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For the processes described by dynamical systems, closed trajectories of dynamical systems are in line with periodic oscillations. Therefore, there is a considerable interest in describing the bifurcations of the generation of closed trajectories from equilibrium when the parameters change. In typical one-parameter and two-parameter families of smooth dynamical systems on a plane, closed trajectories can be generated only from equilibrium – weak focus. In mathematical modeling in the theory of automatic control, in mechanics and in other applications, piecewise smooth dynamical systems are often used. For them, there are other bifurcations of the generation of closed trajectories from equilibrium. The paper describes one of them, which is a typical family of dynamical systems specified by a piecewise smooth vector field on a two-dimensional manifold depending on two small parameters. It is assumed that for zero values of the parameters the vector field has a singular point O on the line of discontinuity of the field, and the point O is stable; in one half-neighborhood of the point O the field coincides with a smooth vector field for which the point O is a weak focus with positive (negative) first Lyapunov value, and in the other half-neighborhood it coincides with a smooth vector field directed at the points of the line of discontinuity inside the first of the semi-neighborhoods. The paper describes bifurcations in the neighborhood of the point O as the parameters change, in particular, indicating the regions of the parameters for which the vector field has a stable closed trajectory.

The paper discusses issues connected with the use of an artificial neural network (ANN) to approximate the experimental data. One of the problems in the development of the ANN is the choice of an appropriate activation function for neurons of the hidden layer and adjusting the parameters of the function in the learning process of the network. The article discusses a three-layer perceptron with one hidden layer, each neuron of which has the activation function in the form of a Gaussian curve. The choice of radial basis activation function allows the use of the direct method of determining the weight coefficients – method of least squares in the process of network training. Thus the quality of the approximation depends on the correct choice of the value parameter of the activation function, which in this case is the width of the Gaussian bell curve. In practice, this parameter is determined by conducting numerical experiments. This is a rather time-taking process. In this paper we propose to define the value of this parameter by the training set, representing the coordinates of the test curve points set with the desired properties. These properties are based on the a priori data of the approximated functions (linear, quadratic, logarithmic, exponential relationship). Because the test curve is given in explicit form, the parameter of activation function is determined from the condition of reaching the minimum of the integral from the squared difference between the values of the test functions and the output of the network. This approach guarantees obtaining the approximating curve with good properties, in particular, it is characterized by the absence of so-called "oscillations" – many inflection points in its graph.

Описаны поля Киллинга на четырехмерных 2-симметрических лоренцевых многообразиях. Поля Киллинга играют важную роль в исследовании солитонов Риччи, которые впервые были рассмотрены Р. Гамильтоном. Солитоны Риччи являются обобщением эйнштейновых метрик на (псевдо)римановых многообразиях. Уравнение солитона Риччи изучалось на различных классах многообразий многими математиками. В частности, было найдено общее решение уравнения солитона Риччи на 2-симметрических лоренцевых многообразиях размерности четыре, доказана разрешимость этого уравнения в классе 3-симметрических лоренцевых многообразий. Описать поля Киллинга удается при помощи нормальных координат Бринкмана, существующих на лоренцевых многообразиях более общего класса - pp-волнах. Система дифференциальных уравнений, соответствующая уравнению киллинговых полей, может быть приведена к значительно более простому виду, что было сделано В. Глобке и Т. Лейстнером. Опираясь на этот результат, найдено общее решение данной системы дифференциальных уравнений и вычислена размерность алгебры киллинговых полей. Результаты, изложенные в настоящей работе, продолжают исследования солитонов Риччи на лоренцевых многообразиях., In the following paper, we describe Killing fields on 2-symmetric Lorentzian manifolds of dimension four. Killing fields play an important role in the study of Ricci solitons which were introduced by R. Hamilton. Ricci solitons are the generalization of the Einstein metrics on (pseudo)Riemannian manifolds. Ricci soliton equation was studied by many mathematicians on different classes of manifolds. In particular, in the recent author’s papers, solvability of the Ricci soliton equation on 3-symmetric Lorentzian manifolds was proved, and the general solution of the Ricci soliton equation on 2-symmetric Lorentzian manifolds was described. We describe Killing fields using Brinkmann normal coordinates which exist on the class of Lorentzian manifolds, the so-called pp-waves. The system of differential equations that corresponds to the Killing equation can be reduced to a simpler form. This was done by W. Globke and T. Leistner. By applying their result, the general solution of the system was found, the dimension of an algebra of Killing fields was calculated. The results stated in this paper continue author’s research on Ricci solitons on Lorentzian manifolds.

