Grzegorz Biernat

Contribution to the Jacobian Conjecture: Polynomial Mapping Having Two Zeros at Infinity

This article contains the theorems concerning the algebraic dependence of polynomial mappings with the constant Jacobian having two zeros at infinity. The work is related to the issues of the classical Jacobian Conjecture. This hypothesis affirm that the polynomial mapping of two complex variables with constant non-zero Jacobian is invertible. The Jacobian Conjecture is equivalent to the fact that polynomial mappings with constant non-zero Jacobian do not have two zeros at infinity, therefore it is equivalent to the two theorems given in the work. The proofs of these theorems proceeds by induction.

Tensor Interpolation of Tribological Wear in Ionitriding of 316L Steel

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.