Asymptotic expansion of a semi-Markov random evolution

We determine the regular and singular components of the asymptotic expansion of a semi-Markov random evolution and show the regularity of boundary conditions. In addition, we propose an algorithm for finding initial conditions for t = 0 in explicit form using the boundary conditions for the singular component of the expansion. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 9, pp. 1234–1248, September, 2006.     

Modeling of the dynamics of a well-stirred reactor with parallel exothermic reactions

The dynamics of a well-stirred reactor in which two parallel exothermic first-order reactions occur was modeled. Phase portraits of the system were plotted, the possibility of realization of oscillatory regimes was revealed, and the stability of the steady states involved was examined. Two types of limit cycle extinction, smooth and sharp, were demonstrated to exist.     

Asymptotic two-phase solitonlike solutions of the singularly perturbed Korteweg-de Vries equation with variable coefficients

We propose an algorithm for the construction of asymptotic two-phase solitonlike solutions of the Korteweg-de Vries equation with a small parameter at the higher derivative. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 3, pp. 388–397, March, 2008.     

Phase transitions in the atomic, electronic, and magnetic subsystems of LaSrMnO films versus the synthesis temperature

The evolution of the cluster structure in amorphous LaSrMnO films as synthesis temperature T _s increases from 20 to 300°C is considered. Two order-disorder phase transitions with different scale parameters are observed. One of them, the aggregation of disordered atoms into small (～20 Å) amorphous clusters at T _s = 100°C, shows up as a sharp increase in the intensity of diffuse X-ray scattering (diffuse halo 1) with a simultaneous suppression of incoherent (background) scattering. At T _s > 150°C, disordering dominates (I _incoh = I _max) until the next stage of ordering sets in at T _s = 250?300°C. At this stage, the crystalline phase forms from large (>100 Å) crystalline clusters. This amorphous-crystalline phase transition is characterized by the appearance of Debye lines and a reduction of the halo intensity. The structural phase transition to long-range order is accompanied by a decrease in the LaSrMnO resistivity from 10¹⁰ to 10 Ω cm and a change from the tunneling mechanism of conductivity involving metallic clusters (which is typical of granulated systems) to the hopping mechanism with a hop variable length following the Mott law ρ ～ exp(T ^?1/4). In the magnetic subsystem, the paramagnetic-ferromagnetic phase transition occurs.     

Algebraic operators and pairs of self-adjoint operators connected by a polynomial relation

Institute of Mathematics, Academy of Sciences of the Ukrainian SSR. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 25, No. 4, pp. 72–74, October–December, 1991.     

Complete integrability of a hydrodynamic Navier-Stokes model of the flow in a two-dimensional incompressible ideal liquid with a free surface

We establish the complete integrability of a nonlinear dynamical system associated with the hydrodynamic Navier-Stokes equations for the flow of an ideal two-dimensional liquid with a free surface over the horizontal bottom. We show that this dynamical system is naturally connected with the nonlinear kinetic Boltzmann-Vlasov equation for a one-dimensional flow of particles with a point potential of interaction between particles.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 45, No. 1, pp. 86–90, January, 1993.