Quasiinvariant deformations of invariant submanifolds of Hamiltonian dynamical systems

We study quasiinvariant deformations of invariant submanifolds of nonlinear Hamiltonian dynamical systems and their small perturbations.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1043–1054, August, 1994.     

On representations of the Heisenberg relations for the quantumE(2) group

All irreducible^* -representations of the involutive algebraA generated by the so-called Heisenberg relations for the quantum deformation of the group of motions of the Euclidean plane are described.Dedicated to the 70th birthday of our teacher, Prof. Yu. M. BerezanskiiPublished in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 5, pp. 689–692, May, 1995.This research was partially supported by the Ukrainian State Committee on Science and Technology, the International Science Foundation, grant No. U6D000, and a grant from the American Mathematical Society.     

Method for studying boundary value problems with nonlinear boundary conditions

A constructive approach to the determination of an approximate solution of a boundary value problem with nonlinear boundary conditions g[z (0), z (T)]=0 is proposed. Existence of the exact solution is proved, and error estimates for the constructed approximate solution are provided.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 7, pp. 951–957, July, 1990.     

On Some Properties of the Behavior of Linear Extensions of Dynamical Systems on a Torus under Perturbation of Phase Variables

We investigate classes of linear extensions of dynamical systems on a torus for which the Lyapunov functions exist for an arbitrary flow on the torus. Linear extensions for which the Lyapunov functions exist only with varying coefficients are considered separately. We investigate the problem of preservation of regularity under perturbation of phase variables.     

Averaging of Nonlinear Oscillation Systems of Higher Approximation with Delay

For oscillation resonance systems with delay on a finite segment or a semiaxis, we justify the averaging method with respect to fast variables. We also obtain error estimates for this method, which explicitly depend on a small parameter.     

Some problems of the linear theory of systems of ordinary differential equations

We consider problems of the linear theory of systems of ordinary differential equations related to the investigation of invariant hyperplanes of these systems, the notion of equivalence for these systems, and the Floquet–Lyapunov theory for periodic systems of linear equations. In particular, we introduce the notion of equivalence of systems of linear differential equations of different orders, propose a new formula of the Floquet form for periodic systems, and present the application of this formula to the introduction of amplitude–phase coordinates in a neighborhood of a periodic trajectory of a dynamical system.     

Multilayer metallization of ferrite Mg<Subscript>0.54</Subscript>Zn<Subscript>0.46</Subscript>Fe<Subscript>2</Subscript>O<Subscript>4</Subscript>

The character of interatomic interaction in ferrite Mg_0.54Zn_0.46Fe₂O₄ was studied using the x-ray spectroscopy technique and theoretically. It was found that the electronic structure of samples is rearranged during annealing at high temperatures (1280°C, 0.5–0.8 h). The electronic structure rearrangement was shown to be associated with multilayered ferrite metallization in which alternating layers with metallic and ionic-covalent bonds form.