No 4 (2013)

The paper is devoted to investigation of the class of linear and quasi-linear systems of ordinary differential equations, the matrix of which can be characterized as polynomially periodic. The main aim of this article is to generate a new algorithm of their splitting in order to create equivalent sets with almost diagonal matrix that are simpler to analyze. Another objective is formulating and proving of sufficient stability conditions or asymptotic stability of their trivial decision. The question is topical since the analysis of a considered class of non-autonomous systems with the use of known methods (for example, the method of functions of Lyapunov) is complicated. In addition, the usage of spectral and other methods while solving non-uniform sets might cause extra difficulties. The authors of the paper develop an analytical method which appears to be a summary of known classical theorems. At the heart of the offered algorithm of reducibility lies one of options of splitting method, which is conducted by a spectrum of a defining matrix in studied non-autonomous system (taking into account its splitting on diagonal and non-diagonal part) lies. The present article shows possibilities of reducibility of sets of the specified class depending on structure of a matrix spectrum. This simplifies the analysis of questions of stability. Theorems of stability or asymptotic stability of the trivial decision of the transformed equivalent systems and the relevant initial systems that is development and generalization of a spectral method of research of stability for the class of non-autonomous systems considered in work are proved.

In this paper, we study the nonexistence of positive solution for some higher-order semilinear elliptic inequality particularly involving polyharmonic operator: Δku(x) ≥ x1 α1 x2 α2… xn αnuq(x), where k ∈ ℕ,q > 1, x = (x1,x2,…,xn) and αi ∈ ℝ,i = 1,2,…,n. The purpose of this paper is to establish conditions on values of αi,i = 1,2,…,n for the nonexistence of positive solution to this problem in a bounded and unbounded domain. The main tools are a priori estimates and integral inequalities. Using the test function method, we derive first a priori estimates for solutions of the inequality based on integral inequalities and on the weak formulation of the problem with an optimal choice of test functions and then we formulate the nonexistence condition of the solution of the problem. The choice of such functions is determined by the nonlinear characters of the problem and depends on the concept of solutions that we are dealing with.

A homogeneous (stationary in wide sense) random field with zero mean and real-valued components is given. The case of discrete parameter is considered. The matrix of second order periodograms, constructed by the sample is given. The asymptotic behaviour of the second moment of the second order spectral estimate of the homogeneous field is considered in the paper.

Investigation results of impact interaction of uncharged metallic nanocluster with metallic surface are presented. Simulation and investigations of impact processes are fulfilled by molecular dynamics methods and suitable software. Characteristic dimensions of surface layer produced by impact as a functions of cluster size, colliding energy and frequency of impulsive nanoclusters source have been analysed. It was found out that penetration depth and deposited layer thickness depend on number of particles in colliding nanoclusters and frequency of impulsive nanoclusters source. It was also discovered that deposited layer thickness in contrast to penetration depth ceases depending on number of particles in colliding nanoclusters N, and frequency of impulsive nanoclusters source ω and colliding energy E with increasing of N, ω and E. And at the same time deposited layer becomes heterogeneous in thickness and gets a funnel-shaped form. It is shown that realization of one of the choice of nanoclusters surface interaction (soft landing, droplet spreading and implantation) should be controlled by means of changing of both nanoclusters beam energy and number of atoms in clusters. Investigation results should be of interest in various fields of technologies developing nanomaterial with new physical and chemical properties.

In this paper the properties of solutions of the geodesic equations for a model of a point source of gravity, radiating heat are studied. Geodesic equations are constructed using a metric which is the solution of equations that represent the zero trace of the Ricci tensor. These equations are a generalization of Einstein’s equations in vacuum. They allow to obtain solutions in the form of non-stationary spherically symmetric metrics, whose components are a function of two variables. The ordinary system of differential equations of second order for surveying natural parameter consists of four equations. It can be partially integrated and reduced to a system of two second order differential equations. By substitution method the system is reduced to a pair of differential equations in partial derivatives of the two unknown variables. Finally, we obtain one quasi-linear equation. In the normal case, equations of this type form gaps with limited solutions. However, the numerical calculations show that the solutions can also become unrestricted due to the pecularities in the right parts.

