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Section
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Mathematics
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Title
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Embedding of phenomenologically symmetric geometries of two sets of rank $(N,M)$ into phenomenologically symmetric geometries of two sets of rank $(N+1,M)$
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Author(-s)
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Kyrov V.A.a
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Affiliations
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Gorno-Altaisk State Universitya
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Abstract
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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.
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Keywords
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phenomenologically symmetric geometry of two sets, metric function, differential equation
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UDC
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517.912, 514.1
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MSC
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35F05, 39B05, 51P99
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DOI
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10.20537/vm170104
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Received
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31 October 2016
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Language
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Russian
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Citation
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Kyrov V.A. Embedding of phenomenologically symmetric geometries of two sets of rank $(N,M)$ into phenomenologically symmetric geometries of two sets of rank $(N+1,M)$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 1, pp. 42-53.
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References
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