#### Results in Journal Geometry Integrability and Quantization: 129

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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.
Ivaïlo M. Mladenov, Marin Drinov Academic Publishing House
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.
Published: 1 January 2021
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.
Tsukasa Takeuchi
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-263-273

Abstract:
Certain ways of characterizing integrable systems with $(1,1)$-tensor field have been investigated, so far. For example, recursion operators and Haantjes operators are known. We show that geometrical examples of four- or six-dimensional symplectic Haantjes manifolds and recursion operators for several Hamiltonian systems. Through these examples, we consider the relation between recursion operators and Haantjes operators.
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-253-262

Abstract:
Mathematicians have shown interest in manifolds endowed with several distributions, e.g., webs composed of different regular foliations and multiply warped products, as well as distributions having variable dimensions (e.g., singular Riemannian foliations). In this paper, we extend our previous study of the mixed scalar curvature of two orthogonal singular distributions for the case of $k>2$ singular (or regular) pairwise orthogonal distributions, prove an integral formula with this kind of curvature, and illustrate it by characterizing autoparallel singular distributions.
Ciprian Sorin Acatrinei
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-35-42

Abstract:
We extend the Feynman derivation of the Maxwell-Lorentz equations to the case in which coordinates do not commute, adding significantly to previous results. New dynamics is pinned down precisely both at the level of the homogeneous equations and for the Lorentz force, for which a complete derivation is given for the first time.
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-154-164

Abstract:
Despite of their importance, the symplectic groups are not so popular like orthogonal ones as they deserve. The only explanation of this fact seems to be that their algebras can not be described so simply. While in the case of the orthogonal groups they are just the anti-symmetric matrices, those of the symplectic ones should be split in four blocks that have to be specified separately. It turns out however that in some sense they can be presented by the even dimensional symmetric matrices. Here, we present such a scheme and illustrate it in the lowest possible dimension via the Cayley map. Besides, it is proved that by means of the exponential map all such matrices generate genuine symplectic matrices.
Makoto Nakamura, Hiroshi Kakuhata, Kouichi Toda
Published: 1 January 2021
Geometry, Integrability and Quantization; https://doi.org/10.7546/giq-22-2021-188-198

Abstract:
Noncommutative phase space of arbitrary dimension is discussed. We introduce momentum-momentum noncommutativity in addition to co-ordinate-coordinate noncommutativity. We find an exact form for the linear transformation which relates a noncommutative phase space to the corresponding ordinary one. By using this form, we show that a noncommutative phase space of arbitrary dimension can be represented by the direct sum of two-dimensional noncommutative ones. In two-dimension, we obtain the transformation which relates a noncommutative phase space to commutative one. The transformation has the Lorentz transformation-like forms and can also describe the Bopp's shift.
Naoko Yoshimi, Akira Yoshioka
Published: 1 January 2021
Geometry, Integrability and Quantization; https://doi.org/10.7546/giq-22-2021-286-300

Abstract:
For given $k$ bodies of collinear central configuration of Newtonian $k$-body problem, we ask whether one can add another body on the line without changing the configuration and motion of the initial bodies so that the total $k+1$ bodies provide a central configuration. The case $k=4$ is analyzed. We study the inverse problem of five bodies and obtain a global explicit formula. Then using the formula we find there are five possible positions of the added body and for each case the mass of the added body is zero. We further consider to deform the position of the added body without changing the positions of the initial four bodies so that the total five bodies are in a state of central configuration and the mass of the added body becomes positive. For each solution above, we find such a deformation of the position of the added body in an explicit manner starting from the solution.
Akira Yoshioka
Published: 1 January 2021
Geometry, Integrability and Quantization; https://doi.org/10.7546/giq-22-2021-301-307

Abstract:
Star product for functions of one variable is given. A deformation of the Mittag-Leffler functions is suggested by means of the star product.
Tihomir Valchev
Published: 1 January 2021
Geometry, Integrability and Quantization; https://doi.org/10.7546/giq-22-2021-274-285

Abstract:
This work is dedicated to systems of matrix nonlinear evolution equations related to Hermitian symmetric spaces of the type $\mathbf{A.III}$. The systems under consideration generalize the $1+1$ dimensional Heisenberg ferromagnet equation in the sense that their Lax pairs are linear bundles in pole gauge like for the original Heisenberg model. Here we present certain local and nonlocal reductions. A local integrable deformation and some of its reductions are discussed as well.
Laarni B. Natividad, Job A. Nable
Published: 1 January 2021
Geometry, Integrability and Quantization; https://doi.org/10.7546/giq-22-2021-209-218

Abstract:
In this work, we perform exact and concrete computations of star-product of functions on the Euclidean motion group in the plane, and list its $C$-star-algebra properties. The star-product of phase space functions is one of the main ingredients in phase space quantum mechanics, which includes Weyl quantization and the Wigner transform, and their generalizations. These methods have also found extensive use in signal and image analysis. Thus, the computations we provide here should prove very useful for phase space models where the Euclidean motion groups play the crucial role, for instance, in quantum optics.
Daisy A. Romeo, Job A. Nable
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-242-252

