# Bulletin of Taras Shevchenko National University of Kyiv Series Physics and Mathematics

Published by Taras Shevchenko National University of Kyiv

Print ISSN: 1812-5409

Published by Taras Shevchenko National University of Kyiv

Print ISSN: 1812-5409

Publications

The article presents an analysis of the educational and professional program "Informatics" of the first (bachelor's) level of higher education in the sphere of knowledge 12 "Information Technology", specialty 122 "Computer Science", implemented at the Faculty of Computer Science and Cybernetics, Taras National University of Kyiv Shevchenko with educational and professional programs of the same level and specialty of other institutions of higher education of Ukraine in terms of program results. During the analysis, they were compared with the approved standard of the first (bachelor's) level of higher education in the specialty 122 "Computer Science". In order to analyze the authors developed a database of educational programs. The ratio of program results in different programs by common specialty is analyzed.

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The article presents a comparative analysis of the educational and professional program "Informatics" of the first (bachelor's) level of higher education in the field of knowledge 12 "Information Technology", specialty 122 "Computer Science", which is implemented at the Faculty of Computer Science and Cybernetics Taras Shevchenko National University of Kyiv with educational and professional programs of the same level and specialties of other institutions of higher education in Ukraine. During the analysis, they were compared with the approved standard of the first (bachelor's) level of higher education in the specialty 122 "Computer Science". In order to conduct a comparative analysis, the authors developed and completed a educational program database. The result of the study is checking the educational program for completeness, that is lack of competencies that are not provided by any discipline and sufficiency, that is the lack of disciplines that do not provide any competence. The ratio of competencies and disciplines in different programs in a common specialty is analyzed.

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A method for determining the characteristics of functional gradient materials (FGM) for providing zero thermal stresses in an infinite layer with given constant thermal loads is proposed. We assume that the classical convective conditions of heat transfer are given on the surfaces of the layer, the temperature field is stationary, the characteristics of the FGM are described by the model of a simple mixture, the characteristics of the thermo-stressed state and the material depend only on the transverse variable. Precise analytical expressions were obtained for the distribution of the concentration of one of the materials on the thickness of the layer in the absence of mass forces and heat sources, which provides zero longitudinal stresses.

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Knowledge of the basic theoretical foundations and possession of the skills of applying Mathematical Analysis, Differential Equations, Linear Algebra, Analytic Geometry, Discrete Mathematics, Numerical Methods, Theory of Probability and Mathematical Statistics are important for the professional training of future specialists in the field of information technologies. Since future specialists in the field of information technology require a deep mathematical training, the curricula of IT specialties usually contain various mathematical disciplines from this list. A specialist of any IT profile must have specific professional features and competencies. These characteristics of future IT professionals should preferably be formed in the process of studying both special and general scientific disciplines. The result of the educational process is formation of both hard and soft skills of students. This article is devoted to the study of some features of the use of information technology in the process of teaching some questions of mathematical disciplines in English to students of IT profile who are not native speakers of this language.

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The formulation of problem on the forced resonant vibration and dissipative heating of layered element of structure containing both piezoelectric and electrically passive layers is considered. The improved problem statement taking account of both shear strain and rotatory inertia as well as geometrical nonlinearity is developed. Particular statement of the problem of axisymmetric vibration and dissipative heating of three layer cylidrical shell is formulated. It is assumed that the core layer of the shell is composed of the electrically passive material while the outer layers are manufactured from the piezoceramics. Theory of coupled thermo-electro-viscoelasticity is used to derive the problem statement in the case of monoharmonic loading. Within this theory, the concept of complex-value modulae is applied to formulate the relations between main field characteristics. It is also supposed that the piezo-active material characteristics do not depend on the temperature. Then the coupled problem is reduced to the problem of mechanics on the forced nonlinear vibrations and dissipative heating of the layered plate. Complete set of complex analogs of motion equations, geometric equations and constitutive relations was used to derive the resolving system of equations. Numerical method to attack this nonlinear system of ordinary differential equations supplemented with necessary boundary conditions is developed.

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The modification of a two-dimensional model of incompressible viscous fluid motion along a deformed thick-walled tube from viscoelastic bioactive material is proposed in connection to the modeling of blood flow along the arterial bed is proposed. The motion of a viscous incompressible fluid is described by a system of equations including the Navier-Stokes equations and the continuity equation. The behavior of the tube wall material is described by a 5-element rheological model with one active element. The solution of the problem is solved setting boundary conditions on the interface of the two media, the outer surface of the tube is considered as non-moving. At the end of the tube, a zero-dimensional Frank model with regulation is considered, as a model of the microcirculatory bed. The dispersion equation for the propagation of wave velocity is obtained for the case of active properties of tube, the amplitudes of fluid velocities, wall displacements, and fluid and tube pressures. Numerical computations have been carried out for the model parameters corresponded to the normal and pathological arterial wall.

