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A methodological method based on the use of the theory of preferred numbers has been proposed in order to improve the most important parameters of the working bodies of noise mufflers. As a result of many years of scientific research, the authors have established a previously unknown theoretical relationship between the main series of preferred numbers, golden ratio and Fibonacci series numbers. A new direction in the development of number theory has been considered in the paper, its classification has been compiled, including the geometric theory of numbers, preferred numbers, containing a new basic series of preferred numbers using the Fibonacci sequence. New formulas have been obtained to determine the denominators of geometric progressions for the series of preferred numbers and the area of a circle. Determining the area of a circle using the new formula allows to get more accurate values. A new formula for determining the circumference of a circle has also been derived. The designs of perforated partitions have been developed, in which the laws of the new basic series of preferred numbers are used. Determining the area of a circle using the new formula allows you to get more accurate values. A new formula for determining the circumference of a circle is also obtained. The designs of perforated partitions have been developed, in which the regularities of the new basic series of preferred numbers have been used. A calculated substantiation of the main geometric and structural dimensions of noise mufflers is given using a mathematical model of a perforated golden partition and new basic series of preferred numbers, which allow to obtain a noise muffler design that has the lowest possible aerodynamic resistance with the maximum possible reduction in the noise level of exhaust gases from internal combustion engines. An innovative model of a noise muffler for reciprocating internal combustion engines with improved hydraulic and acoustic characteristics based on the theory of numbers is proposed in the paper. The theory of preferred numbers applies to any technical device.

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

The Exner equation in conjunction phenomenological sediment transport models is widely used for mathematical modeling non-cohesive river bed. This approach allows to obtain an accurate solution without any difficulty if one models evolution of simple shape bed. However if one models evolution of complex shape bed with unstable soil the numerical instability occurs in some cases. It is difficult to detach this numerical instability from the natural physical instability of bed. This paper analyses the causes of numerical instability occurring while modeling evolution of complex shape bed by using the Exner equation and phenomenological sediment rate models. The paper shows that two kinds of indeterminateness may occur while solving numerically the Exner equation closed by phenomenological model of sediment transport. The first indeterminateness occurs in the bed area where sediment transport is transit and bed is not changed. The second indeterminateness occurs at the extreme point of bed profile when the sediment rate varies and the bed remains the same. Authors performed the closure of the Exner equation by the analytical sediment transport model, which allowed to transform the Exner equation to parabolic type equation. Analysis of the obtained equation showed that its numerical solving does not lead to occurring of the indeterminateness mentioned above. Parabolic form of the transformed Exner equation allows to apply the effective and stable implicit central difference scheme for this equation solving. The model problem of bed evolution in presence of periodic distribution of the bed shear stress is carried out. The authors used the explicit central difference scheme with and without filtration method application and implicit central difference scheme for numerical solution of the problem. It is shown that the explicit central difference scheme is unstable in the area of the bed profile extremum. Using the filtration method resulted to increased dissipation of the solution. The solution obtained by using the implicit central difference scheme corresponds to the distribution law of bed shear stress and is stable throughout the calculation area.

Рассмотрены специальные преобразования декартовых координат, позволяющие строить эффективные (экономичные) алгоритмы численного моделирования многомерных случайных величин. В качестве иллюстративных и практически значимых примеров таких преобразований рассмотрены переходы к полярным, сферическим, параболическим и цилиндрическим координатам., The purpose of the paper was to expand the range of efficient (economical) computer algorithms for simulation of multi-dimensional random variables. The authors noticed that in a number of applied problems (for example, when modelling twoor three-dimensional isotropic vectors), transitions from Cartesian to other coordinate systems (for example, to polar or spherical) are effective. In this regard, the new generalizing notation of the computable simulated transformation of Cartesian coordinates is introduced in the paper. Such transformations allow constructing effective (economical) algorithms for the numerical modelling (simulation) of multi-dimensional random variables. The problem of finding examples of constructive and practical applications for computer simulated transformations of Cartesian coordinates is formulated. Transitions to polar, spherical, parabolic and cylindrical coordinates are considered as illustrative examples of such transformations. The practical applications of computable simulated transformations of Cartesian coordinates found in the scientific literature are described in detail by the authors. These applications are associated with both computer modelling (simulation) of isotropic vectors and Gaussian distribution along with the numerical solution of boundary value problems and problems of radiation transfer. Thus, the introduced notion of the computable simulated transformations of Cartesian coordinates is quite constructive. It opens up prospects for the scientific search for new transformations of this type with the aim of using them in stochastic computer models for important processes and phenomena., Вычислительные технологии, Выпуск 6 2021, Pages 62-84

