Vol. 51 No. 1 (242) (2017)

We introduce the Banach spaces $h_{\infty}(\varphi)$, $h_{0}(\varphi)$ and $h^{1}(\psi)$ functions harmonic in the unit ball $B\subset\mathbb{R}^n$. These spaces depend on weight functions $\varphi$, $\psi$. We prove that if $\varphi$ and $\psi$ form a normal pair, then $h^{1}(\psi)^*\sim h_{\infty}(\varphi)$ and $h_{0}(\varphi)^*\sim h^{1}(\psi)$.

We bring an example of integrable function on $[0;1]^2$, so that the double Fourier–Haar series has a subseries, whose majorant of partial sums does not belong to $ L^{1, \infty}$.

An edge-coloring of a graph G with colors $1,2,...,t$ is an interval $t$-coloring, if all colors are used, and the colors of edges incident to each vertex of $G$ are distinct and form an interval of integers. A graph $G$ is interval colorable, if it has an interval $t$-coloring for some positive integer $t$.  $def (G)$ denotes the minimum number of pendant edges that should be attached to $G$ to make it interval colorable. In this paper we study interval colorings of outerplanar graphs. In particular, we show that if $G$ is an outerplanar graph, then $def(G) \leq (|V(G)|-2)/(og(G)-2)$, where $og(G)$ is the length of the shortest cycle with odd number of edges in $G$.

We prove embedding theorems for multianisotropic spaces in the case when the Newton polyhedron has two vertices of anisotropicity. The case of one anisotropicity vertex of the polyhedron was studied in previous papers of one of the authors. The present paper is the continuation of those.

The Riemann boundary problem in weighted spaces $L^1(\rho) $ on $T = {t; |t| = 1}, $ where $\rho(t) =|t -t_0|^\alpha, t_0 \in T$ and $\alpha > -1$, is investigated. The problem is to find analytic functions $\Phi^+(z)$ and $\Phi^-(z)$, $\Phi^-(\infty)= 0$ defined on the interior and exterior domains of $T$ respectively, such that: $\lim_\limits{ r\rightarrow 1-0} ||\Phi^+(rt)-a(t)\Phi^-(r^1t)- f (t)||_{L^1(\rho) }= 0,$ where $f\in L^1(\rho),  a(t) \in H_0(T;t_1, t_2,...,t_m)$. The article gives necessary and sufficient conditions for solvability of the problem and with explicit form of thr solutions.

On the basis of the equations of generalized plane stress state the problems of planar vibrations of a rectangular thin plate are investigated. It is established possibility of the appearance of vibrations with amplitude decreasing exponentially, when moving from free edge to the opposite fixed edge. The conditions of the appearance of such localized vibrations depending on the size of the plate and the methods of fixation of other three sides are obtained.

In the paper typed and untyped $\lambda$-terms are considered. Typed $\lambda$-terms use variables of any order and constants of order $\leq$1. Constants of order 1 are strong computable functions with indeterminate values of arguments and every function has an untyped $\lambda$-term that $\lambda$-defines it. The so-called canonical notion of $\delta$-reduction is introduced. This is the notion of $\delta$-reduction that is used in the implementation of functional programming languages. For the canonical notion of $\delta$-reduction the translation of typed $\lambda$-terms into untyped $\lambda$-terms is studied.

The paper is devoted to the analysis and programmatic implementation of directed evolution processes in complex networks. We describe the evolution model and present requirements for implementation. Then we display the developed system in architectural form and present the performance evaluation.

In this paper, bias and bias uncertainty in the neutron multiplication factor due to biases and bias uncertainties in the calculated nuclide concentrations in the spent nuclear fuel of WWER-440 type was assessed. To determine isotopic biases and bias uncertainties in the calculated nuclide concentrations, they were compared to the results of the measurements of isotopic compositions from destructive radiochemical assay of WWER-440 spent fuel carried out in the RIAR. In calculations Monte Carlo uncertainty sampling method was used.

Motivated by the prospects of developing DNA field-effect transistors, as tools for a variety of application fields such as medical diagnostics, environmental pollutants monitoring, biological weapons defense, and taking into account that the efficiency of DNA-sensors depends on the precise prediction of experimental parameters responsible for thermostability of nucleic acids duplexes and specific times of formation of DNA duplexes, we analyze the factors influencing both the thermodynamics of hybridization and the stability of DNA–DNA and DNA–RNA duplexes. In this work the case of competition-free DNA hybridization is analyzed. It is shown how the intercalating ligands effectively increase the binding constant for the target sequences and thus affects the sensitivity of the DNA-chip.

Equation of state of strange quark matter has been studied in the framework of MIT bag model, when vacuum pressure $B$ depends on concentration of baryons $n$. The actuality of such studies is conditioned by the increasing of quark matter density from surface to star center. In the literature there exist different representations of function $B(n)$. In the present work Gaussian parametrization is used, which is based on the idea of existence of asymptotic limiting value of this parameter. For four groups of parameters the equations of state of quark matter were determined. The main integral parameters of star configurations were obtained by numerically integrating of star equilibrium equations (the TOV equation). In the considered case it turns that when vacuum pressure dependence on concentration of baryons is taken into account, configurations of strange stars have maximal masses less than two solar masses.