One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and only if the corresponding parallel transport to the connection is a solution of the Laplace equation for the Levy Laplacian or of the d'Alembert equation for the Levy d'Alembertian respectively (see [5, 6]). There are two approaches to define Levy type operators, both of which date back to the original works of Levy (see [7]). The first is that the Levy Laplacian (or Levy d'Alembertian) is defined as an integral functional generated by a special form of the second derivative. This approach is used in the works [5, 6], as well as in the paper [8] of Leandre and Volovich, where stochastic Levy-Laplacian is discussed. Another approach to the Levy Laplacian is defining it as the Cesaro mean of second order derivatives along the family of vectors, which is an orthonormal basis in the Hilbert space. This definition of the Levy Laplacian is used for the description of solutions of the Yang-Mills equations in the paper [10].The present work shows that the definitions of the Levy Laplacian and the Levy d'Alembertian based on Cesaro averaging of the second order directional derivatives can be transferred to the stochastic case. In the article the values of these operators on a stochastic parallel transport associated with a connection (vector potential) are found. In this case, unlike the deterministic case and the stochastic case of Levy Laplacian from [8], these values are not equal to zero if the vector potential corresponding to the stochastic parallel transport is a solution of the Maxwell's equations. As a result, two approaches to definition of the Levy Laplacian in the stochastic case give different operators. This situation is different from the flat deterministic case, which is discussed in [11].It can be expected that the work can be summarized in the non-commutative case of the Yang-Mills theory.DOI: 10.7463/mathm.0615.0822138

The article presents a statement of technique to research performances of multi-channel combo standalone information systems with positional analyzers of the signal states at the channel outputs. In most cases, in considered multi-channel systems there has been impossible to coincide in time the random moments of signals coming from the objects through various channels in all ways of encounter environment and conditions of practical application. The analyzer makes decision on the signal using the discrete operations on the quantized signals of the certain duration from the channel outputs. The analyzer performance is described by a set of Boolean algebra functions defined for all possible states of the signals at the outputs of the channels, and in the general case is specified in a perfect disjunctive normal form. To determine the validity or falsity of functions of the algebra of logics, which are calculated statements concerning the available or unavailable useful signal at the system input, on the authority of the Poretsky’s theorem and the theory of coincidence in time of the random pulse flow of the channels response because of uncorrelated and correlated noise, are obtained dependences to calculate the probabilities of false alarms and omissions of the signals in discrete combined systems. It is shown that the flows of false alarms because of noise at the channel outputs in the system are Poisson streams. On the basis of the ordinary Poisson flows the paper justifies the relationships for calculating the false alarms of the system with uncorrelated and correlated noise in the channels. The paper also justifies the relationships for performance of multichannel combined systems with positional analyzers of the channels states. Based on the obtained relationships was calculated the average coincidence frequency of the extended pulses of the channel response in a dualchannel system, depending on the noise cross-correlation coefficient with different extended pulse durations and relative response thresholds in the channels. The impact of cross-correlation noise has been analysed. The obtained results enable us to examine the performance of the multichannel autonomous information systems with positional analyzers of the signal states at the outputs of channels operating under uncorrelated and correlated noise.