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The paper proposes a methodology for building an effective system of self-diagnostics of information systems on the example of Ukrainian enterprises in the metallurgical, energy and chemical industries. The article shows that if the dependence of the probability of issuing information on the time of execution of the element of elementary checks is known, it is enough to carry out checks within a predetermined time, when a given probability value is given. It is investigated that in the information system of the enterprise the self-control organized by means of elementary checks occurs at arbitrary moments of time of functioning of modules on purpose and the relation of probability of delivery of the information which occurs by comparison of deviations from average values. Criteria for the adequacy of diagnostic information in the absence of restrictions on the implementation of basic tests and in the presence of restrictions on the implementation of basic tests.

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In this paper, the representation of random processes in the form of random series with uncorrelated members obtained in the work by Yu. V. Kozachenko, I.V. Rozora, E.V. Turchina (2007) [1]. Similar constructions were studied in the book by Yu. V. Kozachenko and others. [2] in the general case. However, there are additional difficulties in construction of models of specific process, such as, for example, selection of the appropriate basis in L_2(R). In this paper, models are constructed that approximate the Gaussian process with a stable correlation function $\rho_{\alpha} (h) = E X_{\alpha}(t + h) X_{\alpha}(t) = B^2 \exp{-d|h|^{\alpha}}, \alpha > 0, d > 0$ with parameter $\alpha = 2$, which is a centered stationary process with a given reliability and accuracy in the space L_p ([0,T]). And also the rates of convergence of the models are found, the corresponding theorems are formulated. Methods of representation and main properties of the process with a stable correlation function $\rho_2(h) = B^2 \exp{-d|h|^2}, d > 0$ are considered. As a basis in the space L_2(T) Hermitian functions are used.

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Today, the theory of random processes and time series prediction is widely used in various fields of science, not only in natural fields. That is why one of the urgent problems is to build a mathematical model of a random process and study its properties. Numerical modeling tasks become especially important due to the powerful capabilities of computer technology, which allows you to create software modeling tools and predict the behavior of a random process. There are different methods of modeling random processes and fields. In some works related to the modeling of random processes, the issues of accuracy and reliability have not been studied. In [1, 2, 3] for various stochastic processes and fields this problem was investigated. In this paper the question of accuracy and reliability of the constructed model is considered. This means that we first build the model and then test it using some adequacy tests with known accuracy and reliability. We also find the estimators of the model parameters using methods of moments. All theoretical results are applied to develop software for model construction of stochastic processes.

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Acoustic radiation force effect on a liquid spherical drop placed in the vicinity of an ideal liquid free surface is studied. The problem of determination of the radiation forces acting on an obstacle in ideal liquid is formulated with respect to the Lagrange coordinate system. Thus, the radiation pressure is defined as time-averaged value of the acoustic pressure over the obstacle surface. This approach is adequate if, at determining of the acoustic pressure in a fluid, the deviation of the pressure from the harmonic law in time domain is taken into account in the obstacle vicinity. An action of the acoustic radiation force on a spherical drop of ideal liquid placed in turn in a liquid by its free plane surface is studied here for the case of the incident plane sound wave propagating perpendicularly to the liquid boundary. As a result, the liquid sphere is appeared to be located in the standing sound wave of pressure which has its displacement node at the free surface. Problem solution is obtained as a three step procedure. Initially, solution of the problem of an incident wave scattering at the drop is derived. With making use of the results obtained, the second step encompasses determining of hydrodynamic forces acting on the liquid spherical drop with their subsequent averaging over the suitable time interval at the third step. It is found there frequencies of the incident wave exist that provide zero radiation force acting on the drop which is immobile in this case. These equilibrium positions of the spherical drop could be stable or unstable with respect to the incident wave frequency variation.

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The proposed work analyzes the design features of the acousto-optical deﬂector and ﬁlter on paratelurite. It is shown that under certain conditions the acousto-optical deﬂector can be used as an acousto-optical ﬁlter (as an element that performs spectral ﬁltering of the incident light beam). The fundamental possibility of creating a monochromatic light source with a variable wavelength and a spectrum width of about 5 nm using an acousto-optical deﬂector as an element that adjusts the original wavelength is shown experimentally. As a broadband light source in this system, a semiconductor laser operating in subthreshold mode was used. The dependence of the output wavelength on the acoustic frequency is obtained. The comparison of experimental data with the calculated ones is given, it is shown that they have small diﬀerences.

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There is a thin absolutely rigid inclusion that in a cross-section represents three segments broken line in an infinite elastic medium (matrix) that is in the conditions of antiplane strain. The inclusion is under the action of harmonic shear force Pe^{iwt} along the axis Oz. Under the conditions of the antiplane strain the only one different from 0 z-component of displacement vector W (x; y) satisfies the Helmholtz equation. The inclusion is fully couple with the matrix. The tangential stresses are discontinuous on the inclusion with unknown jumps. The method of the solution is based on the representation of displacement W (x; y) by discontinuous solutions of the Helmholtz equation. After the satisfaction of the conditions on the inclusion the system of integral equations relatively unknown jumps is obtained. One of the main results is a numerical method for solving the obtained system, which takes into account the singularity of the solution and is based on the use of the special quadrature formulas for singular integrals.