Публикация посвящена применению методов вычислительной геометрии, интервального анализа и линейного программирования к задачам физики управляемого термоядерного синтеза. Рассмотрены геометрические аспекты проблемы, получены проекции светимостей различных объемов сферического токамака на плоскость матричного детектора, изучены изображения предполагаемых макроскопических структур и микроскопических включений. Для набора модельных распределений светимости объема токамака поставлена задача восстановления сигнала. Решение получено с использованием задач линейного программирования., The problems of reconstruction of plasma luminosity are important for physics and technology of power plants-tokamaks. The Globus-M research tokamak obtained a large amount of data using a matrix detector in pinhole camera geometry. From the mathematical point of view, finding the luminosity for different regions of the plasma volume according to the matrix detector is an inverse problem related to the field of integral geometry. An essential feature of the particular task is the use of a single fixed camera with a small viewing angle. In this regard, application of methods of harmonic analysis of data is not enough. The paper investigates the geometric aspects of the problem. In the general view, a threedimensional object is projected onto a two-dimensional plane through a diaphragm. Under the assumption of azimuthal symmetry, there is a central projection of the luminosity of the body of rotation onto a flat matrix detector. The initial information for the calculation is the plasma boundary obtained from magnetic sensors. There is no reliable information about the internal structure of the plasma, so its division into regions of the equal luminosity is not unambiguous. The paper presents an algorithm for finding the projections of the luminosity of plasma volumes on the plane of the matrix detector. A set of model direct problems for the construction of algorithms for their recognition according to the detector data was investigated. Images of supposed macroscopic structures and microscopic inclusions were obtained. The methodological basis of the work is the use of interval analysis methods for solving geometric and algebraic problems. This approach allows obtaining qualitative and quantitative results that takes into account the uncertainty of the input data with the minimum amount of computational costs. Algebraic solvability is investigated in the interval formulation using response functionality. Solutions for a set of test problems are obtained, which demonstrate the availability of successful reconstruction for real data. An important result of the study is an information about the presence of uncertainties in geometric data and related calculations by obtaining results about the luminosity of the plasma by solving linear programming problems., №25 (2020)

Актуальность исследования обусловлена необходимостью количественной оценки связности коллектора геолого-гидродинамических моделей для классификации реализаций при многовариантной оценке неопределенностей и оптимизации выбора метода и параметров моделирования куба песчанистости. Цель: ввести понятие коэффициента связности коллектора, оценить репрезентативность предложенного параметра для моделей залежей различной детальности и геометрии; оценка влияния параметров неоднородности геологических моделей на коэффициент связности. Объекты: геолого-гидродинамические модели залежей нефтегазовых месторождений. Методы: геологическое и гидродинамическое моделирование, статистический анализ результатов геологического и гидродинамического моделирования. Результаты. Кратко рассмотрены существующие методы оценки связности коллектора в геолого-гидродинамических моделях, используемые для оценки ресурсов углеводородного сырья и выбора наиболее эффективных методов их разработки. На основе проведенного авторами анализа был предложен новый подход к определению параметра связанности как отношения перетока жидкости в исследуемой модели к модели с единичной песчанистостью. Рассмотрено влияние геометрии и детальности геолого-гидродинамических моделей на величину коэффициент связности. На примере синтетических стохастических моделей рассмотрены зависимости коэффициента связности от таких параметров, как песчанистость, ранги вариограмм, их анизотропия и эффект самородка. Введено понятие коэффициента гидродинамической связности коллектора, который характеризует способность модели фильтровать флюид, без учета влияния фильтрационно-емкостных характеристик. Этот параметр учитывает только распределение коллектора в объеме залежи и то, как ячейки коллектора расположены друг относительно друга. Предложенный способ расчёта позволяет оценить анизотропию перетоков флюида между ячейками по различным направлениям и оценить способность модели фильтровать этот флюид через себя. Коэффициент связанности, в отличие от перколяционных методов, позволяет получать непрерывную оценку гидродинамической связи между ячейками, не требует обязательного наличия истории работы залежи и не зависит от текущей системы разработки, реализуемой на месторождении. The relevance of the research is caused by the need in quantitative evaluation of reservoir connectivity of models for classifying realizations in multivariant modelling and optimization of selection of methods and parameters for reservoir geomodelling. The main aim of the research is to define connectivity coefficient and estimate its representative validity for different size and geometry models; to evaluate impact of reservoir heterogeneity parameters on connectivity coefficient value. Objects of the research are geological and simulation models of oil and gas reservoirs. Methods of the research are geological modelling and reservoir simulation, statistical analysis of the results of geomodeling and reservoir simulation. Results. The paper briefly describes the existing methods for evaluating the connectivity of oil and gas reservoirs and new approach to connectivity estimation. New connectivity coefficient is defined as a ratio of model liquid rate to unit model liquid rate. Unit model means that net to gross ratio value for all cells is 1 and the model provides maximum possible liquid rate. The paper considers the impact of the model geometry and scale on the connectivity coefficient value. The authors have studied the relation of connectivity coefficient to net to gross ratio, variogram ranges and anisotropy, nugget effect. The connectivity coefficient was stated as a variable, which characterizes the ability of reservoir to pass fluid independent of absolute and relative permeability, porosity, transmissibility etc. The connectivity coefficient as opposed to percolation theory methods can provide continuous evaluation of fluid flow between model cells and does not require valid production history of reservoir.

