On the Exponential Dichotomy on $$\mathbb{R}$$ of Linear Differential Equations in $$\mathbb{R}^n$$

We present new results on the exponential dichotomy on the entire axis of linear differential equations in .     

Truncation method for countable-point boundary-value problems in the space of bounded number sequences

We consider possible methods for the reduction of a countable-point nonlinear boundary-value problem with nonlinear boundary condition on a segment to a finite-dimensional multipoint problem constructed on the basis of the original problem by the truncation method. The results obtained are illustrated by examples.__________Translated from Ukrainskyi Matematychnyi Zhurnal, Vol. 56, No. 9, pp. 1203–1230, September, 2004.     

Fine structure and inverse population in the MnKβ1 spectra of MnxZnyFezO4 ferrites

The electronic structure of Mn_xZn_yFe_zO₄ single-crystal ferrites was investigated. The fine structure of their X-ray spectra was found to correlate with the populations of quantum states. A scheme is suggested for the evolution of the electronic structure with strengthening exchange interaction of valence electrons as the chemical composition of the system becomes more complex. The metal-insulator phase transition is possible in the range of compositions corresponding to densities from ρ=5.097 g/cm³ to ρ=5.125 g/cm³, where manganese changes its charge state.     

Self-oscillations in a flow ideal mixing reactor in the region of multiplicity of stationary equilibrium states: a consecutive reaction

A model of a flow ideal mixing reactor for a consecutive two-stage exothermic first-order reaction was considered. The presence of a region of multiplicity of stationary states was established. This multiplicity region can contain regions of three and five stationary states; the region of five states always appears inside the region of three stationary states. Changes in the type of stationary state stability in each of these regions are analyzed. Numerical calculations of phase trajectories in the regions of stationary state stability and instability were performed. A mechanism of the creation of a stable limiting cycle was suggested for the case of three stationary states.     

On the Thermal Explosion Theory of a Heterogeneous Liquid–Liquid System

In this paper, the regularities of a thermal explosion of a heterogeneous system consisting of two immiscible liquids have been studied. Each phase is a solution of A and B reagents. Reagent B is extracted into a solution of reagent A, where the bimolecular exothermic reaction A + B → Products takes place. It has been shown that an exothermic reaction (combustion regime) continues to proceed in the system at high mass-exchange rates between phases after a thermal explosion. As a result, the maximal temperature may significantly exceed the temperature of the thermal explosion. The critical value of the Semenov parameter decreases with an increase in the mass-exchange rate between phases. In the limited range of values of the distribution coefficient of reagent B between phases, the increase of this coefficient is also accompanied by a decrease in the critical value of the Semenov parameter. The concentration of reagent B in the initial phase decreases monotonically due to its extraction into another phase. However, the equilibrium of the extraction of reagent B can shift, due to the temperature dependence of the distribution coefficient during the reaction. Thus, the time dependence of the concentration of reagent B on may be more complex and can pass through a minimum.     

On the Existence of a Smooth Bounded Semiinvariant Manifold for a Degenerate Nonlinear System of Difference Equations in the Space m

Using the Green–Samoilenko function, we construct a bounded Frechét-differentiable semiinvariant manifold for a nonlinear system of difference equations in a Banach space of bounded number sequences.     

Self-organization of the structure in the Cu60Fe40 composite during deformation-thermal treatment

The deformation-thermal stability of a clusterized amorphous-crystalline structure prepared from a Cu₆₀Fe₄₀ powder mixture at a logarithmic strain e = 4.6 and subjected to isochronous (40 min) annealings at T _a = 200–800°C has been investigated. Periodic changes (ΔT = 300°C) in the order and disorder with a maximum ordering at T _a = 300 and 600°C and a maximum disordering at T _a = 400 and 700°C have been observed. The periodicity of the dominant crystallographic order with a period ΔT = 400°C in the annealing temperature has been revealed for face-centered cubic copper phase planes separated by a singular point at T = 500°C characterized by the dominant body-centered cubic iron phase ordering. It has been shown that the sawtooth shape of the size distribution of strain clusters formed within the crystal structure of deformed samples slowly changes with increasing annealing temperature from exponential (T _a = 200–700°C) to linear (T _a = 800°C). This indicates a high density of internal local distortions in structural units.