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The article deals with analytical solution and adaptation to the parameter estimation of the SIR model of the epidemic. By a special replacement of the exponential function by inverse proportionality, the approximate general solution of the SIR model is found. It is spoken in detail about the process of integration of ordinary differential equations of the SIR model. The equality of the sum of the obtained analytical solutions and population size is checked. The obtained solutions are simple and understandable. To parametrically estimate the SIR model, its general solution is adapted to paired linear regressions. The article is of interest for students, graduate students and scientists involved in mathematical epidemiology.

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Model of normal adhesive contact between elastic bodies with stochastic surface roughness is under consideration. Roughness is simulated by Winkler-Fuss nonlinear layer, which can resist to compressive and tensile (in the case of adhesion) contact stresses. Mechanical properties of the layer are determined by statistical theories of adhesive contact between nominally flat rough surfaces. The contact of solids is described by nonlinear boundary integral equations with non-monotonic operators. Their solutions determine reduction of effective thickness of rough layer, contact stresses, contact region, adhesion force. Formulas for adhesion force calculation are presented for the most frequent nominal gap between solids in contact for DMT–theory of contact.

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The article considers the method of evaluating the effectiveness of the textual content of the advertising message. The basis of the proposed method is the linguistic principles, in which the effectiveness of the text is analyzed from the view of the decoding of information by the target recipient of the message. Materials used for the study were the texts used in SEO promotion of information resources, the method is based on the approach of breaking the text into keywords and phrases. The method of evaluating the effectiveness of the text is based on an analogue of the method of mechanical verification of the relevance of the text by the search engine, which was expanded by the parameters of expert evaluation of the quality of text construction.

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In continual mechanics sedimentation of aggregating particles in concentrated suspensions are determined by the mass and momentum conservation laws for each component of the suspension. The resulting quasilinear system of differential equations governing the flow could be hyperbolic, strongly strictly or weakly hyperbolic depending on the model accepted. The type and Eigenvalues of the matrix influence the characteristics of the pattern formation during the sedimentation that is essential for the model application in modern medical, microbiological and nanofluidic technologies. In this paper the hyperbolicity of the three-phase model of aggregation and sedimentation of micro/nanoparticles is studied.

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Sedimentation of the aggregating particles in the gravity field is widely used as an easy and cheap test of the suspension stability of different technical suspensions, blood and nanofluids. It was established the tube inclination makes the test much faster that is known as the Boycott effect. The dependence of the sedimentation rate on the angle of inclination is complex and poorly understood yet. In this paper the two phase model of the aggregating particles is generalized to the inclined tubes. The problem is formulated in the two-dimensional case that corresponds to the narrow rectangle vessels or gaps of the viscosimeters of the cone-cone type. In the suggestion of small angles of inclination the equations are averaged over the transverse coordinate and the obtained hyperbolic system of equations is solved by the method of characteristics. Numerical computations revealed the increase in the initial concentration of the particles, their aggregation rate, external uniform force and inclination angle accelerate the sedimentation while any increase in the fluid viscosity decelerates it that is physically relevant. Anyway, the behaviors of the acceleration are different. Based on the results, a novel method of estimation of the suspension stability is proposed.

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This work presents the analysis of experimental data on studies of optical and nonlinear optical properties of lyotropic ionic liquid crystals of potassium caprylate doped with electrochromic viologen admixtures, and smectic glasses of thermotropic ionic liquid crystals of cobalt alkanoates homologous series (number of carbon atoms in alkanoate chain n = 7, 9, 11) and their multicomponent mixtures. Prior to performing nonlinear optical experiment the optical absorption spectra for all samples were investigated. Laser induced dynamic grating recording under the action of nanosecond laser pulses was realized, observed and analyzed for the proposed absorptive media. It was discovered that studied materials are characterized by cubic optical nonlinearity and have values of cubic nonlinear susceptibility χ(3) and hyperpolarizability γ comparable with the best characteristics of organic dyes. The possible mechanism of nonlinear response in studied systems was considered on the base of obtained data. The nonlinear response mechanism is connected with nonlinear polarization of π-electrons in the field of laser radiation.