Рассмотрена задача выявления аномальных наблюдений в данных больших размерностей на основе метода многомерного шкалирования с учетом возможности построения качественной визуализации данных. Предложен алгоритм модифицированного метода главных проекций Торгерсона, основанный на построении подпространства проектирования исходных данных путем изменения способа факторизации матрицы скалярных произведений при помощи метода анализа кумулятивных кривых. Построено и проанализировано эмпирическое распределение F -меры для разных вариантов проектирования исходных данных, Purpose. Purpose of the article. The paper aims at the development of methods for multidimensional data presentation for solving classification problems based on the cumulative curves analysis. The paper considers the outlier detection problem for high-dimensional data based on the multidimensional scaling, in order to construct high-quality data visualization. An abnormal observation (or outlier), according to D. Hawkins, is an observation that is so different from others that it may be assumed as appeared in the sample in a fundamentally different way. Methods. One of the conceptual approaches that allow providing the classification of sample observations is multidimensional scaling, representing by the classical Orlochi method, the Torgerson main projections and others. The Torgerson method assumes that when converting data to construct the most convenient classification, the origin must be placed at the gravity center of the analyzed data, after which the matrix of scalar products of vectors with the origin at the gravity center is calculated, the two largest eigenvalues and corresponding eigenvectors are chosen and projection matrix is evaluated. Moreover, the method assumes the linear partitioning of regular and anomalous observations, which arises rarely. Therefore, it is logical to choose among the possible axes for designing those that allow obtaining more effective results for solving the problem of detecting outlier observations. A procedure of modified CC-ABOD (Cumulative Curves for Angle Based Outlier Detection) to estimate the visualization quality has been applied. It is based on the estimation of the variances of angles assumed by particular observation and remaining observations in multidimensional space. Further the cumulative curves analysis is implemented, which allows partitioning out groups of closely localized observations (in accordance with the chosen metric) and form classes of regular, intermediate, and anomalous observations. Results. A proposed modification of the Torgerson method is developed. The F1-measure distribution is constructed and analyzed for different design options in the source data. An analysis of the empirical distribution showed that in a number of cases the best axes are corresponding to the second, third, or even fourth largest eigenvalues. Findings. The multidimensional scaling methods for constructing visualizations of multi-dimensional data and solving problems of outlier detection have been considered. It was found out that the determination of design is an ambiguous problem., Вычислительные технологии, Выпуск 3 2020

The given paper touches upon the problem of German migration policy in contemporary period. The on-going processes allow us to conclude that the German migration paradigm has been changing as well as one of the most successful states in the European Union and as a state endowed with the greatest peace load after the crisis of 2008. The study gives the main qualitative and quantitative characteristics of the German migration policy in the 2000s. First of all, it is characterized by changes in the geography of migration flows. In 2004, with ten new countries entering the EU, the inflow of migrants from Eastern Europe countries began to increase. During this period, the number of immigrants from Poland has almost doubled. After Bulgaria and Romania joined the EU, the number of people from these countries also increased by more than three times. At the period of 2008 crisis, the inflow of migrants from Italy, Spain, Greece and the Balkan countries intensified; the number of migrants from Croatirapidly increasedafter this country joined the EU in 2013. The author analyzes the qualitative composition of migrants and concludes that, from the beginning of the 21st century, the working migration, having dominated for many decades, is gradually changing in favor of skilled workers’ migration. The important role of universities in order to attract highly qualified specialists is substantiated. The migration impact on the labor market is studied with a special attention. It is due to the fact that, when arriving in the country and getting the opportunity to apply skills and realize their human capital, a migrant becomes a kind of investor in the hosting country economy, contributing to its growth. One of its main advantages is that, already having a certain set of knowledge and skills (the hosting country does not need to pay for his education), he can generate new knowledge, create innovations, thereby increasing the competitiveness of the recipient country economy. Germany has a policy of attracting highly qualified migrants to the country. The paper touches upon the innovations in the migration policy of Germany in the 21st century. A “Blue Card” has been introduced for highly qualified professionals in accordance with the EU directive and labor market regulation has been changed. The need to reduce the number of legal restrictions for migrants from third countries, and to improve attitudes towards migrants, despite the failure of the multiculturalism policy, is indicated too. There are still many obstacles to immigrants’active participation in the labor market, such language difficulties, differences in educational systems, and lack of information about vacancies.

