Vol. 42 No. 1 (215) (2008)

In the paper the weighted spaces $b^p_\alpha(B)$ of functions harmonic in the unit ball $B\subset\mathbb{R}^n$ are introduced. The reproducing kernel $K\alpha$ is constructed by means of which for functions belonging to $b^p_\alpha(B)$ the integral representation is obtained.

In this paper von Neumann-Fuglede-Putnam’s asymptotic theorem is proved for subnormal elements in complex Banach algebra in topology of wide classes.

In this paper the behavior of solutions of the following initial boundary value problem for a class of Sobolev type equations is considered. $$\begin{cases}  A\left(\dfrac{\partial u}{\partial t}\right)+Bu=0, \\ u|_{t=0}=u_0, \\u|_{\Sigma}=0, \end{cases}$$ where $A$ and $B$ are nonlinear operators of the following form: $$Au=-\displaystyle\sum^n_{i,j=1}\dfrac{\partial}{\partial x_i}a_j(x,u, \triangledown u), ~~Bu= -\displaystyle\sum^{n}_{i,j=1}\dfrac{\partial}{\partial x_i}b_j (x,u,\triangledown u).$$

It’s proved that when functions $a_j (x, u,\triangledown u)$ and $b_j (x, u,\triangledown u)$ specify some conditions, the semigroup generated by this equation has attractor, which is bounded in $^0 W^1_ 2 (\Omega)$.

In the paper a variational inequality is considered for pseudoparabolic operators. The theorem of existence and uniqueness of weak solution is proved. Also it’s proved the solvability of corresponding initial–boundary value problem and is shown it to be equivalent to variational inequality.

In the present paper conditions on moments’ convergence for Waring Distributions are obtained from the point of view of regularly varying functions. Under these conditions a manner on the moments’ evaluation is suggested and the results for mean value and variance are presented.

Control with guidance for $m$ target sets when the system has varying dynamics is considered in this paper. It is assumed that the consequence of the meetings with the target sets is fixed. The procedure consists in introducing a secondary system – a guidance, which moves by the given stable bridge. The movements of the initial and the secondary systems are formed in such a way that in the process of the game they track each other, which guarantees ensures the stability of the solutions with respect to informational disturbances.

In the present work the construction of the cosmological model which takes into account the presence of dark energy is realized within the framework of Jordan–Brans–Dicke tensor-scalar theory with dominating minimal and non minimal coupled scalar fields in the presence of the cosmological scalar.

The free energy of vesicle stretched membrane, on membrane which has transmembrane potential difference and cylindrical pore, was determined. It is shown that, the presence of transmembrane potential difference makes the formation of pore easier. Critical expansion of membrane, pressure drop and difference of osmotic active substance, in the presence of which pore is forming, sharply reduced. It is shown that minimal radius of pore does not depend on transmembrane potential difference.

A new class super-broadband, nano-scale resolution position sensor is created. It may be used as a sensor in seismographs. It enables to extend the frequency band of the available technique. Combined with such a sensor traditional technique may enable to study quasi-static deformations & low-order free oscillations of Earth crust precursor to earthquakes. It allows to transfer mechanical vibrations with amplitudes over 2--3 nm, into detectable signal in a frequency range starting practically from quasi-static movements. We discuss test data of such position sensor, installed in a Russian SM-3 seismometer, as the additional pick-up component, showing its advantages compared with the traditional technique.

Recording dynamics of holographic gratings in polymer dispersed liquid crystals for thick layers has been investigated in this work. In contrast to existing theoretical models, besides the diffusion of monomer molecules, it have been taken into account the diffusion of polymer molecules. Our analysis shows that the profile of recorded diffraction grating becomes fuzzy when considering the diffusion processes of polymer molecules.

In this paper we characterize the {2, 3}-hyperidentities of associativity in invertible {2, 3}-algebras.

Possibility to create planar defects in cholesteric liquid crystal via external electric field has been investigated as well as dependences of transmission and reflection coefficients on the temperature and dependence of the step of cholesteric liquid crystal helix on the temperature have been obtained in this work.

The parameters of DNA denaturation, the temperature of DNA denaturation (T_m) and its interval (ΔT) has been calculated by denaturation differential curves. It was shown that during the radiation of DNA solution by the frequency of 64.5 GHz the temperature of DNA denaturation (T_m) increases by 1.2ºC and its interval (ΔT) decreases by 0.35ºC.