#### Geometry, Integrability and Quantization

Journal Information
ISSN / EISSN: 13143247 / 23677147
Total articles ≅ 129

#### Latest articles in this journal

Ramon González Calvet
Published: 1 January 2022
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.
Takeshi Hirai
Published: 1 January 2022
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.
Published: 1 January 2022
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.
Published: 1 January 2022
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.
Ramon González Calvet, Marin Drinov Academic Publishing House
Published: 1 January 2022
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.
Daniele Corradetti, Marin Drinov Academic Publishing House, Alessio Marrani, David Chester, Raymond Aschheim
Published: 1 January 2022
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.
Jumpei Gohara, Marin Drinov Academic Publishing House, Yuji Hirota, Keisui Ino, Akifumi Sako
Published: 1 January 2022
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.
Dragan S. Djordjevic
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-13-32

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.
Jan Krizek, , Patrik Peska, Lenka Ryparova
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-136-141

Abstract:
In the paper we study the extremals and isoperimetric extremals of the rotations in the plane. We found that extremals of the rotations in the plane are arbitrary curves. By studying the Euler-Poisson equations for extended variational problems, we found that the isoperimetric extremals of the rotations in the Euclidian plane are straight lines.