Our paper is devoted to the description of the superposition operators which
map on function spaces of finite variation. We present the results which develop and generalize
the recent researches by J. Matkowski, J. Mis, D. Waterman and V.V. Chistyakov:
we introduce the notion of a total (two-dimensional) -variation for functions of two
real variables and show that the Waterman class of these functions with finite total variation
is a Banach space. Also, we give the description of the Lipschitzian superposition
(Nemytskii) operator mapping the Wateman class into itself.

In this article the strongly continuous semigroup is considered, this semigroup
is hyperbolic (or admits exponential dichotomy). For investigated semigroup the Greens
function is built, which plays a great role in presentation of weak bounded solution of
differential equations. Using frequency characteristic of operator and integral performance
criterion of dichotomy the article provides estimations of Green's function based on
hyperbolic semigroup of operators. The results of this article are obtained with use of
harmonic analysiss methods.

The singularly perturbed quasilinear hyperbolic equations are considered. The
existence of the solutions of moving front type, having contrast structure is proved. The
asymptotic expansion of the moving interior layer solution is built based on the boundary
layer function method. The equation for moving front velocity was obtained. Important
case of quasi-discrete nonlinearities was considered.

Exact solution of similar problems on the pressure field in the vicinity of the
propagating fracture are demonstrated. It is assumed that the crack develops on squareroot
law in a permeable elastically deformable porous medium with a constant disjoining
forces acting on the crack coasts. The considered problem for the piezo-conductivity
equation by changing of variables has been reduced to solving a mixed boundary problem
for the Helmholtz equation. Solution is expressed by the integral of Bessel functions. The
resulting solution is valid for any self-similar pressure distribution on the crack.

Article is devoted to developing physico-chemical model of erosion-corrosion
of metals in the two-phase flow in the lower zone of the absorber of installation amine
gas desulfurization. Author proposes a model which allows to develop mathematical
model of the erosion-corrosion processes of metal in the two-phase flow.

Statistical functions of distribution for the gas-like system of the dispersed particles
in the periodical field were found from Liouville - Gibbs equation. The kinetic description
of the evolution of the system for a special case was given. The main physical
laws of the evolution of the system were revealed and their dependence from the task parameters
was explored.

In the paper orbital movements of a relativistic nucleon and antinucleon in the
nonlinear field of a massive nucleus are studied. The problem is reduced to a system of
nonlinear differential equations of the second order with respect to the radial coordinate
and polar angle of the particle. Numerical computations for this system of equations are
carried out. It is shown that periodic orbits can occur in the case of movements of antiprotons
about nuclei. Conditions of their existence in the relativistic case are investigated.

The article is devoted to the analysis of polarization reversal processes in ferroelectric
crystals in the mode of injection of the electron bunches under the sample surface.
The simulation model of ferroelectric repolarization dynamics for 2-D domain configuration
geometry is supposed. The calculation of polarization switching current in the injection
mode is also presented.

The structure of physical fields (density anomaly, horizontal and vertical velocity)
of localized stationary full-nonlinear disturbances in the symmetric three-layer fluid
is examined. Wavelength-amplitude and phase speed-amplitude relationship depending
on vertical coordinate are analyzed for these waves. The comparison with weaklynonlinear
theory is also carried out.

Expediency of spatial light phase modulator for forming quasi-random realization
sampling of irradiation field for study behavior of correlation function of sixth order
was explained. The way of creation the spatial light phase modulator based on usage of
liquid crystal cell with variable cell-gap was suggested. Variation of the cell-gap in necessary
range is easy to produce by usage of vee-type gap. This type of cell provides required
amount realizations of magnitude phase shift. System of sequentially liquid crystals
cells can provide enlarge number of statistically independent realization of sample of
field irradiation intensity.

The paper describes a method of detection orientational structure of liquid
crystals inside a pore of porous film. Measurement of electric capacity of
composite material (porous film is filled with liquid crystals) in dependence on
temperature is main principle of this method. Porous polyethylene terephthalate
films (track membranes) were used in experiments because of fabrication of different
pores radius and orientation is easy. The porous films with different pores
radius 75, 170, 210, 300 and 425nm were investigated. New type of liquid crystal
cell was used in experiments. Surfaces of the porous films were covered by conducting
electrodes. Results of a theoretical calculation based on the experimental
data shown that escaped radial structure of liquid crystal formed in pores of polyethylene
terephthalate films.

The mathematical model of probing of a two-dimensional micro-field by electron
beam is investigated. Some relations for electron beam parameters that depend on the
distribution of the field along its trajectory are obtained. The definition of inverse problem
of evaluation of harmonic function describing distribution of magnetic (or electrical)
field is considered.

The paper proposes an approach to adaptation of neural network models for
creating OCR-systems, designed to work with uncjnstrained handwriting. The approach is
based on the rejection of recognition of continuous characters and the transition to the
recognition of individual strokes, which are then going to the characters and / or words of
text. This approach can significantly reduce the dimension of neural networks used in the
OCR-systems that will enhance their productivity and quality of recognition.

In this paper we propose and study a model of handwriting strokes. The model
consider handwritten text as a collection of strokes of several known classes. Model
allows us to abstract from the initial conditions of the writing, such as angle, velocity,
density, etc. Thanks to the procedure of tracing the model is irrelevant to noise of the
original image and the way it was received. We conducted an experiment, confirming the
applicability of the model for use in pattern recognition problems

The article discusses the development of methods how the expert committee should be
formed, and the final experts estimation and evaluation of the accuracy of the solution should be obtained.
Solution of the problem was considered in the context of the developing method for the remote,
individual, and anonimous expert evaluation with a variable number of experts. The author suggested
a model of formation of the expert committee; method of determining final experts evaluation for the
following issues: choice of options, order of the options within the list, the numerical score; method of estimating the solution accuracy. The proposed method is based on Kendall coefficient of concordance
as a criteria for the accuracy of the obtained solution.

Sharing demonstration of modern computer and multimedia equipment opens
up great opportunities for the teacher to learn physics, both in school and University. Using
these funds, the teacher has an opportunity for a new show, it would seem, the longknown
phenomenon. And it, undoubtedly, increases the interest of students to study the
topic in general.