Vol. 41 No. 1 (212) (2007)

Basing on the methods of the theory of boundary-value problems for analytic functions, the paper constructs generalized eigenfunctions of integral operator generating by Cauchy kernel in a finite interval. Further formulas for generalized integral Fourier transform by these eigenfunctions are obtained. Then the results are applied to construct solutions of singular integral equations with Cauchy kernel and constant coefficients.

In the paper elementary expression for the chord length distribution function of a regular hexagon is obtained. The formula is derived using        $\sigma$-formalism in Pleijel identity.

In the present work MAP|$G$|1 type queue with MAP stream of customers and two-stage service is considered. The busy period duration, queue length and the probability of empty queue state in non-stationary situation are obtained. The analysis is done with the help of additional variable method.

In this work the mechanical system of rigid body with flywheels inside is considered. The system is under the operation of dissipative forces. It is shown, that irrespective of rotation of flywheels, the position of equilibrium of the system is asymptotically steady. It is also investigated the stability of the system at a full and partial dissipation, when the system rotates around one axis.

In this paper proof normal forms are given for some non-traditional systems of classical propositional logic. On basis of that forms the upper and lower bounds of proof complexity are given using the notion of  $\varphi$-determinative disjunctive normal form.

The problem of the equilibrium distribution of potential in the contact two dimensional electron gas–two dimensional metal in the presence of dielectric inhomogeneity of surrounding medium has been considered. It is shown that such inhomogeneity can be taken into account by introducing in the final expressions the effective dielectric constant, equal to the arithmetic average of the dielectric constants of upper and lower mediums surrounding the plane of two dimensional gas.

By application of X-ray Pendellosung fringes method was shown, that for all studied region of dislocation monocrystall the distribution of Pendellosung fringes coincided with the theoretical isostrain lines. This fact allows the application of Pendellosung fringes method when studying the mechanical stresses around the dislocations in Si-monocrystalls.