The interaction between the two eddy streams of a gas of rough spheres is investigated. A bimodal distribution with a Maxwellian modes of a special form is used. Different sufficient conditions for the minimization of the integral discrepancy between the sides of the Bryan-Piddaсk equation is obtained.

Keywords: spheres, bimodal distribution, eddies, pure integral discrepancy.

The stability problem of the trivial solution of the systems of differential equations with unfixed times of impulse effect was studied by means of Lyapunov functions. The new conditions of asymptotic stability and instability were obtained.

Keywords: stability, impulsive systems, Lyapunov functions.

The interpolation quadrature formula for calculation of the integral of a function containing irreparable rupture of the rst kind had been obtained. This formula is exact for polynomials, the degree of which less than the number of interpolation points. The convergence of approximate values of integrals of the exact values of integrals had been shown. The estimates of the convergence rates had been given.

Keywords: quadrature formulas, the speed of convergence, the polynomials of Chebyshev.

The constructive conditions for the existence of solutions have been studied and generalized Green's operator for the linear periodic boundary value problem for system of ordinary differential equations with impulse perturbation have been searched.

Keywords: generalized Green's operator, periodic boundary value problems, impulse perturbation.

Let $\mu$ be a complex borelian measure on the unit circle $\bf T$ and let $\hat{\mu}(n)$ be its Fourier coefficients. The sum of the series $\sum\limits_{n=-\infty}^\infty \frac{\hat{\mu}(n)}{n+z}$ is calculated. There are different criteria a sequence $c_n$, $n\in(-\infty,\infty)$, to be a sequence of Fourier coefficients of some complex borelian measure on $\bf T$. One more criterion of such type is proved.

Keywords: algebra $M(\bf T)$, Fourier coefficients, reducing formula, Geza theorem.

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: controllability, linear non-autonomous inhomogeneous system, canonical form.