Geometry Integrability and Quantization

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ISSN / EISSN : 1314-3247 / 2367-7147
Total articles ≅ 129
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Ramon González Calvet
Geometry, Integrability and Quantization, Volume 24; https://doi.org/10.7546/giq-24-2022-39-64

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.
Ramon González Calvet, Marin Drinov Academic Publishing House
Geometry, Integrability and Quantization, Volume 23; https://doi.org/10.7546/giq-23-2022-1-38

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.
Jumpei Gohara, Marin Drinov Academic Publishing House, Yuji Hirota, Keisui Ino, Akifumi Sako
Geometry, Integrability and Quantization, Volume 23, pp 59-73; https://doi.org/10.7546/giq-23-2022-59-74

Abstract:
We discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finite-dimensional vector spaces over a fixed field. For a fixed chain complex with zero maps as an object, a chain map from the object to another chain complex is defined, and the chain map introduce a mapping cone. The fixed object is isomorphic to the (co)homology groups of the codomain of the chain map if and only if the chain map is injective to the cokernel of differentials of the codomain chain complex and the mapping cone is homotopy equivalent to zero. On the other hand, it was found that the fixed object can be regarded as a categorified eigenvalue of the chain complex in the context of the categorical diagonalization, recently. It is found that (co)homology groups are constructed as the eigenvalue of a chain complex.
Clementina D. Mladenova,
Geometry, Integrability and Quantization, Volume 24; https://doi.org/10.7546/giq-24-2022-65-83

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 $2\times 2$ matrix. Last, but not least, we have derived their explicit parameterizations.
Daniele Corradetti, Marin Drinov Academic Publishing House, Alessio Marrani, David Chester, Raymond Aschheim
Geometry, Integrability and Quantization, Volume 23; https://doi.org/10.7546/giq-23-2022-39-57

Abstract:
In this work we present a useful way to introduce the octonionic projective and hyperbolic plane $\mathbb{O}P^{2}$ through the use of Veronese vectors. Then we focus on their relation with the exceptional Jordan algebra $\mathfrak{J}_{3}^{\mathbb{O}}$ and show that the Veronese vectors are the rank-one elements of the algebra. We then study groups of motions over the octonionic plane recovering all real forms of $\text{G}_{2}$, $\text{F}_{4}$ and $\text{E}_{6}$ groups and finally give a classification of all octonionic and split-octonionic planes as symmetric spaces.
Takeshi Hirai
Geometry, Integrability and Quantization, Volume 24; https://doi.org/10.7546/giq-24-2022-1-37

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.
Ivaïlo M. Mladenov, Marin Drinov Academic Publishing House
Geometry, Integrability and Quantization, Volume 23; https://doi.org/10.7546/giq-23-2022-75-98

Abstract:
A plethora of new explicit formulas that parameterize all three types of the Cassinian ovals via elliptic and circular functions are derived from the first principles. These formulas allow a detailed study of the geometry of the Cassinian curves which is persuaded to some extent here. Conversion formulas relating various sets of the geometrical parameters are presented. On the way some interesting relationships satisfied by the Jacobian elliptic functions were found. Besides, a few general identities between the complete elliptic integrals of the first and second kind were also established. An explicit universal formula for the total area within the Cassinians which is valid for all types of them is derived. Detailed derivation of the formulas for the volumes of the bodies obtained as a result of rotations of the Cassinian ovals is presented.
Edward Anderson
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-43-63

Abstract:
The problem of time - a foundational question in quantum gravity - is due to conceptual gaps between GR and physics' other observationally-confirmed theories. Its multiple facets originated with Wheeler-DeWitt-Dirac over 50~years ago. They were subsequently classified by Kucha\v{r}-Isham, who argued that most of the problem is facet interferences and posed the question of how to order the facets. We show the local classical level facets are two copies of Lie theory with a Wheelerian two-way route therebetween. This solves facet ordering and facet interference. Closure by a Lie algorithm generalization of Dirac's algorithm is central.
Kensaku Kitada
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-121-135

Abstract:
We investigate a version of Yang-Mills theory by means of general connections. In order to deduce a basic equation, which we regard as a version of Yang-Mills equation, we construct a self-action density using the curvature of general connections. The most different point from the usual theory is that the solutions are given in pairs of two general connections. This enables us to get nontrivial solutions as general connections. Especially, in the quaternionic Hopf fibration over four-sphere, we demonstrate that there certainly exist nontrivial solutions, which are made by twisting the well-known BPST anti-instanton.
Rutwig Campoamor-Stursberg
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-88-106

Abstract:
Various structural properties of semidirect sums of the rotation Lie algebra of rank one and an Abelian algebra described in terms of real representations with at most two irreducible constituents are obtained. The stability properties of these semidirect sums are studied by means of the cohomological and the Jacobi scheme methods.
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