Abstract:
This work presents quantization of time of arrival functions using generalized Stratonovich-Weyl quantization. We take into account the ordering problems involved, mainly the Born-Jordan and the symmetric ordering schemes. We call attention to the combination of the group theoretic methods usually employed in Weyl quantization with the implementation of different ordering schemes via integral kernel factors. It is possible to, and we do, apply the Pegg-Barnett method to the quantization of time to address physical issues such as boundedness and self-adjointness.
Abdigappar Narmanov, Xurshid Sharipov
Published: 1 January 2021
Geometry, Integrability and Quantization; https://doi.org/10.7546/giq-22-2021-199-208

Abstract:
Subject of present paper is the geometry of foliation defined by submersions on complete Riemannian manifold. It is proven foliation defined by Riemannian submersion on the complete manifold of zero sectional curvature is total geodesic foliation with isometric leaves. Also it is shown level surfaces of metric function are conformally equivalent.
, Alexey Kozhedub
Published: 1 January 2021
Geometry, Integrability and Quantization; https://doi.org/10.7546/giq-22-2021-165-187

Abstract:
We developed the theory of superalgebraic spinors, which is based on the use of Grassmann densities and derivatives with respect to them in a pseudo-continuous space of momenta. The algebra that they form corresponds to the algebra of second quantization of fermions. We have constructed a vacuum state vector and have shown that it is symmetric with respect to $P$, $CT$ and $CPT$ transformations. Operators $C$ and $T$ transforms the vacuum into an alternative one. Therefore, time inversion $T$ and charge conjugation $C$ cannot be exact symmetries of the spinors.
Published: 1 January 2021
Geometry, Integrability and Quantization; https://doi.org/10.7546/giq-22-2021-219-241

Abstract:
As the title itself suggests here we are presenting extremely reach two/three parametric families of non-bending rotational surfaces in the three dimensional Euclidean space and provide the necessary details about their natural classifications and explicit parameterizations. Following the changes of the relevant parameters it is possible to trace out the evolution'' of these surfaces and even visualize them through their topological transformations. Many, and more deeper questions about their metrical properties, mechanical applications, etc. are left for future explorations.
Miroslav Kures
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-142-153

Abstract:
A detailed derivation of the jet composition in local coordinates for jet (differential) groups is presented. A suitable faithful representation in matrix groups is demonstrated. Furthermore, Toupin subgroups which occur in continuum mechanics are demonstrated as an example in which representations can be used effectively.
Volodymyr Berezovski, Yevhen Cherevko, Svitlana Leshchenko,
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-78-87

Abstract:
In the paper we consider almost geodesic mappings of the first type of spaces with affine connections onto generalized 2-Ricci-symmetric spaces. The main equations for the mappings are obtained as a closed system of linear differential equations of Cauchy type in the covariant derivatives. The obtained result extends an amount of research produced by Sinyukov, Berezovski and Mike\v{s}.
Adina V. Crisan, Ion V. Vancea
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-107-120

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.
Rutwig Campoamor-Stursberg
Published: 1 January 2021
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.
Paolo Aniello
Published: 1 January 2021
Geometry, Integrability and Quantization, Volume 22; https://doi.org/10.7546/giq-22-2021-64-77

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.
Edward Anderson
Published: 1 January 2021
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.
Rafael Lopez, Marin Drinov Academic Publishing House
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-13-54

Nikolay Dukov, Marin Drinov Academic Publishing House,
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-127-137

Kuralay Yesmakhanova, Marin Drinov Academic Publishing House, Zhanar Umurzakhova, Gaukhar Shaikhova
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-320-327

Zoya Tsoneva, Marin Drinov Academic Publishing House
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-280-290

Svilen Popov, Marin Drinov Academic Publishing House, Vassil Vassilev, Daniel Dantchev
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-242-250

Josef Mikes, Marin Drinov Academic Publishing House, Patrik Peska, Lenka Ryparova
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-181-185

Detelina Kamburova, Marin Drinov Academic Publishing House, Diana Nedelcheva
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-163-169

Prolet Deneva, Marin Drinov Academic Publishing House,
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-118-126

Jumpei Gohara, Marin Drinov Academic Publishing House, Yuji Hirota, Akifumi Sako
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-138-148

Eugenio Aulisa, Marin Drinov Academic Publishing House, Anthony Gruber, Magdalena Toda, Hung Tran
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-57-65

Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-251-264

Ismet Yurduşen, Marin Drinov Academic Publishing House
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-334-348

Rutwig Campoamor-Stursberg, Marin Drinov Academic Publishing House
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-75-88

Viorel Laurentiu Cartas, Marin Drinov Academic Publishing House
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-89-99

Vassil Vassilev, Marin Drinov Academic Publishing House, Daniel Dantchev, Svilen Popov
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-302-309

Clementina D. Mladenova, Marin Drinov Academic Publishing House, Ivaïlo M. Mladenov
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-186-220

Diana K. Nedelcheva, Marin Drinov Academic Publishing House
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-221-231

Ildefonso Castro, Marin Drinov Academic Publishing House, Ildefonso Castro-Infantes, Jesus Castro-Infantes
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-100-117

Iliyan Boychev, Marin Drinov Academic Publishing House
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-66-74

Gulmira Yergaliyeva, Marin Drinov Academic Publishing House, Tolkynay Myrzakul, Gulgassyl Nugmanova, Ratbay Myrzakulov
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-310-319

Gergana Spasova, Marin Drinov Academic Publishing House
Published: 1 January 2020
Geometry, Integrability and Quantization, Volume 21; https://doi.org/10.7546/giq-21-2020-272-279

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