In this paper we discuss lower bounds for convergence of convex optimization methods of high order and attainability of this bounds. We formulate a hypothesis that covers all the cases. It is noticeable that we provide this statement without a proof. Newton method is the most famous method that uses gradient and Hessian of optimized function. However, it converges locally even for strongly convex functions. Global convergence can be achieved with cubic regularization of Newton method [Nesterov, Polyak, 2006], whose iteration cost is comparable with iteration cost of Newton method and is equivalent to inversion of Hessian of optimized function. Yu.Nesterov proposed accelerated variant of Newton method with cubic regularization in 2008 [Nesterov, 2008]. R.Monteiro and B. Svaiter managed to improve global convergence of cubic regularized method in 2013 [Monteiro, Svaiter, 2013]. Y.Arjevani, O. Shamir and R. Shiff showed that convergence bound of Monteiro and Svaiter is optimal (cannot be improved by more than logarithmic factor with any second order method) in 2017 [Arjevani et al., 2017]. They also managed to find bounds for convex optimization methods of p-th order for $p 2$. However, they got bounds only for first and second order methods for strongly convex functions. In 2018 Yu.Nesterov proposed third order convex optimization methods with rate of convergence that is close to this lower bounds and with similar to Newton method cost of iteration [Nesterov, 2018]. Consequently, it was showed that high order methods can be practical. In this paper we formulate lower bounds for p-th order methods for $p 3$ for strongly convex unconstrained optimization problems. This paper can be viewed as a little survey of state of the art of high order optimization methods.

In this paper differences between Fisher Information Matrix (FIM) and inverse covariation matrix of normalized correlation estimations for white and colored noise are investigated. It’s shown that implementation of normalized correlation function estimation leads to modification of maximum likelihood estimation FIM elements, so in case of arbitrary energy affected parameter vector, variance of estimation by normalized correlation function maximization is not equal to Cramer–Rao lower bound. Statistical characteristics of joint Doppler stretch and delay estimation by maximization of normalized correlation function for signal with nuisance parameters are derived in this paper. It’s shown that normalized correlator is equal to wideband ambiguity function, but this method of estimation follows from Cauchy–Schwarz inequality without using energy conservation assumptions. Besides, it is proved that estimation of Doppler stretch and delay by normalized correlation function or WBAF of signal with random initial phase and gain is asymptotically unbiased and effective.

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.

We study polynomial solutions of gyrostat motion equations under potential and gyroscopic forces applied and of gyrostat motion equations in magnetic field taking into account Barnett-London effect. Mathematically, either of the above mentioned problems is described by a system of non-linear ordinary differential equations whose right hand sides contain fifteen constant parameters. These parameters characterize the gyrostat mass distribution, as well as potential and non-potential forces acting on gyrostat. We consider polynomial solutions of Steklov-Kovalevski-Gorjachev and Doshkevich classes. The structure of invariant relations for polynomial solutions shows that, as a rule, on top of the fifteen parameters mentioned one should add no less than twenty five problem parameters. In the process of solving such a multi-parametric problem in this paper we (in addition to analytic approach) apply numeric methods based on CAS. We break our studies of polynomial solutions existence into two steps. During the first step, we estimate maximal degrees of polynomials considered and obtain a non-linear algebraic system for parameters of differential equations and polynomial solutions. In the second step (using the above CAS software) we study the solvability conditions of the system obtained and investigate the conditions of the constructed solutions to be real. We construct two new polynomial solutions for Kirchhoff-Poisson. The first one is described by the following property: the projection squares of angular velocity on the non-baracentric axes are the fifth degree polynomials of the angular velocity vector component of the baracentric axis that is represented via hypereliptic function of time. The second solution is characterized by the following: the first component of velocity conditions is a second degree polynomial, the second component is a polynomial of the third degree, and the square of the third component is the sixth degree polynomial of the auxiliary variable that is an inversion of the elliptic Legendre integral. The third new partial solution we construct for gyrostat motion equations in the magnetic field with Barnett-London effect. Its structure is the following: the first and the second components of the angular velocity vector are the second degree polynomials, and the square of the third component is a fourth degree polynomial of the auxiliary variable which is found via inversion of the elliptic Legendre integral of the third kind. All the solutions constructed in this paper are new and do not have analogues in the fixed point dynamics of a rigid body.

Investigations of arithmetic properties of the hyper-geometric function values make it possible to single out two trends, namely, Siegel’s method and methods based on the effective construction of a linear approximating form. There are also methods combining both approaches mentioned. The Siegel’s method allows obtaining the most general results concerning the abovementioned problems. In many cases it was used to establish the algebraic independence of the values of corresponding functions. Although the effective methods do not allow obtaining propositions of such generality they have nevertheless some advantages. Among these advantages one can distinguish at least two: a higher precision of the quantitative results obtained by effective methods and a possibility to study the hyper-geometric functions with irrational parameters. In this paper we apply the effective construction to estimate a measure of the linear independence of the hyper-geometric function values over the imaginary quadratic field. The functions themselves were chosen by a special way so that it could be possible to demonstrate a new approach to the effective construction of a linear approximating form. This approach makes it possible also to extend the well-known effective construction methods of the linear approximating forms for poly-logarithms to the functions of more general type. To obtain the arithmetic result we had to establish a linear independence of the functions under consideration over the field of rational functions. It is apparently impossible to apply directly known theorems containing sufficient (and in some cases needful and sufficient) conditions for the system of functions appearing in the theorems mentioned. For this reason, a special technique has been developed to solve this problem. The paper presents the obtained arithmetic results concerning the values of integral functions, but, with appropriate alterations, the theorems proved can be adapted to the case of the hyper-geometric series with a finite radius of convergence. Subsequently, it will be possible to consider also a case of the irrational parameters, but there are still a number of difficulties, which are specific for the problems of such type and which are to be overcome.

