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## Archive of Issues

Russia Gorno-Altaisk
Year
2017
Volume
27
Issue
1
Pages
42-53
 Section Mathematics Title Embedding of phenomenologically symmetric geometries of two sets of rank $(N,M)$ into phenomenologically symmetric geometries of two sets of rank $(N+1,M)$ Author(-s) Kyrov V.A.a Affiliations Gorno-Altaisk State Universitya 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 UDC 517.912, 514.1 MSC 35F05, 39B05, 51P99 DOI 10.20537/vm170104 Received 31 October 2016 Language Russian Citation 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. References Kulakov Yu.I. The one principle underlying classical physics, Soviet Physics Doklady, 1971, vol. 15, no. 7, pp. 666-668. Mikhailichenko G.G. Two-dimensional geometry, Soviet Mathematics. Doklady, 1981, vol. 24, no. 2, pp. 346-348. Mikhailichenko G.G. The solution of functional equations in the theory of physical structures, Soviet Mathematics. Doklady, 1972, vol. 13, no. 5, pp. 1377-1380. Kyrov V.A. Embedding of phenomenologically symmetric geometries of two sets of the rank $(N,2)$ into phenomenologically symmetric geometries of two sets of the rank $(N+1,2)$, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 2016, vol. 26, issue 3, pp. 312-323 (in Russian). DOI: 10.20537/vm160302 Kyrov V.A. Functional equations in pseudo-Euclidean geometry, Sib. Zh. Ind. Mat., 2010, vol. 13, no. 4, pp. 38-51 (in Russian). Kyrov V.A. Functional equations in symplectic geometry, Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk, 2010, vol. 16, no. 2, pp. 149-153 (in Russian). Kyrov V.A. On some class of functional-differential equation, Vestnik Samarskogo Gosudarstvennogo Tekhnicheskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, 2012, vol. 26, no. 1, pp. 31-38 (in Russian). DOI: 10.14498/vsgtu986 Mikhailichenko G.G. Gruppovaya simmetriya fizicheskikh struktur (The group symmetry of physical structures), Barnaul: Barnaul State Pedagogical University, 2003, 204 p. Elsgolts L.E. Differentsial'nye uravneniya i variatsionnoe ischislenie (Differential equations and the calculus of variations), Moscow: Nauka, 1969, 424 p. Mikhailichenko G.G. Functional equations in geometry of two sets, Russian Mathematics, 2010, vol. 54, issue 7, pp. 56-63. DOI: 10.3103/S1066369X10070066 Full text