No 3 (2013)

The generalization of paracompact spaces via so-sets, sets are unions of open and nowhere dense sets, was studied. The aim of this paper is to establish the relationship between so-paracompact spaces and other generalizations of paracompact spaces and clarify the conditions under which the so-paracompact space is compact. The problem is solved by methods of general topology. It is proved that the sequentially compact so-paracompact space is compact. It is proved that the so-paracompactness saved when multiplied by the compact. Previously, other authors introduced the concept of S-paracompact space, based on the semi-open sets. Class of so-paracompact spaces wider than the class S-paracompact spaces. This paper shows that there are so-paracompact spaces which are not S-paracompact.

This article is devoted to the study of the uniqueness of solutions of linear integral equations of the first kind with two independent variables in which the operator generated by the kernel, is not compact operator. The relevance of the problem is due to the needs in development of new approaches for the regularization and uniqueness of the solution of linear integral equations of the first kind with two independent variables. For approximate solutions of such tasks, stable to small variations of the initial data, we use the solutions derived by the method of regularization and belonging to the class of incorrectly formulated tasks. One of the classes of such ill-posed problems are integral equations of the first kind with two independent variables. The aim of the work is to prove the theorems of uniqueness for solving linear integral equations of the first kind with two independent variables. In the paper a theorem of the uniqueness of the solution of integral equations of the first kind with two independent variables is proved. To obtain the results formulated in the article the methods of functional analysis and method of nonnegative quadratic forms are used. The obtained results are new. The reliability of the result is set by prooves and illustrated by examples. The work has a theoretical character. The obtained theoretical results can be used in various fields of science and technology.

The purpose of the present paper is to investigate the potentiality of the differential difference operator with deviant arguments and to construct the functional, if the given operatoris a potential on a given set relatively to the some special bilinear form, i.e. the problem ofexistence of solutions of inverse problems of the calculus of variations for partial differentialdifference operators is investigated. Let ..,.. be normed linear spaces over the field of real numbers R. Take any operator .. : ..(..) > ..(..), where ..(..) . .., ..(..) . .. . A limit if it exists, is called the G.ateaux differential of .. at the point ... The operator ....(.., ·): .. > .. is called the G.ateaux derivative of .. at .. and will be denoted by ..... . Its domain of definition ..(..... ) consists of elements . . .. such that (.. + ...) . ..(..) for all .. sufficiently small. We obtain necessary and sufficient conditions for the adjusted partial differential difference op..,.. erator of ..class. The nonlinear differential operator of the second order and the nonlinear ..,.. differential operator of the second order with deviant arguments is consider as an example. Using the obtained conditions of potentiality corresponding functionals are constructed.

The quantum model of a cluster, consisting of A identical particles, coupled by the internal pair interactions and affected by the external field of a target, is formulated in the new symmetrized coordinates. A new method and symbolic algorithm for generating (A − 1)-dimensional oscillator eigenfunctions, symmetric or antisymmetric with respect to permutations of A identical particles, is elaborated and implemented using the MAPLE computer algebra system. Examples of generating the symmetrized coordinate representation for composite systems of several identical particles in one-dimensional Euclidean space are given and their symmetry properties are analyzed. The systems composed from three to six particles in one dimensional Euclidean space were analyzed a correspondence between the representations of the symmetry groups D3 and Td for A = 3 and A = 4 and symmetric or antisymmetric oscillator functions was found. It is shown that the transformations of (A− 1)-dimensional oscillator functions from the symmetrized coordinates to the Jacobi coordinates, reducible to permutations of coordinates and (A − 1)-dimensional finite rotation, are implemented by means of the (A − 1)-dimensional oscillator Wigner functions. The examples of construction of the symmetric or antisymmetric oscillator functions in closed analytical form by means of mathematical induction and the algorithm are given. The approach is aimed at solving the problem of tunnelling the clusters, consisting of several identical particles, through repulsive potential barriers of a target.

he article sets forth the methodological approach to the problem of formalization of the process of observation and measurement of parameters of dynamic systems. The necessity of selection of the measured parameters of the system for the assessment of the control is shown. The wording of the concept of security system is given and the formalization of the quantitative assessment of the safety of the operation and status changes of complex technical systems (STS) in time are proposed. Important process of ensuring the safety of the technical system, particularly for the special purposes, is the measurement and observation of the parameters. The process of the state and dynamics of the studied object (system) is described by the model in the form of dynamic equations in vector form. Dynamic management system and its measuring complex represent a linear stationary system, which is described by the model in a matrix form. When assessing the safety of technical systems different algorithms of measurement of parameters may be used. In any case allowed operations are implemented by the operator, which for linear stationary dynamical system is also linear. The method of determining the optimal set of parameters observations characterizing the state of the system which is proposed in the article significantly affects the safety of the study of the dynamical system in case of internal disturbances. By varying these parameters, one can determine the time and the monitoring interval, when the current state of the dynamic system is determined with precision.