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Investigation of sub-gaussian random processes are of special interest since obtained results can be applied to Gaussian processes. In this article the properties of trajectories of a sub-Gaussian process drifted by a curve a studied. The following functionals of extremal type from stochastic process are studied: $\sup_{t\in B}(X(t)-f(t))$, $\inf{t\in B}(X(t)-f(t))$ and $\sup_{t\in B}|X(t)-f(t)|$. An alternative estimate of exceeding by sub-Gaussian process a level, given by a continuous linear curve is obtained. The research is based on the results obtained in the work \cite{yamnenko_vasylyk_TSP_2007}. The results can be applied to such problems of queuing theory and financial mathematics as an estimation of buffer overflow probability and bankruptcy

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The damping of vibrations of rectangular plates by means of both viscoelastic layers and using piezoelectric inclusions is considered. For modeling viscoelastic properties of passive and piezoelectric materials, linear models of integral type viscoelasticity are used, which are most effective for simulating the dissipative properties of materials in the linear region. In the case of taking into account the influence of the piezoelectric inclusions on the rigid characteristics of the passive plate and in other types of boundary conditions (for example, with rigid fixing of the ends), the finite element method was used to solve the problem of damping. The solutions of concrete problems of damping of stationary and non-stationary vibrations of plates using analytical and finite element methods are given. On the basis of the aforementioned approach, algorithms for solving dynamical problems with both fully and partially electrodes are implemented.

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The study of the analytical properties of random processes and their functionals, without a doubt, was and remains the relevant topic of the theory of random processes. The first result from which the study of the local properties of random processes began is Kolmogorov’s theorem on sample continuity with probability one. The classic result for Gaussian random processes is Dudley’s theorem. This paper is devoted to the study of local properties of sample paths of random processes that can be represented as a sum of squares of Gaussian random processes. Such processes are called square-Gaussian. We investigate the sufficient conditions of sample continuity with probability 1 for square-Gaussian processes based on the convergence of entropy Dudley type integrals. The estimation of the distribution of the continuity module is studied for square-Gaussian random processes. It is considered in detail an example with an estimator (correlogram) of the covariance function of a Gaussian stationary random process. The conditions on continuity of correlogram’s trajectories with probability one are found and the distribution of the continuity module is also estimated.

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The rings torsion theory that is based on the assumption about flat rigid cross-section was suggested by the authors in the previous papers. The analytical expressions of torsional stiffness have been derived for different kind of loads: pure moment, shear force and surface pressure. In the present paper the analytical model of flange with attached cylindrical shell deforming under internal pressure is suggested. The mechanical system is split into two parts (flange and shell) with the help of imaginary section method. An unknown shear force and bending moments are applied to both parts according to this method. Therefore flange is loaded under internal pressure, shear force and bending moments. As mentioned above, for all these loads the angle of flange cross-section rotation can be presented in analytical form based on the rings torsion theory. Full rotation angle is presented as a sum of these angles. The radial displacement of imaginary section was determined on the basis of the assumption about flat rigid cross-section. On another hand, the rotation angle and radial displacement of imaginary section are determined on the base of the cylindrical shell bending theory too. Two linear equations in the unknown shear force and bending moment were derived by equating corresponding expressions. In such а way the analytical model of flange with attached shell deforming was built. The comparison calculations by finite element methods confirmed the adequacy of proposed model.

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The previously developed direct cutting-out method in application to isotropic materials, in particular to bodies with thin inhomogeneities in the form of cracks and thin deformable inclusions is extended to the case of taking into account the possible anisotropy of the material. The basis of the method is to modulate the original problem of determining the stress state of a limited body with thin inclusions by means of a technically simpler to solve problem of elastic equilibrium of an infinite space with a slightly increased number of thin inhomogeneities, which in turn form the boundaries of the investigated body. By loaded cracks we model the boundary conditions of the first kind, and by absolutely rigid inclusions embedded into a matrix with a certain tension – the boundary conditions of the second kind. Using the method of the jump functions and the interaction conditions of a matrix with inclusion, the problem is reduced to a system of singular integral equations, the solution of which is carried out using the method of collocations. Approbation of the developed approach is carried out on the problem of elastic equilibrium of anisotropic (orthotropic in direction of shear) half-space with a symmetrically loaded very flexible inclusion (a crack) at jammed half-space boundary. The influence of inhomogeneity orientation and the half-space material on the generalized stress intensity factors were studied.

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Investigating the Stokes vector of light reflected from the surface of the optical glass, the presence of an anisotropic surface layer was established. The phase difference between radiation p- and s-components is revealed, which varies depending on the angle of incidence. This shows a weak anisotropy. Assuming that the anisotropic layer has an increased refractive index due to its chemical-mechanical treatment, it can be considered as some near-surface weakly guiding gradient burried waveguide. The possibility of coming radiation into such a planar waveguide using a coupling total reflection prism was investigated. The inspection showed a violation of the total internal reflection, unequal for p- and s-polarizations, which confirms the presence of the subsurface layer and its anisotropy. The absorption of radiation, which could be compared with the excitation of modes, is small. This is due to the properties of the prism material. Also, higher order modes are absorbed better. There is no significant angular dependence of the polarization degree of the output beam; however, it is higher in the case of s-polarization.