The paper deals with identifying linear dynamical systems from the experimental data obtained through applying the test signals to the system. The paper objective is to determine both the form and the coefficients of the transfer function retrieved from the hodograph samples experimentally at bench test. The order of the frequency transfer function of the system being identified was assumed to be unknown. It was expected that in obtaining the frequency characteristics of a real system there would be noise during the experiment as a result of which the points of the experimentally obtained hodograph would be randomly shifted. As a model, a certain transfer function of the system was adopted. The authors proposed to find a solution of the identification problem in the class of hodographs specified by the model of the system. The search for unknown coefficients of the transfer function of the system model is carried out by minimizing a proximity criterion (measure) - described and published earlier by one of the authors - between the experimentally received system hodograph and the system model on an entire set of the experimental points of the system hodograph and the hodograph of the system model. The solution of linear dynamic system identification from the frequency hodograph was reduced to solving a system of equations of the system model frequency transfer function that is linear with respect to unknown parameters.The proposed identification algorithm allows us to determine the order of the frequency transfer function of the identified system from the experimentally obtained samples of the frequency hodograph of the system. For dynamic systems of the fifth order at most there is software developed to simulate the process providing the pseudo-experimental data with random errors and determining the parameters of such systems.A computational experiment has been carried out to evaluate the error with which the proposed algorithm determines the parameter values of the system to be identified. The illustrative computational experiment has shown that using the proposed algorithm for identifying a linear dynamic system from the frequency hodograph the error in determining the coefficient values of the frequency transfer function of the system is comparable with a range of measuring error in the experimental samples of the hodograph of this system. In known sources on identification of linear dynamic systems there is no method of identification this publication describes. This identification method of linear dynamic systems can find application in experimental testing, verification tests in situ and iron bird tests for vehicles of various purposes.

WENO schemes (weighted, essentially non oscillating) are currently having a wide range of applications as approximate high order schemes for discontinuous solutions of partial differential equations. These schemes are used for direct numerical simulation (DNS) and large eddy simmulation in the gas dynamic problems, problems for DNS in MHD and even neutron kinetics. This work is dedicated to clarify some characteristics of WENO schemes and numerical simulation of specific tasks. Results of the simulations can be used to clarify the field of application of these schemes. The first part of the work contained proofs of the approximation properties, stability and convergence of WENO5, WENO7, WENO9, WENO11 and WENO13 schemes. In the second part of the work the modified wave number analysis is conducted that allows to conclude the dispersion and dissipative properties of schemes. Further, a numerical simulation of a number of specific problems for hyperbolic equations is conducted, namely for advection equations (one-dimensional and two-dimensional), Hopf equation, Burgers equation (with low dissipation) and equations of non viscous gas dynamics (onedimensional and two-dimensional). For each problem that is implying a smooth solution, the practical calculation of the order of approximation via Runge method is performed. The influence of a time step on nonlinear properties of the schemes is analyzed experimentally in all problems and cross checked with the first part of the paper. In particular, the advection equations of a discontinuous function and Hopf equations show that the failure of the recommendations from the first part of the paper leads first to an increase in total variation of the solution and then the approximation is decreased by the non-linear dissipative mechanics of the schemes. Dissipation of randomly distributed initial conditions in a periodic domain for one-dimensional Burgers equation is conducted and a comparison with the spectral method is performed. It is concluded that the WENO7-WENO13 schemes are suitable for direct numerical simulation of turbulence. At the end we demonstrate the possibility of the schemes to be used in solution of initial-boundary value problems for equations of non viscous gas dynamics: Rayleigh-Taylor instability and the reflection of the shock wave from a wedge with the formation a complex configuration of shock waves and discontinuities.

