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9 pages, 428 KiB

Open AccessArticle

Nonrelativistic Quantum Mechanical Problem for the Cornell Potential in Lobachevsky Space

by Laszlo Jenkovszky, Yurii Andreevich Kurochkin, N. D. Shaikovskaya and Vladimir Olegovich Soloviev

Universe 2024, 10(2), 76; https://doi.org/10.3390/universe10020076 - 5 Feb 2024

Cited by 2 | Viewed by 1588

In Friedmann–Lobachevsky space-time with a radius of curvature slowly varying over time, we study numerically the problem of motion of a particle moving in the Cornell potential. The mass of the particle is taken to be a reduced mass of the charmonium system. In contrast to the similar problem in flat space, in Lobachevsky space the Cornell potential has a finite depth and, as a consequence, the number of bound states of the system is finite and motion with a continuum energy spectrum is also possible. In this paper, we study the bound states as well as the scattering states of the system. Full article

(This article belongs to the Special Issue The Friedmann Cosmology: A Century Later)

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17 pages, 7913 KiB

Open AccessArticle

Toward Interactions through Information in a Multifractal Paradigm

by Maricel Agop, Alina Gavriluț, Claudia Grigoraș-Ichim, Ștefan Toma, Tudor-Cristian Petrescu and Ștefan Andrei Irimiciuc

Entropy 2020, 22(9), 987; https://doi.org/10.3390/e22090987 - 4 Sep 2020

Cited by 3 | Viewed by 2548

Abstract

In a multifractal paradigm of motion, Shannon’s information functionality of a minimization principle induces multifractal–type Newtonian behaviors. The analysis of these behaviors through motion geodesics shows the fact that the center of the Newtonian-type multifractal force is different from the center of the multifractal trajectory. The measure of this difference is given by the eccentricity, which depends on the initial conditions. In such a context, the eccentricities’ geometry becomes, through the Cayley–Klein metric principle, the Lobachevsky plane geometry. Then, harmonic mappings between the usual space and the Lobachevsky plane in a Poincaré metric can become operational, a situation in which the Ernst potential of general relativity acquires a classical nature. Moreover, the Newtonian-type multifractal dynamics, perceived and described in a multifractal paradigm of motion, becomes a local manifestation of the gravitational field of general relativity. Full article

(This article belongs to the Special Issue Ring, Phases, Self-Similarity, Disorder, Entropy, Information)

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28 pages, 636 KiB

Open AccessFeature PaperArticle

Construction of Fullerenes and Pogorelov Polytopes with 5-, 6- and one 7-Gonal Face

by Nikolai Erokhovets

Symmetry 2018, 10(3), 67; https://doi.org/10.3390/sym10030067 - 15 Mar 2018

Cited by 8 | Viewed by 4573

Abstract

A Pogorelov polytope is a combinatorial simple 3-polytope realizable in the Lobachevsky (hyperbolic) space as a bounded right-angled polytope. These polytopes are exactly simple 3-polytopes with cyclically 5-edge connected graphs. A Pogorelov polytope has no 3- and 4-gons and may have any prescribed [...] Read more.

k \geq 7

. Any simple polytope with only 5-, 6- and at most one 7-gon is Pogorelov. For any other prescribed numbers of k-gons,

k \geq 7

, we give an explicit construction of a Pogorelov and a non-Pogorelov polytope. Any Pogorelov polytope different from k-barrels (also known as Löbel polytopes, whose graphs are biladders on

2 k

vertices) can be constructed from the 5- or the 6-barrel by cutting off pairs of adjacent edges and connected sums with the 5-barrel along a 5-gon with the intermediate polytopes being Pogorelov. For fullerenes, there is a stronger result. Any fullerene different from the 5-barrel and the

(5, 0)

-nanotubes can be constructed by only cutting off adjacent edges from the 6-barrel with all the intermediate polytopes having 5-, 6- and at most one additional 7-gon adjacent to a 5-gon. This result cannot be literally extended to the latter class of polytopes. We prove that it becomes valid if we additionally allow connected sums with the 5-barrel and 3 new operations, which are compositions of cutting off adjacent edges. We generalize this result to the case when the 7-gon may be isolated from 5-gons. Full article

(This article belongs to the Special Issue Mathematical Crystallography)

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5 pages, 209 KiB

Open AccessConference Report

Geometry of Bigravity

by Vladimir Soloviev

Universe 2018, 4(2), 19; https://doi.org/10.3390/universe4020019 - 31 Jan 2018

Cited by 1 | Viewed by 2517

Abstract

The non-Euclidean geometry created by Bolyai, Lobachevsky and Gauss has led to a new physical theory—general relativity. In due turn, a correct mathematical treatment of the cosmological problem in general relativity has led Friedmann to a discovery of dynamical equations for the universe. And now, after almost a century of theoretical and experimental research, cosmology has a status of the most rapidly developing fundamental science. New challenges here are problems of dark energy and dark matter. As a result, a lot of modifications of general relativity appear recently. The bigravity is one of them, constructed with a couple of interacting space–time metrics accompanied by some coupling to matter. We discuss here this approach and different kinds of the coupling. Full article

(This article belongs to the Special Issue BGL2017: 10th Bolyai-Gauss-Lobachevsky Conference on Non-Euclidean Geometry and Its Applications)

20 pages, 320 KiB

Open AccessArticle

Fractal Information by Means of Harmonic Mappings and Some Physical Implications

by Maricel Agop, Alina Gavriluţ, Viorel Puiu Păun, Dumitru Filipeanu, Florin Alexandru Luca, Constantin Grecea and Liliana Topliceanu

Entropy 2016, 18(5), 160; https://doi.org/10.3390/e18050160 - 26 Apr 2016

Cited by 9 | Viewed by 4668

Abstract

Considering that the motions of the complex system structural units take place on continuous, but non-differentiable curves, in the frame of the extended scale relativity model (in its Schrödinger-type variant), it is proven that the imaginary part of a scalar potential of velocities can be correlated with the fractal information and, implicitly, with a tensor of “tensions”, which is fundamental in the construction of the constitutive laws of material. In this way, a specific differential geometry based on a Poincaré-type metric of the Lobachevsky plane (which is invariant to the homographic group of transformations) and also a specific variational principle (whose field equations represent an harmonic map from the usual space into the Lobachevsky plane) are generated. Moreover, fractal information (which is made explicit at any scale resolution) is produced, so that the field variables define a gravitational field. This latter situation is specific to a variational principle in the sense of Matzner–Misner and to certain Ernst-type field equations, the fractal information being contained in the material structure and, thus, in its own space associated with it. Full article

(This article belongs to the Special Issue Wavelets, Fractals and Information Theory II)

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