This paper presents a derivation of the dispersion equation for a three-layer integrated-optical Luneburg lens based on the method of adiabatic waveguide modes. From this equation there follows the relationship between the coefficient of phase deceleration and function, which determines the thickness of the irregular waveguide layer. The dispersion equation is represented in the form of non-linear partial differential equation of the first order with coefficients, depending on parameters. Among these parameters are regular waveguide layer thickness and optical parameters of the pending Luneburg lens. To represent the dispersion equation in the form of differential equations in partial derivatives, it is necessary to calculate a symbolic form the determinant of a matrix of 12th order, which determines the solubility of the system of linear algebraic equations, resulting from the boundary conditions. To calculate this determinant in analytical form a procedure of reduction of the system of linear algebraic equations with the use of the computer algebra system Maple is proposed.

In the present paper the method of least squares with T-elements for solving linear boundary value problems with Laplace and Poisson’s equations is developed. In this approach it is offered to use discontinuous basis functions of a high-order approximation from special functional spaces, elaborated by the authors earlier. Advantage of the algorithm in comparison with Galerkin’s standard method is that, in the process of adaptive solving, it makes possible to condense economically a mesh and, moreover, to use different order of approximation of the solution on each cell of partition of calculated region. In contrast to Galerkin’s method with discontinuous basis functions, a penalty parameter here is not required, and the matrix of a discretized problem also is symmetric and positively definite. Examples of calculations by means of the schemes providing computer accuracy of the solution of boundary value problems for polynomials up to seventh order inclusive are given. In a three-dimensional case h − p-convergence of approximate solution to the exact one is shown.

Hydrogenated amorphous silicon is found to be a leading candidate for the fabrication of low cost solar cells. However presently there are two main factors that limit the large scale applications of a-Si:H solar cells as power sources. One of the central technological obstacle is the low conversion efficiency of the cells. The other obstacle for the large scale technological application of a-Si:H solar cells is degradation of critical material properties under the light exposure. In our experiments we have performed light soaking tests on pinpin structure samples to see if the stability of a-SI:H solar cells is improved in comparison with the stability of pin structures. The pattern of light induced degradation, i.e. the degree of degradation of a-Si:H pinpin solar cell parameters was studied on different i-layer thickness using high intensity ( ∼10 AM 1.5) illumination. It was found that stacked cells do not show a uniform degradation pattern as in the case of single junction solar cells. In particular, the degradation in short-circuit current Isc of stacked cells shows a big difference for thick ( ∼500 nm) and thin ( ∼400 nm) pinpin cells. It was found the degradation of the stacked cells with thick bottom layers exhibit a degradation pattern similar to that of single junction cells, i.e. the degradation in efficiency comes from the fill factor and the short circuit current, while open circuit voltage being degraded slightly. The degradation in short circuit current of cells with thin bottom layers is negligibly small.

The neutrino charge with its gauge field introduced in [Kopysov Yu. S., Stozhkov Yu. I., Korolkov D. N. (2001)] for the purpose of decreasing counting rates in solar neutrino detectors generates a lot of new phenomena in astrophysical objects. The physics of the new phenomena is determined by the value of the neutrino charge eν which carriers are neutrinos, quarks and neutrons, and also by almost degenerate neutrino condensate in substance of macroscopic objects. It is shown that the strongest restriction on the value of eν can be obtained by a method of thermal balance of the Sun developed in [Domogatsky G. V. (1968)]. The new interaction generated by a new gauge (“neutromagnetic”) field, gives rise to the neutrino Dirac’s magnetic moment of new type. Restriction on its value at the obtained restrictions on eν is only 2 ÷ 3 orders of magnitude lower than the electronic Bohr magneton and on many orders of magnitude exceeds all possible estimates of the traditional anomalous neutrino magnetic moment! The new scenario of formation of solar activity at which new interaction can play a key role is offered. The new model assumes two-story structure of a convective zone: external with the developed thermal convection and internal — the solar troposphere, — in which under the influence of tidal forces of planets whirls like a tornado of the terrestrial troposphere are formed. In these whirlwinds magnetic fields of the new (neutromagnetic) type are generated which interaction with substance generates also usual magnetic fields. The new class of the phenomena arises due to inclusion of the neutrino charge into the theory of collapsing and neutronizing stars. New opportunities for solving old problems are being opened on this pathway. In this regard it is desirable to have theoretical justification of need of introduction of the neutrino charge. In this work the problem of extension of the standard unified model of electroweak interaction by means of inclusion of the second charge in the right sector of extended model is put forward. The possible solution of this problem is planned.