Рассмотрена задача маршрутизации транспорта с ограничениями по временным окнам. Требовалось составить план доставки товара клиентам, построив маршруты движения идентичных транспортных средств так, чтобы общая длина пройденного пути была минимальной. Для решения задачи разработан гибридный алгоритм. Он состоит из методов построения исходных решений, муравьиного алгоритма и локального поиска. В муравьином алгоритме в процессе формирования маршрутов разрешается нарушение временных ограничений при условии добавления штрафа в целевую функцию. Предложенный метод показал высокую эффективность при решении задач кластерного типа и задач с долгосрочным горизонтом планирования., The purpose of this paper is to improve the performance of a hybrid method based on ant colony optimization (ACO) that finds approximate solutions of the vehicle routing problem with time windows (VRPTW). In order to solve this problem it is required to design a plan for goods delivery to the customers generating the routes of identical vehicles so that the total travelled distance is minimal. For the VRPTW solving, the hybrid method is developed in which a usage of trial solutions makes it possible to explore the most promising parts of the search space. The initial methods for solution construction, an ant colony optimization (ACO) algorithm and local search are proposed in the framework of the hybrid method. In the ACO algorithm, when generating the routes, it is allowed to violate the time window constraints. A method to restore the feasibility of solutions is implemented within the relaxation scheme under “returns in time” principle. Numerical results for solving all problems with 25, 50 and 100 customers from the Solomon test set are obtained. We provide the results on the time and deviation of the solution of these problems in comparison with the results of other authors. Some problems and their classes were solved much faster by the algorithm proposed in this paper. Relative deviations from optimal values of the objective function for the most complex tasks decrease with increasing decision time. The proposed approach can be considered to be an additional or an alternative algorithm for solving the cluster type and the long-term planning horizon problems of the VRPTW., №5(23) (2018)

A large class of systems that have found application in various industries and households, electrified transportation facilities and energy sector has been classified as electrical engineering systems. Their characteristic feature is a combination of continuous and discontinuous modes of operation, which is reflected in the appearance of a relatively new term “hybrid systems”. A wide class of hybrid systems is pulsed DC converters operating in a pulse width modulation, which are non-linear systems with variable structure. Using various methods for linearization it is possible to obtain linear mathematical models that rather accurately simulate behavior of such systems. However, the presence in the mathematical models of exponential nonlinearities creates considerable difficulties in the implementation of digital hardware. The solution can be found while using an approximation of exponential functions by polynomials of the first order, that, however, violates the rigor accordance of the analytical model with characteristics of a real object. There are two practical approaches to synthesize algorithms for control of hybrid systems. The first approach is based on the representation of the whole system by a discrete model which is described by difference equations that makes it possible to synthesize discrete algorithms. The second approach is based on description of the system by differential equations. The equations describe synthesis of continuous algorithms and their further implementation in a digital computer included in the control loop system. The paper considers modeling of a hybrid electrical engineering system using differential equations. Neglecting the pulse duration, it has been proposed to describe behavior of vector components in phase coordinates of the hybrid system by stochastic differential equations containing generally non-linear differentiable random functions. A stochastic vector-matrix equation describing dynamics of the processes has been obtained in the paper. The equation contains both continuous and discrete components, which characterize an amplitude signal modulation. An equation for probability density of phase coordinate distribution in the system has been developed on the basis of a mathematical model for a hybrid system.

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

In 1955 A.B. Shidlovski's general theorems were published. They allow us to reduce the problem of algebraic independence of the analytic function values, belonging to the specific class, to a simpler problem of algebraic independence of these functions. Since the abovementioned general theorems can be applied to the generalized hyper-geometric functions with rational parameters, there appeared many works in which the algebraic independence of such functions (and their derivatives) had been established. The A.B. Shidlovski's results generalize and develop a Siegel's method well known in the theory of transcendental numbers. Besides the Siegel's method to solve the problems concerning the arithmetic nature of the values of analytic functions one also applies methods based on the effective construction of linear approximating forms. Such methods enabled finding the most accurate estimates of linear forms and obtaining the numerous results concerning the arithmetic properties of the values of hyper-geometric functions with irrational parameters. This shows that effective methods are of some value for the development of the theory of transcendental numbers.Recently, in the context of studied arithmetic nature of the values of differentiated hypergeometric functions with respect to parameter, there was a need in results concerning the linear independence of such functions over the field of rational fractions. Similar investigations were also conducted earlier because of applications of A.B. Shidlovski's general theorems, but in that case a more difficult problem of algebraic independence had to be solved, and therefore only the simplest functions were considered. The paper studies the issue of linear independence of hypergeometric functions, differentiated with respect to parameter, and this parameter is included both in the numerator and in the denominator of the common member of the appropriate power series. The paper defines a condition (in some cases, it is necessary and sufficient) of linear independence of such functions, which is very convenient for checking in concrete cases. The paper results are obtained by calculating some determinants, which, naturally, arise from the problems under consideration. In the future, the theorems proved in this paper can be used to have the diverse statements concerning the arithmetic nature of the values of the appropriate functions.DOI: 10.7463/mathm.0415.0817328

