Vol. 41 No. 3 (214) (2007)

A Dirichlet problem in the half-plane in the weighted spaces is investigated. Conditions of solvability of the problem are found. Under these conditions an explicit solution is found. A theorem of uniqueness in the class of harmonic functions is also obtained.

The model $M|G|1|\infty$ with unreliable server and «negative» customers, which destroy regular customers, is considered. Distribution number of customer, the busy period and the period of regeneration of model are investigated.

The concept of convexity of the set of points of the plane in certain sense of direction is defined and it is proved that for the finite number of the given directions there exists a prime polygon, which is convex only for those directions.

One generalization of formulae of Abel concerning to the Fermat's last theorem is received. Supposing that the theorem is not true, for the solution of the equation the necessary condition is obtained. In the case when none of the numbers is divided by p, we get a simple proof of the theorem for p = 3 and p = 5.

The paper is aimed at finding the extrinsic (or secondary) non-numeric and numeric roots to the algebraic equations of various orders. Further, the idea of algebraic root is generalized by exploiting algebraic operations with noncommutative addition law.

In the present paper commutative Abelian binary algebras are studied by applying the construction of the algebra of semiterms. We have found the necessary condition when any hyperidentity (i.e. $\forall(\forall)$-identity) is satisfied in all Abelian commutative binary algebras.

In this work, on the basis of classical theory by Kirchhoff and S.A. Ambartsumyan’s specified theory problems on the bending of the plate are investigated. It is shown that in a case when the plate is leaned free from both sides, and on two others is leaned in a way of mobile connection of two parts, so the accuracy of Kirchhoff's hypothesis is the neglecting of related thickness in comparison with unit. Formulas for deflection, moments, also for cutting forces are received. In case of narrow and wide plates the approaches for maximal deflection, cutting and generalized cutting forces are made.

The ²³²Th and ²³⁸U photofission yields were measured by using bremsstrahlung at end-point energy of 50 and 3500 MeV. The nuclear charge distributions of the fission fragments were investigated and necessary parameters were determined: the most probable charge (Zр) and the width parameter (С). The obtained data were compared with accounts on models UCD and МРЕ. The universal picture of fission processes with different particles was confirmed.

A semiconductor detector of charged particles was made from silicum monocrystal. It was tried to detect separate particles of the stream. Inlike to the traditional two-side covering of the model, total metallic covering of the background surface, cоntоur metallic covering of the frontal surface was done. The investigations show that the detector is not sensitive to the detection of gamma quantum, beta particles and μ mesons. While Alfa particles are accurately (clearly) detected by the detector, because they form signals which are clearly seen on the background of the noise signals.