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In the presented paper, the limiting state of the orthotropic plates weakened by the periodic system of collinear cracks under biaxial external loading is studied on the basis of the modified crack model of the Leonov-Panasyuk-Dagdale. The material of plate satisfies the strength condition of the general form. On the basis of the solution of a similar problem for an orthotropic plate with one crack, we obtain the relations for determining the basic parameters of a crack model, such as the size of the process zones, the stresses in these zones, and the opening at the top of the cracks. The criterion of critical crack opening is selected as a fracture criterion. On the example of a material satisfying Hoffman strength criterion (generalization of the Mises–Hill criterion, which takes into account the dependence of the difference between the tensile and compressive strength of unidirectional composite materials), the fracture mechanism of a plate weakened by the periodic system of collinear cracks was investigated. The influence of the degree of material anisotropy and biaxiality of external loading on the fracture process and the limiting state of the plate are shown.

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Small dimensional transition metal carbides (MXenes) are promising materials for the development of photocatalysts and are highly efficient cocatalysts for industrial TiO_2 (P25). Thus, in the Ti_3C_2@TiО_2 nanocomposite obtained by layering Ti_3C_2 nanoplates, the ability to separate charge carriers increases due to the high electrical conductivity of TiC_{1-х}. The task of forming the TiC_{1-х}@TiО_{2-х} nanocomposite by direct synthesis with n-TiO_2 is promising, which allows to increase the quality of contact between the shell and the nanocomposite core and to reduce the number of intermediate stages of synthesis. In addition, highly dispersed TiC has high values of hardness, melting point, modulus of elasticity and shear and has the prospect of use in materials science in plasma spraying coatings. In work ТіС was synthesized on the surface of TiO_2 - the shell of the modified micropowder TiH_2/TiO_2/С during reductive annealing in vacuum using TiH_2 as a source of atomic hydrogen. After a series of annealing at 535 ºС - 600 ºС, the Ti2p- C1s- and O1s- spectra of surface atoms were obtained. The main stages of TiC synthesis in the TiO_2/С conversion reaction were established by the XPS method. The use of TiH_2 as a source of atomic hydrogen in nanosystems of the «core/shell» type is proposed for local synthesis on the surface of nanoobjects in a vacuum or inert atmosphere.

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Among the old results, there are only some results on the representation type of semigroups, namely, for a finite quite simple semigroup (I. S. Ponizovsky) and some semigroups of all transformations of a finite set (I. S. Ponizovsky, C. Ringel); these papers were discussed on finite representation type. If we talk about new results, and even for semigroup classes, then it should be noted works on representations of the semigroups generated by idempotents with partial zero multiplication (V. M. Bondarenko, O. M. Tertychna), semigroups generated by the potential elements (V. M. Bondarenko, O. V. Zubaruk) and representations of direct products of the symmetric second-order semigroup (V. M. Bondarenko, E. M. Kostyshyn). Such semigroups can have both a finite and infinite representation type. V. M. Bondarenko and Ja. V. Zatsikha described representation types of the third-order semigroups over a field, and indicate the canonical form of the matrix representations for any semigroup of finite representation type. This article is devoted to the study of similar problems for oversemigroups of commutative semigroups.

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The article is devoted to the review of conditional test generation, one of the most promising fields of natural language processing and artificial intelligence. Specifically, we explore monolingual local sequence transduction tasks: paraphrase generation, grammatical and spelling errors correction, text simplification. To give a better understanding of the considered tasks, we show examples of good rewrites. Then we take a deep look at such key aspects as publicly available datasets with the splits (training, validation, and testing), quality metrics for proper evaluation, and modern solutions based primarily on modern neural networks. For each task, we analyze its main characteristics and how they influence the state-of-the-art models. Eventually, we investigate the most significant shared features for the whole group of tasks in general and for approaches that provide solutions for them.

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In the present work we considered the solution of one periodic optimal regulated boundary value problem by the asymptotic method. For the solution of the problem with extended functional writing, boundary conditions and Euler-Lagrange equations were found. The approach to the solution of the problem depending on a small parameter by seeking a system of nonlinear differential equations and solving Euler-Lagrange equations, the solution of the general problem in the first approach comes down to solving two nonlinear algebraic equations.

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We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function $f є W^r [0,1]$ by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function $f є C^r [0,1] \cap \Delta^0$ where $\Delta^0$ is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider $r є , r > 2$. In [8] is consider $r є , r > 2$. It was proved that for monotone approximation estimates of the form (1) are fails for $r є , r > 2$. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6]). In [5] is consider $r є , r > 2$. In [6] is consider $r є , r > 2$. It was proved that for convex approximation estimates of the form (1) are fails for $r є , r > 2$. In this paper the question of approximation of function $f є W^r \cap \Delta^1, r є (3,4)$ by algebraic polynomial $p_n є \Pi_n \cap \Delta^1$ is consider. The main result of the work generalize the result of work [8] for $r є (3,4)$.