The paper is aimed at determining the marginal distributions of the estimates of model parameters for a nonlinear dynamic model of the cell population dynamics. The cell population system evolves in the laboratory (in vitro) and comprises two types of cells.Basic estimates of model parameters are obtained on a single limited sample of experimental data. The value of the number of populations of each cell type is obtained from the experiment at equal intervals of time. This paper proposes a technique for determining the marginal distributions of the parameter estimates using numerical modeling.This technique includes the identification of parameters of nonlinear models and test of the obtained model adequacy with base parameter estimates, the identification of the initial data and finding the reference trajectory. The initial data for the trajectory are found using the least squares method, while minimizing the deviation from the experimental trajectory. Data sets on measured densities of the populations at specific points in time are generated using the reference trajectory and the normally distributed random numbers generator. The problem of obtaining estimates of system parameters is solved for each data set. Tests of hypotheses about the marginal distribution of each parameter are carried out based on the calculated set of estimated parameters. To prove hypothesis, the Kolmogorov test is used. The description of a numerical example is included. The obtained marginal distributions of the parameter estimates can be further used to evaluate the probabilities of different scenarios of the population system development.DOI: 10.7463/mathm.0415.0812686

Многопользовательские схемы цифровых водяных знаков (ЦВЗ) предназначены для защиты авторских прав на цифровую продукцию. В данной работе рассматриваются две схемы ЦВЗ Боне—Шоу: базовая и каскадная, проводятся исследование и сравнительный анализ этих схем с целью определения более эффективной схемы из двух по длине, а также указаны границы значений параметров схем, для которых одна схема эффективнее другой., Multiuser digital watermarking is a powerful tool for copyright protection of widely replicable digital products. This paper addresses Boneh—Shaw fingerprinting schemes. The algorithms proposed by D. Boneh and J. Shaw allow a distributor to detect any unauthorized copy and trace it back to user and they work correctly for any size of coalition. The main idea of this paper is a comparative analysis of two Boneh—Shaw digital watermarks schemes (cascade and basics), which makes clear which one of these schemes is more effective in terms of the length of the label. Using the comparative analysis and numerical experiments, we obtain the specific values of the boundaries for input parameters in which one scheme will be more efficient than another. From the results of the comparative analysis it can be concluded that for large number of users of the scheme and a small size of the coalition (in relation of users number) it is advisable to use the cascade scheme. In the case of a small number of users and a large size of the coalition (compared to the number of users) it is more expedient to use the basic scheme., №3(23) (2018)

Предложен теоретико-информационный подход к оценке производительности суперкомпьютеров. Этот подход основан на характеристике “вычислительная способность”, которую можно определить теоретически, опираясь исключительно на описание архитектуры исследуемой системы. Предложен метод определения вычислительной способности суперкомпьютеров с учетом влияния межпроцессорного взаимодействия. Метод применяется в работе для оценки производительности пяти суперкомпьютеров среди первых 50 позиций в списке TOP500 (ноябрь 2016). Полученные результаты сравниваются с аналогичными значениями бенчмарка LINPACK и характеристиками Theoretical peak, используемыми для формирования рейтинга TOP500. По результатам сравнения делаются выводы относительно применимости указанного подхода для оценки производительности суперкомпьютеров на этапе разработки., In this paper we present the informational-theoretical approach to the evaluation of performance for supercomputers. This approach is based on the computer capacity characteristic which can be estimated theoretically (without any experiments over the working model of the investigated computer) relying on the description of the architecture of the system under study. We describe the method for estimation of the supercomputers Computer capacity which takes into account the influence of the interprocess communication. The described method is used in this paper to estimate the performance of the five supercomputers among the first fifty positions in TOP500 list (according to November 2016 data). The obtained results are compared with the values of LINPACK benchmark and Theoretical peak characteristic which are used to form the TOP500 rating. Based on the comparison we draw conclusions about the applicability of the presented approach to the evaluation of supercomputers performance at the design stage., №1(23) (2018)

The paper considers an issue of updating a turbine blade finite element model. The turbine blade is fixed in the special rig and the unknown contact conditions strongly influences on the blade dynamics. Hence, there is a difference between the real blade dynamic characteristics and the results of the finite element simulation. The task is to update the blade FE model on the basis of the modal test data to obtain fine dynamic model of the structure. The experimental modal parameters are known with some scatter, which is the result of measurement errors, and some differences from blade to blade within the kit of parts.The proposed algorithm is capable to update the turbine blade finite element model by variation of elasticity modulus in contact areas taking into account the experimental modal parameters scatter. It is worth noting that elasticity modulus of finite elements near the contact area should be considered as unknown contact stiffness parameters.The stochastic updating algorithm based on the sensitivity analysis is proposed. The paper presents equations to define sensitivity coefficients and provides the special recurrence relations, on which the proposed algorithm is based. The eigenvalue problem is resolved repeatedly with system parameters changing during the algorithm implementation. The system parameters (elasticity modulus in the contact areas) are recalculated according to the recurrence relations, mentioned above. To implement the proposed algorithm, a special software in C++ language has been developed.The proposed algorithm has demonstrated its efficiency for updating the finite element model of the complex structure. Thus, the differences between the calculated eigenfrequencies and those of identified from modal testing have been reduced from 13% to 3%. The updated model of the turbine blade could be used for accurate simulation of the blade dynamic behavior under different operating conditions.

