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DOI: 10.46698/d7752-5993-6789-y Solution of a System of Functional Equations Associated with an Affine Group
Abstract:
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.
Keywords: geometry of two sets, Jordan form of a matrix, system of functional equations, system of differential equations
Language: Russian
For citation: Bogdanova, R. A. and Kyrov, V. A. Solution of a System of Functional Equations Associated with an Affine Group, Vladikavkaz Math. J., 2024, vol. 26, no. 3, pp. 24-32 (in Russian).
DOI 10.46698/d7752-5993-6789-y
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