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We discuss whether on not it is possible to have interpolatory estimates in the approximation of a function f \in W^r [0,1] by polynomials. The problem of positive approximation is to estimate the pointwise degree of approximation of a function f \in C^r [0,1] \Wedge \Delta^0, where \Delta^0 is the set of positive functions on [0,1]. Estimates of the form (1) for positive approximation are known ([1],[2]). The problem of monotone approximation is that of estimating the degree of approximation of a monotone nondecreasing function by monotone nondecreasing polynomials. Estimates of the form (1) for monotone approximation were proved in [3],[4],[8]. In [3],[4] is consider r \in N, r>2. In [8] is consider r \in R, r>2. It was proved that for monotone approximation estimates of the form (1) are fails for r \in R, r>2. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is that of estimating the degree of approximation of a convex function by convex polynomials. The problem of convex approximation is consider in ([5],[6],[11]). In [5] is consider r \in N, r>2. It was proved that for convex approximation estimates of the form (1) are fails for r \in N, r>2. In [6] is consider r \in R, r\in(2;3). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(2;3). In [11] is consider r \in R, r\in(3;4). It was proved that for convex approximation estimates of the form (1) are fails for r \in R, r\in(3;4). In [9] is consider r \in R, r>4. It was proved that for f \in W^r [0,1] \Wedge \Delta^2, r>4 estimate (1) is not true. In this paper the question of approximation of function f \in W^r [0,1] \Wedge \Delta^2, r>4 by algebraic polynomial p_n \in \Pi_n \Wedge \Delta^2 is consider. It is proved, that for f \in W^r [0,1] \Wedge \Delta^2, r>4, estimate (1) can be improved, generally speaking.

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An approach for approximating unknown densities of potentials in the study of the stressed state of a flat viscoelastic piecewise homogeneous body with inclusions, bounded by piecewise smooth contours, is proposed. The method is based on the construction of a system of boundary-time integral equations to determine the unknown densities of potentials along the contours of the inclusions. The approximation of the unknown densities of potentials was performed taking into account the singularity of the stressed state of a flat viscoelastic body near the angular point of the dividing line of the regions.

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Generalized instantaneous image were introduced by V.K. Dzyaduk [1] in 1981 and proved to be a convenient tool for constructing and studying the Padé approximants and their generalizations (see [2]). The method of generalized instantaneous images proposed by Dzyadyk made it possible to construct and study rational Padé approximants and their generalizations for many classes of special functions from a single position. As an example, the Padé approximants is constructed for a class of basic hypergeometric series, which includes a q-analogue of the exponential function. In this paper the construction of the Pade approximants for the function of two variables is investigated. A two-dimensional functional sequence is constructed, which has a generalized instantaneous image, and rational approximants are determined, which will be generalizations of one-dimensional Padé approximants. The function of the two variables is entirely related to the basic hypergeometric series.

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The problem of pure bending of strip (beam) with transverse rectilinear crack, edges of which are free from acuter load, is investigated in this paper. Under bending moment its edges may not contact or smoothly contact throughout its area length or part. Dependently on where it is located.Using methods of theory of functions of complex variable and complex potentials the problem at issue has been reduced to the problems of linear conjugation, their analytical solution is found. Explicit expressions on complex potentials is written. Based on the energy criterion of destruction stress intensity factors are determined. Limit value of moment when the crack begins to propagate is found. For the case when crack edges partially contact, area length of contact of her edges is determined. Numerical analysis of critical moment of failure of strip (beams) is performed under various parameters of the problem, which are related to the mechanical state of crack. The corresponding graphic dependencies are constructed.

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E-Government is a set of pervasive technologies and automated processes now. The open data plays a crucial role in the successful implementation of this concept. The Open Data Platform (ODP) architecture is described here as the framework for the open data access systems implementation, including specific requirements. The proposed architecture and its components were discussed in this paper in detail for its availability, productivity, and reliability. The open data subsystem based on the architecture presented here was developed for the Jordan Government and was successfully implemented and tested. Thus, this architecture showed its viability. The focus of the paper is the detailed analysis of the proposed ODP architecture and its characteristics. The ODP is a significant system for the mature e-Government. We propose here the architecture for it with usage-proven characteristics. This fact adds the value to the e-Government framework stability, and significant characteristics and improves the overall quality of the system.

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Online learning under delayed feedback has been recently gaining increasing attention. Learning with delays is more natural in most practical applications since the feedback from the environment is not immediate. For example, the response to a drug in clinical trials could take a while. In this paper, we study the multi-armed bandit problem with Bernoulli distribution in the environment with delays by evaluating the Explore-First algorithm. We obtain the upper bounds of the algorithm, the theoretical results are applied to develop the software framework for conducting numerical experiments.

