Abstract. This paper contains the application of the Finite Difference Method in the two-dimensional Fourier equation using Robin's boundary condition (the third boundary condition).

The research results for polynomial tensor interpolation of tribological wear of austenitic 316L steel are analyzed for the first time in this paper. A tribological study was carried out on the tester T-05, both for samples after the process of glow discharge nitriding and for samples of 316L steel in the initial state [1, 2]. The influence of the nitriding parameters on the weight loss of the samples during the abrasion test and the friction coefficient of the abrasion pair were determined. In this paper the actual results which were obtained so far during the research of tribological wear in an interpolative area were put together with the theoretically determined and experimentally verified real research results for tribological wear in the area beyond the scope of interpolation. It was determined that each of the selected nitriding parameters significantly affects abrasive wear resistance of 316L steel, at the same time, in the nitriding process, as time and temperature increase the resistance to wear abrasion becomes better. The mathematical model for planning of experiments with different glow discharge nitriding processes combined with polynomial interpolation of the obtained research results of many variable functions will enable to choose the nitriding process parameters to obtain the assumed and required property of useful nitrified elements. The developed computer program enables multidimensional interpolation of the research results and is a universal tool which can be used in any research field.

Abstract. In the paper a nontrivial example of non-Keller mapping is considered. It is shown that the Jacobian of rare mapping, having one zero at infinity, being constant must vanish.

Abstract. The work is a continuation of the method of calculating the determinant of the block matrix in the two-dimensional case. In this paper we use the Finite Differences Method and the symmetric polynomials.