One of the main reasons for interest in the Levy Laplacian and its analogues such as Levy d'Alembertian is a connection of these operators with gauge fields. The theorem proved by Accardi, Gibillisco and Volovich stated that a connection in a bundle over a Euclidean space or over a Minkowski space is a solution of the Yang-Mills equations if and only if the corresponding parallel transport to the connection is a solution of the Laplace equation for the Levy Laplacian or of the d'Alembert equation for the Levy d'Alembertian respectively (see [5, 6]). There are two approaches to define Levy type operators, both of which date back to the original works of Levy [7]. The first is that the Levy Laplacian (or Levy d'Alembertian) is defined as an integral functional generated by a special form of the second derivative. This approach is used in the works [5, 6], as well as in the paper [8] of Leandre and Volovich, where stochastic Levy-Laplacian is discussed. Another approach to the Levy Laplacian is defining it as the Cesaro mean of second order derivatives along the family of vectors, which is an orthonormal basis in the Hilbert space. This definition of the Levy Laplacian is used for the description of solutions of the Yang-Mills equations in the paper [10].The present work shows that the definitions of the Levy Laplacian and the Levy d'Alembertian based on Cesaro averaging of the second order directional derivatives can be transferred to the stochastic case. In the article the values of these operators on a stochastic parallel transport associated with a connection (vector potential) are found. In this case, unlike the deterministic case and the stochastic case of Levy Laplacian from [8], these values are not equal to zero if the vector potential corresponding to the stochastic parallel transport is a solution of the Maxwell's equations. As a result, two approaches to definition of the Levy Laplacian in the stochastic case give different operators. This situation is different from the flat deterministic case, which is discussed in [11]. It can be expected that the work can be summarized in the non-commutative case of the Yang-Mills theoryDOI: 10.7463/mathm.0515.0820322

The paper systematically describes an approach to solution of initial and initial-boundary value problems for evolution equations based on the representation of the corresponding evolution semigroups with the help of Feynman formulae. The article discusses some of the methods of constructing Feynman formulae for different evolution semigroups, presents specific examples of solutions of evolution equations. In particular, Feynman formula is obtained for evolution semigroups generated by multiplicative perturbations of generators of some initial semigroups. In this case semigroups on a Banach space of continuous functions defined on an arbitrary metric space are considered; Feynman formulae are constructed with the help of operator families, which are Chernoff equivalent to the initial unperturbed semigroups. The present result generalizes the author's paper \Feynman formula for semigroups with multiplicative perturbed generators" and some of the results of the joint with O.G. Smolyanov and R.L. Schilling paper \Lagrangian and Hamiltonian Feynman formulae for some Feller processes and their perturbations". The approach to the construction of Feynman formulae for semigroups with multiplicative and additive perturbed generators is illustrated with examples of the Cauchy problem for the Schrodinger equation, the approximation of transition probabilities of some Markov processes.Further, a wider class of additive and multiplicative perturbations of a particular generator | the Laplace operator | is considered in the paper. And Feynman formula for the solution of the Cauchy problem for a second order parabolic equation with unbounded variable coefficients is proved. In addition, the article describes a method for constructing Feynman formulae for solutions of the Cauchy | Dirichlet problem for parabolic differential equations. The method is also illustrated by a second order parabolic equation with variable coefficients. These results generalize some of the results of the work by Butko, Grothaus and Smolyanov \Lagrangian Feynman formulae for Second Order Parabolic Equations in Bounded and Unbounded Domains". The article also discusses some of the Feynman | Kac formulae and Feynman integrals related to the obtained Feynman formulae.

Introduction. The paper deals with the calculation of wooden arches taking into account the nonlinear relationship between stresses and instantaneous deformations, as well as creep and geometric nonlinearity, are considered. The analysis is based on the integral equation of the viscoelastoplastic hereditary aging model, originally proposed by A.G. Tamrazyan [1] to describe the nonlinear creep of concrete. Materials and Methods. The creep measure is taken in accordance with the work of I.E. Prokopovich and V.A. Zedgenidze [2] as a sum of exponential functions. The transition from the integral form of the creep law to the differential form is shown. The relationship between stresses and instantaneous deformations for wood under compression is determined from the Gerstner formula, and elastic work is assumed under tension. The solution is carried out using the finite element method in combination with the Newton-Raphson method and the Euler method according to the scheme that involves a stepwise increase in the load with correction of the stiffness matrix taking into account the change in the coordinates of the nodes with the sequential calculation of additional displacements of the nodes, which are due to the residual forces. The proposed approach for increasing the accuracy of determination of creep deformations at each step provides using the fourth-order Runge-Kutta method instead of the Euler method. Results. Based on the Lagrange variational principle, expressions are obtained for the stiffness matrix and the vector of additional dummy loads due to creep. The method developed by the authors is implemented in the form of a program in the MATLAB environment. Calculation examples are given for parabolic arches simply supported at the ends without an intermediate hinge and with an intermediate hinge in the middle of the span under the action of a uniformly distributed load. The results obtained are compared in the viscoelastic and viscoelastic formulation. The reliability of the results is validated through the calculation in the elastic formulation in the ANSYS software package. Discussion and Conclusions. For the arches considered, it is found that even with a load close to the instant critical, the growth of time travel is limited. Thus, the nature of their work under creep conditions differs drastically from the nature of the deformation of compressed rods.

