Approximate bimodal solutions for the integro-differential Boltzmann equation for the model of hard spheres are built in the case when the Maxwellian modes are screws with different degrees of infnite-simality of their angular velocities. Some suffcient conditions to minimization of integral remainder between the sides of the Boltzmann equation are obtained.

Keywords: Boltzmann equation, integral remainder, asymmetrical screws.

In terms of a kinematic model of mechanical suffosion the problems of leaching and settling of small soil particles under the influence of filtrational fluid flow are considered. Motions with plane, cylindric and spheric waves are studied. Self-similar solutions of these problems are obtained, which contain surfaces of strong and weak discontinuities.

Keywords: fluid filtration, suffosion, strong discontinuity, self-similar solutions.

The Fourier coefficients of series for potentials of first electrostatic problem are found explicitly. The singular part of the matrix operator of second electrostatic problem was separated and inverted for seek of the Fourier coefficients of series. The contour integration and Abel integral transformation were used. As a result the Fredholm integral equation of the second kind was obtained.

Keywords: electrostatics, charge, spheres, cone, integral equations.

We find necessary and sufficient conditions for existence of solution to a nonlinear autonomous periodic problem for a Hill type equation in a particular critical case, and also constructed a scheme according the least square technique.

Keywords: autonomous periodic problem, a Hill type equation, the least square technique.

We consider a real vector bundle $\E$ of rank $p$ and a unit sphere bundle $\E_1\subset \E$ over the Riemannian $M^n$ with the Sasaki-type metric. A unit section of $\E_1$ gives rise to a submanifold in $\E_1$. We give some examples of local minimal unit sections and present a complete description of local totally geodesic unit sections of $\E_1$ in the simplest non-trivial case $p=2$ and $n=2$.

Keywords: Sasaki metric, unit sphere bundle, totally geodesic unit section.

Step by step process solution matrix moment problem on a compact interval has been considered. Explicit formulas for generalized Schur parameters has been obtained.

Keywords: the matrix moment problem, Blaschke-Potapov product, generalized Schur parameters, step by step Shur process.

The article deals with the mathematical model of diffraction of Е-polarized electromagnetic waves on periodic ideally conducting circular cylindrical surfaces. Discrete mathematical model build above mentioned problems of diffraction based on Discrete features.The numerical experiment has been made.

Keywords: diffraction problem, the boundary integral equation.