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In article we consider a problem of optimal investment strategy by a commercial bank building. This task is actual and the development of a procedure to solve it can help in making investment banking decisions. The general formulation of the problem consists of two criteria. The first one is to maximize the expected return, and the second is to minimize the risk of the investment transaction. Mathematical formulation of the problem is considered as a problem of nonlinear programming under constraints. The procedure for solving such a two-criteria optimization problem allows to obtain many solutions, which requires further steps to make a single optimal solution. According to the algorithm proposed in the work, the problem is divided into two separate problems of single-criteria optimization. Each of these tasks allows to obtain the optimal values of the investment vector both in terms of its expected return and in terms of investment risk. Additional constraints in the mathematical formulation of the problem, make it possible to take into account factors that, from the point of view of the investor, may influence decision-making. The procedures presented in this work allow to obtain analytical representations of formulas that describe the optimal values of the investment distribution vector for both mathematical formulations of the problem.

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In the article within the complex-associative model of liquid systems the nonlinear diffusion for a number of binary solutions, such as acetone-chloroform, tetrachlorethane-chloroform, diethyl ether-chloroform and benzene-chloroform, is considered: Real binary solutions are replaced by ideal three-component ones, which consist of averaged two associates of substance and solvent and an effective averaged complex, which is the result of quasi-chemical reactions of molecular solutions. The coefficient of mutual diffusion, which nonmonotonically depends on the concentration of the solvent, is represented as a matrix of partial coefficients of mutual diffusion, which are constant values and represent the material parameters of the considered solutions. The method of analytical calculation of numerical values of such quantities when considering the simplest type of one averaged complex is developed. It is shown that the partial coefficients are constant values and the analysis of their values for the considered solutions depending on the structure of the molecules of the substance, enthalpy and temperature is carried out. Based on the proposed approach, the deviation of the calculated «Fick’s» coefficient of mutual diffusion through the matrix of partial coefficients in comparison with experimental data is less than 2.5%.

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We study the sums of identically distributed random variables whose indices belong to certain sets of a given family A in R^d, d >= 1. We prove that sums over scaling sets S(kA) possess a kind of the uniform in A strong law of large numbers without any assumption on the class A in the case of pairwise independent random variables with finite mean. The well known theorem due to R. Bass and R. Pyke is a counterpart of our result proved under a certain extra metric assumption on the boundaries of the sets of A and with an additional assumption that the underlying random variables are mutually independent. These assumptions allow to obtain a slightly better result than in our case. As shown in the paper, the approach proposed here is optimal for a wide class of other normalization sequences satisfying the Martikainen–Petrov condition and other families A. In a number of examples we discuss the necessity of the Bass–Pyke conditions. We also provide a relationship between the uniform strong law of large numbers and the one for subsequences.

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The modelling of the fatigue fracture process of the thin isotropic infinite plates with cracks under external biaxial asymmetric cyclic loading is considered. The solution of the problem is based on the joint consideration of the fracture mechanics and continuous damage mechanics concepts and using two types of equivalent stress criteria’s. The first one reduces an asymmetrical cyclic load to the equivalent symmetric cyclic load in time of the rupture. The second one reduces a plane stress state in the vicinity of the top crack to a single-axial one. The obtained system of equations of the model a relatively equivalent stress intensity factor allows us to determine the duration of the incubation stage and the rate of fatigue crack propagation in plates with different stress concentrators. The calculated dependences of the crack length, which extends from the circular hole, from the number of load cycles in the infinite aluminum plate with a circular hole at the variation of the parameters of the asymmetrical cycle and the coefficient of the biaxiality loading are constructed.

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Asymptotic properties of Koenker - Bassett estimators of linear regression model parameters with discrete observation time and random noise being nonlinear local transformation of Gaussian stationary time series with singular spectrum are studied. The goal of the work lies in obtaining the requirements to regression function and time series that simulates the random noise, under which the Koenker - Bassett estimators of regression model parameters are asymptotically normal. Linear regression model with discrete observation time and bounded open convex parametric set is the object of the studying. Asymptotic normality of unknown parameters Koenker - Bassett estimators are obtained. For getting these results complicated concepts of time series theory and time series statistics have been used, namely: local transformation of Gaussian stationary time series, stationary time series with singular spectral density, spectral measure of regression function, admissibility of singular spectral density of stationary time series in relation to this measure, expansions by Chebyshev - Hermite polynomials of the transformed Gaussian time series values and it‘s covariances, central limit theorem for weighted sums of the values of such a local transformation.

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The paper deals with the singularly perturbed Korteweg-de Vries equation with variable coefficients. An algorithm for constructing asymptotic one-phase soliton-like solutions of this equation is described. The algorithm is based on the nonlinear WKB technique. The constructed asymptotic soliton-like solutions contain a regular and singular part. The regular part of this solution is the background function and consists of terms, which are defined as solutions to the system of the first order partial differential equations. The singular part of the asymptotic solution characterizes the soliton properties of the asymptotic solution. These terms are defined as solutions to the system of the third order partial differential equations. Solutions of these equations are obtained in a special way. Firstly, solutions of these equations are considered on the so-called discontinuity curve, and then these solutions are prolongated into a neighborhood of this curve. The influence of the form of the coefficients of the considered equation on the form of the equation for the discontinuity curve is analyzed. It is noted that for a wide class of such coefficients the equation for the discontinuity curve has solution that is determined for all values of the time variable. In these cases, the constructed asymptotic solutions are determined for all values of the independent variables. Thus, in the case of a zero background, the asymptotic solutions are certain deformations of classical soliton solutions.