The total probability formula for continuous random variables is the integral of product of two probability density functions that defines the unconditional probability density function from the conditional one. The need for calculation of such integrals arises in many applications, for instant, in statistical decision theory. The statistical decision theory attracts attention due to the ability to formulate the problems in a strict mathematical form. One of the technical problems solved by the statistical decision theory is the problem of dual control that requires calculation of integrals connected with the multivariate probability distributions. The necessary integrals are not available in the literature. One theorem on the total probability formula for vector Gaussian distributions was published by the authors earlier. In this paper we repeat this theorem and prove a new theorem that uses more familiar form of the initial data and has more familiar form of the result. The new form of the theorem allows us to obtain the unconditional mathematical expectation and the unconditional variance-covariance matrix very simply. We also confirm the new theorem by direct calculation for the case of the simple linear regression.

The technic of the processing of the meteorological data for conclusion on the presence of the time trends in the quantitative characteristics of the weather on the example of the analysis of the average yearly atmospheric temperature change at the meteorological station Minsk from 1989 is presented. The average yearly atmospheric temperature received from the measurements is approximated by the least square error method in the linear time dependence regression function. The linear time dependence regression function received in such a way has some positive growing (positive trend). The aim of this paper is to clarify the significance of this growth. For this aim, the usage of the regression analysis with its procedures of hypotheses testing is proposed. First of all, the performing of the demands presented to the regression analysis is checked: normality of the distribution of the disturbance and the homogeneity of the variance (dispersion) of the disturbance. The normality of the distribution of the disturbance was checked and confirmed by the Kolmogorov test. The homogeneity of the dispersion of the disturbance was checked and confirmed both by checking the hypotheses on the equality of the dispersions of two normal distributions and by the Smirnov test for checking the hypotheses on the equality of two distributions. For checking the significance of the positive trend of the yearly mean temperature, the hypotheses on the significance of the coefficients of the linear regression function by the Student t-statistics and the hypothesis on the linear connection presence by the analysis of variance were checked. As the result, the insignificance of the positive linear trend from 1998 to 2016 and from 1998 to 2017 and its significance from 1998 to 2018 and from 1998 to 2019 on the level of significance 0.05 for mean average yearly atmospheric temperature at the meteorological station Minsk was stated.

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.

Последнее время становится актуальным изучение (псевдо)римановых многообразий с различными афинными связностями, отличными от связности ЛевиЧивита. Метрическая связность с векторным кручением (также известная как полусимметрическая связность) является одной из часто рассматриваемых связностей. Связь между конформными деформациями римановых многообразий и метрическими связностями с векторным кручением на них была установлена в работах К. Яно. А именно, риманово многообразие допускает метрическую связность с векторным кручением, тензор кривизны которой равен нулю, тогда и только тогда, когда оно является конформно плоским. В данной работе впервые исследуется уравнение Эйнштейна на трехмерных локально симметрических (псевдо)римановых многообразиях с метрической связностью с инвариантным векторным кручением. Получена теорема о том, что все такие многообразия либо являются многообразиями Эйнштейна относительно связности Леви-Чивита, либо являются конформно плоскими., The study of (pseudo)Riemannian manifolds with different metric connections different from the Levi-Civita connection has become a subject of current interest lately. A metric connection with vectorial torsion (also known as a semi-symmetric connection) is a frequently considered one of them. The correlation between the conformal deformations of Riemannian manifolds and metric connections with vectorial torsion on them was established in the works of K. Yano. Namely, a Riemannian manifold admits a metric connection with vectorial torsion, the curvature tensor of which is zero, if and only if it is conformally flat. In this paper, we study the Einstein equation on three-dimensional locally symmetric (pseudo)Riemannian manifolds with metric connection with invariant vectorial torsion. We obtain a theorem stating that all such manifolds are either Einstein manifolds with respect to the Levi-Civita connection or conformally flat., №4(104) (2020)

Hardy spaces play an important role in the complex analysis and its numerous applications. However, unlike a one-dimensional case, the Hardy-type spaces in a polydisc are investigated a little. In this paper, the integral representations of the classes of n-harmonic in a polydisc Un functions are received, in particular the characterization of n-harmonic functions in a polydisc which can be represented as a multiple Poisson integral from functions, measurable on a skeleton of a polydisc, from a class where 1 < pi < + ∞, is given. At the proof of the main result the general methods of the complex and functional analysis, theory of Hardy classes is used.

