi.org/10.7546/giq-20-2019-227-238","url":null,"abstract":". We variationally characterize the profile curves of rotational linear Weingarten surfaces as planar p-elastic curves. Moreover, by evolving these planar p-elasticae under the binormal flow with prescribed velocity, we describe a procedure to construct all rotational linear Weingarten surfaces, locally. Finally, we apply our findings to two well-known family of surfaces.","PeriodicalId":53425,"journal":{"name":"Geometry, Integrability and Quantization","volume":"12 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2019-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"90166959","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：