Estimation for the region of attraction of nonlinear difference equations

Summary. We consider nonlinear systems of difference equations with and , where A is any matrix, is a continuous vector-function, and is a numeral parameter. The spectrum of A belongs to the unit circle . We give the estimations for the region of attraction and the speed of convergence solutions to the zero solution of the systems. We indicate a set M such that for solutions of the system with parameter the limit is true. Received April 7, 2000 / Revised version received January 8, 2001 / Published online November 15, 2001     

Systems of differential equations with periodic coefficients

We consider linear systems of differential equations with periodic coefficients. We prove the solvability of nonhomogeneous systems in the Sobolev space W ₂ ¹ (R) and establish estimates for the solutions. This result implies a perturbation theorem for the exponential dichotomy of systems of differential equations with periodic coefficients.     

On estimates of solutions to systems of differential equations of neutral type with periodic coefficients

We study the systems of differential equations of neutral type with periodic coefficients. We establish sufficient conditions for the asymptotic stability of the zero solution and obtain estimates for solutions which characterize the decay rate at infinity.     

Additive coloring of CaF<Subscript>2</Subscript> optical ceramic

The specificity of additive coloring of CaF₂ optical ceramic (formation of color centers in it and photothermochemical transformation of these centers in colored ceramic samples) has been considered. Under the same coloring conditions, this process occurs more slowly in ceramics rather than in crystals; at the same time, the limiting concentration of color centers that can be introduced into ceramics is much higher. The photothermochemical transformations of color centers in crystals and ceramics, which occur under illumination at different wavelengths and upon heating, have been studied. The specific features of introduction of color centers into ceramic and their transformation under illumination and heating are likely to be related to the mass twinning of ceramic grains.     

Crystal structure and electrical properties of GdxMn1 ? xS and TixMn1 ? xSe solid solutions

Regions of the existence of sulfide Gd _x Mn_{1 − x} S and selenide Ti _x Mn_{1 − x} Se solid solutions have been identified. Their electrical and thermoelectric properties have been studied in the temperature range 80–900 K. It has been established that the substitution of Gd²⁺ and Ti²⁺ ions for Mn²⁺ cations initiates reversal of the type of charge carrier with respect to the starting compounds MnS and MnSe. The cation substitution in solid solutions brings about a change from the hole conduction (α > 0) characteristic of the manganese monosulfide and monoselenide to the electronic conduction (α < 0).