Constant development of modern society is setting higher requirements to specialist training. In this connection, riskmanagement concepts need to be developed in order to take important desicions for educational establishment management. To create a qualitative instrument for managing educational risks quantitative techniques for risks assessment in higher education are considered in the paper. Risk assessment has been made by experts. The data received has been used for minimizing educational risks in managerial decision making. Determining an expert panel absence of personal interest in the matter has been taken into account to increase the quality of decision-making. Expert grouping has been based on the reasonableness evaluation of the issue in question. Then experts have assessed the educational risks on the proposed scale. Overall expert assessments have been calculated using mathematical statistics and dimension of agreement has been determined. For this purpose, the average rank and the average rank deviation from the risk universe have been determined and a multivariable rank correlation coefficient has been calculated. The given coefficient shows the dimension of the expert agreement. And the importance of the multivariable rank correlation coefficient has been assessed for evaluating the quality of the decision made and making conclusions on the data obtained. As a result, the most relevant risks in education have been identified and adequate measures have been taken to minimize those risks.

Проведен сравнительный анализ вычислительных свойств модифицированного метода крупных частиц на примере одномерных тестовых задач газовой динамики в широком диапазоне параметров течения. Численные результаты сопоставлены с автомодельными решениями и данными, полученными по схемам высокой разрешающей способности от второго до шестого порядков аппроксимации. Представленная схема продемонстрировала вычислительную эффективность и конкурентоспособность., The paper presents a comparative analysis of the computational properties of a modified large-particle method on one-dimensional gas dynamics test problems in a wide range of flow parameters. The numerical results are compared with self-similar solutions and data obtained by high-resolution schemes from the second to the sixth order of approximation. It is shown that the presented scheme is numerically efficient and competitive., ВЫЧИСЛИТЕЛЬНЫЕ МЕТОДЫ И ПРОГРАММИРОВАНИЕ: НОВЫЕ ВЫЧИСЛИТЕЛЬНЫЕ ТЕХНОЛОГИИ, Выпуск 2 2019

This paper considers the dynamics of the abscissa and ordinate points in the far zone meridian six degree conformal Gauss-Kruger projection. Performed calculations and graphical representation of the classic formulas of conformal projection.

Ранее была рассмотрена задача: на части границы области Ω ⊂ R3 находится нагреватель, имеющий регулируемую температуру. Требуется найти такой режим работы нагревателя, чтобы средняя температура в некоторой подобласти D области Ω принимала заданное значение. В данной работе рассматривается аналогичная задача граничного управления связанная с псевдопараболическим уравнением на отрезке. На части границы рассматриваемого отрезка задается значение решения, которое содержит параметр управления. Ограничение для допустимого управления задано в таком виде, что среднее значение решения в некоторой части рассматриваемого отрезка принимает заданное значение. Решается вспомогательная задача методом разделения переменных. Искомая задача сводится к интегральному уравнению Вольтерра второго рода. Методом преобразования Лапласа доказываются теоремы о существовании и единственности допустимого управления., Previously, a mathematical model for the following problem was considered. On a part of the border of the region Ω ⊂ R3 there is a heater with controlled temperature. It is required to find such a mode of its operation that the average temperature in some subregion D of Ω reaches some given value. In this paper, we consider a similar boundary control problem associated with a pseudo-parabolic equation on a segment. On the part of the border of the considered segment, the value of the solution with control parameter is given. Restrictions on the control are given in such a way that the average value of the solution in some part of the considered segment gets a given value. The auxiliary problem is solved by the method of separation of variables, while the problem in consideration is reduced to the Volterra integral equation of the second kind. By Laplace transform method, the existence and uniqueness theorems for admissible control are proved., №2(98) (2018)

In this paper, using the semi-continuous summation methods, we build a class of means of Fourier series, that converge almost everywhere, as well as in metrics of weighted spaces and the space С of continuous periodic functions. It was found that a sufficient condition for such convergence, and the equity of weighting estimates of corresponding maximal operators, is the generalized condition of B.Nagy. The results are applied to study of behavior of exponential means of Fourier series and include the cases of classical means of Cesaro, Riesz, and Abel-Poisson.