Comparative analysis of two diffraction schemes $d-c$ (diaphragm-crystal) and $c-d$ (crystal-diaphragm) for widely divergent beam of X-ray radiation is conducted. It is justified that in $c-d$ scheme the diffraction image is a topographic map of the studied area of the crystal.

In the work third-order nonlinear Takagi’s equations for X-ray monochromatic waves are investigated for the forbidden reflection case. The forbidden dynamical diffraction in the nonlinear case is related to the presence in the nonlinear equations of the terms proportional to the zero order and the second order nonzero Fourier coefficients of the third-order nonlinear susceptibility. As a consequence, in the third-order nonlinear Bragg diffraction case, a nonlinear analogue of the well known Renninger effect takes place, which is considered theoretically and numerically. The numerical calculations show that in the Bragg geometry the nonlinear reflection curve’s behavior is the same as for not forbidden reflection, but for forbidden reflection the rocking curves are by several orders more sensitive to the input intensity value.

Hydrogenic donor impurity states in the 2D GaAs=Ga_0.7Al_0.3As concentric double quantum rings have been investigated under the action of intense laser field. An analytical expression of the effective lateral confinement potential induced by the laser field is obtained. The laser dressed effect has been considered both on electron confining and electron-impurity Coulomb interaction potentials. The single electron energy spectrum and wave functions have been found using the effective mass approximation and exact diagonalization technique. The accidental degeneracy of the impurity states has been observed.

Excitonic absorption in monolayer and bilayer graphene systems with opened energy gap in the field of laser radiation is investigated. The obtained value of excitonic binding energy in monolayer is in good agreement with the exact analytical solution. It is shown that the account of all tight binding parameters in bilayer graphene leads to an increase of exciton binding energy.

The refractive index of nematic liquid crystal (NLC) E7 in the X-ray spectral range using the X-ray interferometric method, was measured experimentally. It was shown that NLC E7 is anisotropic medium for X-rays. This method is sensitive to reorientation of the molecules of liquid crystals and allows to measure the refractive index of homogeneously aligned NLC layer, when the molecules are reoriented under the influence of external electric field.

The earlier represented three-block Fresnel Zone Plate interferometer based hard X-ray phase-contrast imaging setup designed for the initial plane wave radiation has been adapted to the laboratory sources of radiation. The limited distance between the source and the interferometer as well as the finite size of the source and the limited degree of monochromaticity of the illumination have been taken into account. It was shown that the decrease of the spatial coherency of the initial radiation does not worsen, and even improves the quality of the phase-contrast imaging, by suppressing the distortions caused by the X-ray diffraction at the edges of the Fresnel zone plates and the knifes of the interferometer.

Applying the X-ray diffraction technique, we have investigated the dependence of the residual stress components on ettringite crystals in expansive cement paste depending on the content of ye’elimite admixture. The results of above stress analysis were used to analyze how the shrinkage-compensating effect and the content of ye’elimite admixture are correlated with each other.

The problem of electromagnetic absorption in the electronic gas, which is localized in asymmetric biconvex thin quantum lens, is discussed. It is shown, that in the case of thin lens, in the plane of quantum lens electron gas is localized in two-dimensional parabolic confining potential. Thus, in this system the conditions for the realization of the generalized Kohn’s theorem are satisfied.

The problem of a dynamic diffraction of spherical X-ray wave on a short period superlattice is considered theoretically. It is shown that a focusing of satellites occurred both at different depths of the superlattice and at different distances from the crystal in vacuum depending on the structural factors of the superlattice. The larger the structural factor of the satellite, the more deeply the wave in the crystal is focused and, respectively, the less is the focal length out of the crystal in vacuum.

Electronic states and optical transitions of a ZnO quantum ring containing few interacting electrons in an applied magnetic field are found to be very different from those in a conventional semiconductor system, such as a GaAs ring. Strong Zeeman and Coulomb interactions of the ZnO system, exert a profound influence on the electron states and on the optical properties of the ring. In particular, our results indicate that the Aharonov– Bohm effect in a ZnO quantum ring strongly depends on the electron number.

The elastic strain distribution in one layer InAs/GaAs quantum ring superlattice is calculated using the Green function method. The dependencies of the strain tensor components on spatial coordinate in three different directions are obtained. The comparison of the obtained results with ones for a single quantum ring shows that the effect of the matrix material on the strain distribution inside the ring is more pronounced in the case of single quantum ring. The behavior of the strain tensor components in different directions is strongly different both for the single quantum ring and quantum ring superlattice due to the anisotropy of the crystal lattice.

The influence of an image charge effect on the impurity-related energies, the oscillator strengths of transitions from the 1s-like state to 2p_x and 2p_z states as well as the optical absorption coefficients in a spherical quantum dot in the presence of an external electric field have been studied. The results show that the image charges lead to significant changes in the optical coefficients of the system.