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In this article, the asymptotic behavior of the mathematical expectation of the total energy of a harmonic oscillator without friction under the influence of an energy pump with a controlling element of the form of a stochastic harmonic oscillator without friction with a white noise perturbation in resonant and non-resonant cases is found. During the analytical solving the problem of finding the mathematical expectation of the total energy of a harmonic oscillator with random perturbation, the properties of the Wiener process, the stochastic Ito integral, and the mathematical expectation of the product of stochastic integrals are used.

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The least squares estimator asymptotic properties of the parameters of trigonometric regression model with strongly dependent noise are studied. The goal of the work lies in obtaining the requirements to regression function and time series that simulates the random noise under which the least squares estimator of regression model parameters are asymptotically normal. Trigonometric regression model with discrete observation time and open convex parametric set is research object. Asymptotic normality of trigonometric regression model parameters the least squares estimator is research subject. For obtaining the thesis results complicated concepts of time series theory and time series statistics have been used, namely: local transformation of Gaussian stationary time series, stationary time series with singular spectral density, spectral measure of regression function, admissibility of singular spectral density of stationary time series in relation to this measure, expansions by Chebyshev-Hermite polynomials of the transformed Gaussian time series values and it’s covariances, central limit theorem for weighted vector sums of the values of such a local transformation and Brouwer fixed point theorem.

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We study the Cauchy problem for a wave equation in three-dimensional space driven by a general stochastic measure. Under some assumptions, we prove that the mild solution tends to zero almost surely as the absolute value of the spatial variable tends to infinity.

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The structural diagram of an automated information and measurement system for monitoring the characteristics of atmospheric ozone, the purpose and interaction of the main components of the system are presented. System management software is considered. The results of determination of the total ozone content (TOC), which were obtained by comparing simultaneous data, determined manually by the operator and using the layout of the information-measuring system, with a relative error of measurement difference did not exceed 4.3%. The results of measurements with automatic averaging of the values obtained during the day showed that there was no need to choose windows of cloudless or homogeneous sky. The development can be recommended for use as a basis for the creation of a modern automated information and measurement system for monitoring the characteristics of atmospheric ozone.

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The estimation problem of slowly time-varying parameter matrices is considered for bilinear discrete dynamic system in the presence of disturbances. The least squares estimate with variable forgetting factor is investigated for this object in non-classical situation when this estimate may be not unique and additionally ‘attraction’ points for unknown parameter matrices are given at any moment. The set of all above-mentioned estimates of these unknown matrices is defined through the Moore-Penrose pseudo-inverse operator. The least squares estimate with variable forgetting factor and least deviation norm from given ‘attraction’ point at any moment is proposed as unique estimate on this set of all estimates. The explicit form of representation is obtained for this unique estimate of the parameter matrices by the least squares method with variable forgetting factor and least deviation norm from given ‘attraction’ points under non-classical assumptions. The recurrent algorithm for this estimate is also derived which does not require the usage of the matrix pseudo-inverse operator.

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The optical properties of ultrathin Au and Sn islet films, obtained by the methods of magnetron sputtering and thermal evaporation, respectively, are considered in this paper. By measuring the Stokes vector of the beam reflected from the samples, polarized and depolarized radiation components were separated. The conditions of the polarization degree dependence on the surface structure for a series of islet films with different morphologies are analyzed. To determine the morphological structure of the metal layer, methods of atomic force microscopy and resistivity measurement were also employed. The parameters of discontinuous film, obtained by optical and non-optical methods, are compared. It is established that with an increase in the angle of radiation incidence onto the samples, the polarization degree of the reflected beam decreases. Such behavior can be explained by the Mie theory of light scattering by particles. The magnitude of depolarizing action of the samples also depends on the morphology of their surface, correlating with the number of inequalities on it. The applied method of Stokes polarimetry, thus, allows one to obtain additional information on the structure of the surface, which is its advantage.

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In this paper we provided the definition of the Audience overlap network, as well as proposed a simple algorithm to compute overlap between two users on social media based on public data about their followers. There was proposed an alternative approach for computing overlaps based only on public data about users. This approach allows to include content overlap and activity patterns signals to be incorporated into more general statistical models featuring other covariates such as influencers’ direct engagement in shared conversations; relative influencer sizes and histories and links to similar third-party content to recover otherwise censored network structures and properties. For validate results there was designed a calibration process which utilizes Evolution Strategies algorithm to find a set of conditions which will make Audience overlap network built using similarity measures structurally equivalent to the Audience overlap network build on full information about followers.

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