A development of the direct Lyapunov method for the analysis of transient stability of a system of synchronous generators based on the determination of non- stable equilibria on a multidimensional sphere.We consider the problem of transient stability analysis for a system of synchronous generators under the action of strong perturbations. The aim of our work is to develop methods to analyze a transient stability of the system of synchronous generators, which allow getting trustworthy results on reserve transient stability under different perturbations. For the analysis of transient stability, we use the direct Lyapunov method.One of the problems for this method application is to find the Lypunov function that well reflects the properties of a parallel system of synchronous generators. The most reliable results were obtained when the analysis of transient stability was performed with a Lyapunov function of energy type. Another problem for application of the direct Lyapunov method is to determine the critical value of the Lyapunov function, which requires finding the non-stable equilibria of the system. Determination of the non-stable equilibria requires studying the Lyapunov function in a multidimensional space in a neighborhood of a stable equilibrium for the post-breakdown system; this is a complicated non-linear problem.In the paper, we propose a method for determination of the non-stable equilibria on a multidimensional sphere. The method is based on a search of a minimum of the Lyapunov function on a multidimensional sphere the center of which is a stable equilibrium. Our method allows, comparing with the other, e.g., gradient methods, reliable finding a non-stable equilibrium and calculating the critical value. The reliability of our method is proved by numerical experiments. The developed methods and a program realized in a MATLAB package can be recommended for design of a post-breakdown control system of synchronous generators or as a component of a real-time control system.

Let $A_j,B_j$ $(j=0,1,\ldots)$ be $m \times m$ matrices, whose elements are complex numbers, $A_j$ are selfadjoint matrices and $B_j^{-1}$ exist. We study the deficiency index problem for minimal closed symmetric operator $L$ with domain $D_L$, generated by the Jacobi matrix $\textbf{J}$ with entries $A_j,B_j$ in the Hilbert space $l_m^2$ of sequences $ u=(u_0,u_1, \ldots), u_j \in C^m$ by mapping $u \rightarrow \textbf{J}u$, i.e. by the formula $Lu=lu$ for $u \in D_L$, where $lu=((lu)_0,(lu)_1, \ldots)$ and $$ (lu)_0:=A_0u_0+B_0u_1, \quad (lu)_j:=B^*_{j-1}u_{j-1}+A_ju_j+B_ju_{j+1}, \;\; j=1,2, \ldots $$ It is well known that the case of the minimal deficiency numbers of the operator $L$ corresponds to the determinate case, and the case of the maximal deficiency numbers of this operator corresponds to the completely indeterminate case of the matrix power moment problem. In this paper we obtain new conditions of the minimal, maximal and not maximal deficiency numbers of the operator $L$ in terms of the entries of the matrix $\textbf{J}$. The special attention is paid to the case $m=1$, i.e. we present some conditions on the elements of the numerical tridiagonal Jacobi matrix, which ensure the realization of the determinate case of the classical power moment problem., 9 pages, in Russian

This paper summarizes the main results of [1] for the spectral asymptotics of the damped wave equation. We define the notion of a high frequency abscissa, a sequence of eigenvalues with imaginary parts going to plus or minus infinity and real parts going to some real number. We give theorems on the number of such high frequency abscissas for particular conditions on the graph. We illustrate this behavior in two particular examples.

In this paper we describe an implementation of a method for blurred pictures restoring. We consider uniform, linear photo-camera shift (movement) while picture is taken. In this case, a picture, formed on a chip of a camera, is not static, but moving along some straight line, and as a result photo is blurred. Blurred picture can be seen as a convolution of two functions – one describes initial picture, another is point-spread (or blurring) function (PSF) [1]–[3]. “Deblurred” picture can be found as a deconvolution of the given picture with known PSF [1], [2]. To do this it is convenient to use Fourier transform and apply the convolution theorem. But it leads to division by 0, and thus the method is unstable. We suggest two heuristics to avoid division by 0 and compare them with the known one.