Solution of a system of functional equations associated with an affine group

R. A. Bogdanova   ; V. A. Kyrov

Vladikavkazskij matematičeskij žurnal, Tome 26 (2024) no. 3, pp. 24-32

Résumé

Solution of the embedding problem for a two-metric phenomenologically symmetric geometry of rank $(3,2)$ with the function $ g (x, y, \xi, \eta) = (g^{1}, g^{2 })= (x\xi+y\ mu,x\eta + y\nu)$ into an affine two-metric phenomenologically symmetric geometry of rank $(4,2)$ with the function $f(x,y,\xi,\eta,\mu,\nu)=(f^{1},f^{2})=(x\xi+y\mu+\rho,x\eta + y\nu+\tau)$ leads to the problem of establishing the existence of non-degenerate solutions to the corresponding system $f(\bar{x},\bar {y},\bar{\xi},\bar{\eta},\bar{\mu},\bar{\nu})=\chi(g(x,y,\xi,\eta),\mu,\nu)$ of two functional equations. This system is solved based on the fact that the functions $g$ and $f$ are previously known. This system is written explicitly as follows: $\bar{x}\bar{\xi }+\bar{y}\bar{\mu}+\bar{\rho}= \chi^{1}(x\xi +y\mu,x\eta+y\nu,\mu,\nu),$ $\bar{x}\bar{\eta }+\bar{y}\bar{\nu }+\bar{\tau}= \chi ^{2}(x\xi+y\mu,x\eta + y\nu,\mu,\nu).$ The main goal of this work is to find a general non-degenerate solution to this system. To solve the problem, we first differentiate with respect to the variables $x$, $y$ and $\xi$, $\eta$, $\mu$, $\nu$, as a result we obtain a system of differential equations with a matrix of coefficients $A$ of the general form. It is proved that the matrix $A$ can be reduced to Jordan form. Then a system of differential equations with such a Jordan matrix is solved. Returning to the original original system of functional equations, we find the additional restrictions. As a result, we arrive at a non-degenerate solution to the original system of functional equations.

Bibliographie

[1] Mikhailichenko, G. G., Group Symmetry of Physical Structures, Barnaul, 2003, 203 pp. (in Russian)

[2] Mikhailichenko, G. G., “Bimetric Physical Structures of Rank $(n+1,2)$”, Siberian Mathematical Journal, 34:3 (1993), 513–522 | DOI

[3] Kulakov, Yu. I., “A Mathematical Formulation of the Theory of Physical Structures”, Siberian Mathematical Journal, 12:5 (1971), 822–824 | DOI

[4] Mikhailichenko, G. G., “The Solution of Functional Equations in the Theory of Physical Structures”, Doklady Akademii Nauk SSSR, 206:5 (1972), 1056–1058 (in Russian)

[5] Kyrov, V. A., “On the Embedding of Two-Dimetric Phenomenologically Symmetric Geometries”, Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika, 2018, no. 56, 5–16 (in Russian) | DOI

[6] Bogdanova, R. A., Mikhailichenko, G. G. and Muradov R. M., “Successive in Rank $(n+ 1, 2)$ Embedding of Dimetric Phenomenologically Symmetric Geometries of Two Sets”, Russian Mathematics, 64:6 (2020), 6–10 | DOI | DOI

[7] Kyrov, V. A., “Nondegenerate Canonical Solutions of Some System of Functional Equations”, Vladikavkaz Mathematical Journal, 24:1 (2022), 44–53 (in Russian) | DOI

[8] Kyrov, V. A. and Mikhailichenko, G. G., “Nondegenerate Canonical Solutions of one System of Functional Equations”, Russian Mathematics, 65:8 (2021), 40–48 | DOI | DOI

[9] Kyrov, V. A. and Mikhailichenko, G. G., “Solving Three Systems of Functional Equations Associated with Complex, Double, and Dual Numbers”, Russian Mathematics, 67:7 (2023), 34–42 | DOI | DOI

[10] Kostrikin, A. I., Introduction to Algebra, Nauka, M., 1977, 495 pp. (in Russian)