This paper presents an investigation of modeling and solving system of differential equations in the study of mechanical systems with holonomic constraints. A method is developed for constracting equation of motion for mechanical system with constraints. A technique is developed how to approximate the solution of the problem that is obtained from modeling of kinematic constraint equation which is stable. A perturbation analysis shows that velocity stabilization is the most efficient projection with regard to improvement of the numerical integration. How frequently the numerical solution of the ordinary differential equation should be stabilized is discussed. A procedure is indicated to get approximate solution when the systems of differential equations can’t be solved analytically. A new approach is applied for constructing and stabilyzing Runge-Kutta numerical methods. The Runge-Kutta numerical methods are reformulated in a new approach. Not only the technique of formulation but also the test developed for its stability is new.Finally an example is presented not only to demonstrate how the stability of the solution depends on the variation of the factor but also how to find an approximate solution of the problem using numerical integration.

The procedure for building auto-adjustment control vector to bring the state of the mechanical systems without impact in a given manifold for a finite period of time in the face ofuncertainty is described. Previously obtained the solution of the problem of bringing the phase state of the system ina given neighborhood of the manifold formed by the non-stationary holonomic program constraints. In this paper we extend this approach to the task of bringing non-impact phase of thesystem for a finite period of time in the manifold formed by the holonomic and nonholonomicprogram constraints. In this case, even the mechanical system can have besides stationary and non-stationary communications. Received a lot of control vectors that provide a solution to this problem of self-adapting control of feedback on the quasi-accelerations at discretepoints in time. Then this set of control vectors allocated dimension smaller than the numberof degrees of freedom of the system, including the minimum dimension vector. In cases where the vectors control more than the minimum, stand vectors with minimal Euclidean norm. The obtained results allow us to meet the challenges of an applied nature, such as processcontrol unstressed docking surface, swimming, aircraft and spacecraft as they move freely inspace, but also a process of unstressed landers landing on the moving platform, the nature of the movement are not fully known. To illustrate the effectiveness of the proposed method for solving such problems an exampleof a process control of non-impact bringing of position of the body in a predetermined orientation with haunting movement of the center of mass of the body on the basis of proportionalnavigation is proposed.

Using the quantum mechanical Bogolubov’s Hamiltonian hierarchy for localized electronicexcitations of the crystal system, we have obtained the Hamiltonian of the spin excitations ofthe Heisenberg model, taking into account the surface energy. This Hamiltonian is obtained in the zero approximation in the spin-spin interaction for a ferromagnetic crystal, in case ofrigid fixing of ions in the lattice sites. The corresponding expressions for the displacement ofions in the crystal lattice are also shown.

We have considered in the nonrelativistic approach the movement of the three types of the fluid with negative pressure (cosmic vacuum, quintessence, Chaplygin gas) under the action of the own gravitational field. Introduction in cosmological models of a substance with negative pressure is one of alternative approaches to the explanation of existence of the accelerated expansion of the Universe. The space vacuum possesses not only a certain density of energy, but also a pressure. If density of space vacuum is positive, its pressure is negative. Connection between pressure and density, i.e. the equation of state, has an appearance for vacuum P + = 0. This equation of state is compatible with the definition of vacuum as energy form with everywhere and always constant density, irrespective of frame of reference. Study of properties of such fluids represents certain scientific interest from the point of view of existence of usual hydrodynamic properties, in particular, existence of wave movements under the action of the own gravitational field. The movement of the fluids with constant negative pressure is considered in spherical coordinates when only radial component of velocity u(r,t) is considered. We have established that for fluid type space vacuum with constant negative pressure the movement is possible only if the source function doesn’t depend on coordinates. In this case the velocity of the fluid is linear function of the distance from the beginning of coordinates that reminds Hubble’s law in cosmology. For ideal fluid with the equation of state of the type of quintessential we have established that movement of the fluid under the action of the own gravitational field for one-dimensional movement is possible in the case if its density exceeds some critical value, the movement of the fluid takes place in some bounded region 0 ≤ x ≤ xmax and its velocity changes from some critical value ucr to u = 0. We also studied the movement of the medium with the equation of state of the Chaplygin gas under its own gravitational field in one-dimensional case, and show that there are three different flow regimes.