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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, Volume 27, Issue 1, Pages 42–53 (Mi vuu567)

MATHEMATICS

Embedding of phenomenologically symmetric geometries of two sets of rank $(N,M)$ into phenomenologically symmetric geometries of two sets of rank $(N+1,M)$

V. A. Kyrov

Gorno-Altaisk State University, ul. Lenkina, 1, Gorno-Altaisk, 649000, Russia

Abstract: In this paper, a classification of phenomenologically symmetric geometries of two sets of rank $(n+1,m)$ with $n\geqslant 2$ and $m\geqslant 3$ is constructed by the method of embedding. The essence of this method is to find the metric functions of phenomenologically symmetric geometries of two high-rank sets by the known phenomenologically symmetric geometries of two sets of a rank which is lower by unity. By the known metric function of the phenomenologically symmetric geometry of two sets of rank $(n+1,n)$, we find the metric function of the phenomenologically symmetric geometry of rank $(n+1,n+1)$, on the basis of which we find later the metric function of the phenomenologically symmetric geometry of rank $(n+1,n+2)$. Then we prove that there is no embedding of the phenomenologically symmetric geometry of two sets of rank $(n+1,n+2)$ in the phenomenologically symmetric geometry of two sets of rank $(n+1,n+3)$. At the end of the paper, we complete the classification using the mathematical induction method and taking account of the symmetry of a metric function with respect to the first and the second argument. To solve the problem, we write special functional equations, which reduce to the well-known differential equations.

Keywords: phenomenologically symmetric geometry of two sets, metric function, differential equation.

DOI: https://doi.org/10.20537/vm170104

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Bibliographic databases:

UDC: 517.912, 514.1
MSC: 35F05, 39B05, 51P99

Citation: V. A. Kyrov, “Embedding of phenomenologically symmetric geometries of two sets of rank $(N,M)$ into phenomenologically symmetric geometries of two sets of rank $(N+1,M)$”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 27:1 (2017), 42–53

Citation in format AMSBIB
\Bibitem{Kyr17}
\by V.~A.~Kyrov
\paper Embedding of phenomenologically symmetric geometries of two sets of rank $(N,M)$ into phenomenologically symmetric geometries of two sets of rank $(N+1,M)$
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2017
\vol 27
\issue 1
\pages 42--53
\mathnet{http://mi.mathnet.ru/vuu567}
\crossref{https://doi.org/10.20537/vm170104}
\elib{http://elibrary.ru/item.asp?id=28808554}

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Kyrov, “Analiticheskii metod vlozheniya mnogomernykh psevdoevklidovykh geometrii”, Sib. elektron. matem. izv., 15 (2018), 741–758
2. V. A. Kyrov, G. G. Mikhailichenko, “Vlozhenie additivnoi dvumetricheskoi fenomenologicheski simmetrichnoi geometrii dvukh mnozhestv ranga $(2,2)$ v dvumetricheskie fenomenologicheski simmetrichnye geometrii dvukh mnozhestv ranga $(3,2)$”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:3 (2018), 305–327
3. V. A. Kyrov, “Analiticheskoe vlozhenie nekotorykh dvumernykh geometrii maksimalnoi podvizhnosti”, Sib. elektron. matem. izv., 16 (2019), 916–937
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