We develop a method of averaging for the study of linear systems of dynamic equations on time scales. We use the obtained results to construct the asymptotics for the solutions of some equations of discrete adiabatic oscillators.

We have built a quasinormal form for the evolution equation that generalizes the equation
with large delay.

We study the local dynamics of a model for the passive mode locking in semiconductor
lasers. In critical cases special equations (normal forms) were built. The existence of
multistability is shown.

The asymptotics of spatially inhomogeneous periodic solutions of a complex spatially
distributed Hutchinson equation with periodic boundary conditions are presented. It is
shown that such solutions are not observed in a numerical experiment.

It is considered the dynamics of a distributed rotor from a material with nonlinearhereditary
properties, one of its supports being under a periodical vibration. The mathematical
model of the considered mechanical system is a system of differential equations
in partial derivatives with infinite delay of the argument. It is found the conditions of
chaotic fluctuations. The Lyapunov indices and Lyapunov dimension are calculated.

The paper deals with a linear formulation of the problem of plane vibrations in a
supersonic gas flow. Fluctuations of the plane are studied in the case when one of its
ends is rigidly fixed, while the second is free. The flow rate is found for which internal
resonance eigenfrequencies 1:1, 1:2, 1:3 are implemented. The determination of the
speed and the corresponding frequency is reduced to the system of two transcendental
equations. The system is numerically studied.

The dynamics of the system what describes the interaction between two similar
species is investigated. The normal form is constructed, it is reduced to a normal form
of a system of diffused weakly connected Hutchinson equations.

We study the local dynamics of a nonlinear complex DDE with large delay in the
vicinity of the zero solution. The quasinormal forms method is used for the problem
analysis. We show that parabolic type GL-equations act as normal forms in critical
cases of the infinite dimension.

The dynamics of a generalized impulse neuron equation with two delays is studied.
A local analysis of a loss of stability for a nonzero equilibrium state has been made.
Phase reorganizations have been numerically analyzed with the help of the obtained
asymptotic formulas.

In the paper we analyze the generalized model of the V. V. Mayorov and I. Yu. Myshkin
pulsed neuron. The neuron equation contains a delay of the sodium conductance with
respect to the transmembrane potential as well as the potassium one. We proved the
exisence and stability of the periodic solution and obtained its asymptotic expansion.

The paper discusses some general rules for organizing database objects into hierarchical
structures. It describes different kinds of relations between objects, and the role that
relations play in the organization of such structures. It also defines principles of executing
object copy and delete operations that do not break the logic of relations between
objects.

Almost simple SM_m-groups are considered. A group G is called a SM_m-group if
the tensor square of any irreducible representation is decomposed into the sum of its
irreducible representations with multiplicities not greater than m. In the first part of
this article we consider simple groups. It turned out that among them only groups L_2(q), q = 2^t, t > 1, are SM_2-groups.

We prove some new inequalities for the norms of projections due to the polynomial
interpolation of continuous functions of n variables.