Vol. 43 No. 1 (218) (2009)

In the present paper we prove the following theorem. For any  $\epsilon > 0$ there exists a measurable set $G\subset S^3$ with measure mes $G>4\pi-\epsilon$ , such that for each $f(x)\in L^1 (S^3)$ there is a function $g(x)\in L^1 (S^3)$, coinciding with $f (x)$  on $G$ with the following properties. Its Fourier–Laplace series converges to $g(x)$ in metrics $L^1 (S^3)$ and the inequality holds $$\sup\limits_N \parallel\sum\limits^N_{n=1}Y_n[g,(0,\phi)]\parallel_{L^1 (S^3)}\ll 3\parallel g\parallel _{L^1 (S^3)}\leq 12\parallel f\parallel.$$

Two obstacle problem in abstract form is considered in this paper. We prove two side estimates for the solution of this problem.

In the present paper an Euler type equation is considered and it is proved that for $\alpha\in[0,1](\alpha\in(2n-1, 2n])$ the characteristic polynomial has 2n real roots. For other values of $\alpha$ the issue concerning the number of the real roots of this polynomial is investigated.

In the present report it is proved that for a priori given numbers $\rho\in(1, +\infty)$ and $L\in R^+=(0, \infty)$ there is a distribution $\{p_n\}_1^\infty$ with the following properties: $\{p_n\}_1^\infty$ varies regularly as $n\rightarrow +\infty$ with exponent $(-\rho )$, exhibits the constant slowly varying component $L$, and $\{\log p_n\}_1^\infty$ is downward convex.

In the present paper for the sum of random number of independent identically distributed random variables the hazard rate increase is proven under the summanders hazard rate increase condition. With the help of obtained result the stationary waiting time hazard rate increase in the $M |G|1|\infty|$ model is established under the service time hazard rate increase contition.

In the present paper the concepts of $q$-lattice and $q$-algebra introduced in [1] and [2] are studied. The relationship between lattices (Boolean algebras) and $q$-lattices ($q$-algebras) is stated, and the results of [2] are improved.

A fast signal-processing system has been presented in this work for processing of frequency-shift signals obtained from the improved (by a flat pick-up coil based SFCO-method) seismic detector, capable to register also rapid vibrations (sharp shakings) of the Earth crust. It might be used also for detection and explanation of the nature of nuclear explosion and other types of blast explosions.

The process of irradiation of light energy from optical fiber’s conic top covered with metal and dielectric layers is studied. As a dielectric layer is used the meniscus formed during an output of conic top from a solution. Peculiarities registered peak of radiation allow to explore structural features of solution.

Presented technique allowed to measure the phase position of electron bunch relative to accelerating wave.

It is shown, that the transmembrane difference of potentials applied to vesicles causes the decrease of the swelling time. Vesicle swelling time exponentially decreases with the increase of the potential. The swelling time of vesicle is greatly increased with the increase of work for the porous perimeter unit formation.

The data of radioactive emission from the Armenian NPP in atmosphere in a mode of normal operation and in emergency situations are presented. The technique of the azimuths (sectors) radiation risk calculation for the population, living in area of nuclear power plants location, is proposed. The values of azimuths radiation risk for area of the Armenian NPP location are calculated. The analysis of the obtained values is carried out. The azimuths, most significant from the point of view of radiation monitoring organization in area of the Armenian NPP location are determined, as well as concentration of forces and means for the population protection in case of possible accidents.

In the present note it is shown that for a set of positive integers $\wedge$ a Müntztype condition holds if and only if there exists a lacunary power series $f(z)=\sum\limits_{v\in\wedge} f_v/z^v$ that allows an analytic and bounded continuation to the complement of a closed logarithmical sector with vertex at the origin.