The paper describes a new algorithm of construction of the nonlinear arithmetic triangle on the basis of numerical simulation and the binary system. It demonstrates that the numbers that fill the nonlinear arithmetic triangle may be binomial coefficients of a new type. An analogy has been drawn with the binomial coefficients calculated with the use of the Pascal triangle. The paper provides a geometrical interpretation of binomials of different types in considering the branching systems of rays.

In our paper we provide some direct extentions of our recent sharp results on traces in the analytic function spaces, which we proved earlier in case of the unit ball in C^n, to the case of the bounded strongly pseudoconvex domains with a smooth boundary. To be more precise we consider the analytic Bloch space in the strongly pseudoconvex domains with a smooth boundary, mixed norm spaces and so-called the new Herz type spaces of analytic functions in the domains of the same type. The Bloch spaces, for various complicated domains, were studied by many authors, but the various Herz type spaces are introduced in this paper, as far as we know, for the first time. The role of so-called r-lattices and their new properties are essential for our proofs. These techniques based on the lattices in the strongly pseudoconvex domains were invented and heavily used in the recent papers of Abate and coauthors. The arguments in the proofs in the case of the unit ball and the strongly pseudoconvex domains have some similarity.

The following result of G. Pisier contributed to the appearance of this paper: if a convolution operator ★f : M(G) → C(G), where $G$ is a compact Abelian group, can be factored through a Hilbert space, then f has the absolutely summable set of Fourier coefficients. We give some generalizations of the Pisier's result to the cases of factorizations of operators through the operators from the Lorentz-Schatten classes Sp,q in Hilbert spaces both in scalar and in vector-valued cases. Some applications are given.

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 paper describes homotopy classes of open continuous functions on finite open topological graphs

This paper is about a generalization of celebrated Inoue's surfaces. To each matrix M in SL(2n+1,ℤ) we associate a complex non-Kähler manifold TM of complex dimension n+1. This manifold fibers over S1 with the fiber T2n+1 and monodromy MT. Our construction is elementary and does not use algebraic number theory. We show that some of the Oeljeklaus-Toma manifolds are biholomorphic to the manifolds of type TM. We prove that if M is not diagonalizable, then TM does not admit a Kähler structure and is not homeomorphic to any of Oeljeklaus-Toma manifolds.

Currently, an complete topological classification has been obtained with respect to the topological equivalence of Morse-Smale flows, [9, 7], as well as their generalizations of Ω-stable flows on closed surfaces, [4]. Some results on topological conjugacy classification for such systems are also known. In particular, the coincidence of the classes of topological equivalence and conjugacy of gradient-like flows (Morse-Smale flows without periodic orbits) was established in [3]. In the classical paper [8], it was proved that in the presence of connections (coincidence of saddle separatrices), the topological equivalence class of a Ω-stable flow splits into a continuum of topological conjugacy classes (has moduli). Obviously, each periodic orbit also generates at least one modulus associated with the period of that orbit. In the present work, it was established that the presence of a cell in a flow bounded by two limit cycles leads to the existence of an infinitely many stability moduli. In addition, a criterion for the topological conjugation of flows on such cells was found.

The paper treats a particular type of pseudo-Riemannian spaces, namely quasi-Einstein spaces with gradient dening vector. These spaces are a generalization of well-known Einstein spaces. There are three types of these spaces that permit locally geodesic mappings. Authors proved "a theorem of disappearance" for compact quasi-Einstein spaces of main type.

The Harry Dym equation is the third-order evolutionary partial differential equation. It describes a system in which dispersion and nonlinearity are coupled together. It is a completely integrable nonlinear evolution equation that may be solved by means of the inverse scattering transform. It has an infinite number of conservation laws and does not have the Painleve property. The Harry Dym equation has strong links to the Korteweg – de Vries equation and it also has many properties of soliton solutions. A connection was established between this equation and the hierarchies of the Kadomtsev – Petviashvili equation. The Harry Dym equation has applications in acoustics: with its help, finite-gap densities of the acoustic operator are constructed. The paper considers a generalization of the Harry Dym equation, for the study of which the methods of the theory of finite-dimensional dynamics are applied. The theory of finite-dimensional dynamics is a natural development of the theory of dynamical systems. Dynamics make it possible to find families that depends on a finite number of parameters among all solutions of evolutionary differential equations. In our case, this approach allows us to obtain some classes of exact solutions of the generalized equation, and also indicates a method for numerically constructing solutions.

The paper studies linear differential operators in derivatives with respect to one variable. Such operators include, in particular, operators defined on infinite prolongations of evolutionary systems of differential equations with one spatial variable. In this case, differential operators in total derivatives with respect to the spatial variable are considered. In parallel, linear differential operators with one independent variable are investigated. The known algorithms for reducing the matrix to a stepwise or diagonal form are generalized to the operator matrices of both types. These generalizations are useful at points, where the functions, into which the matrix components are divided when applying the algorithm, are nonzero.In addition, the integral operator is defined as a multi-valued operator that is the right inverse of the total derivative. Linear operators that involve both the total derivatives and the integral operator are called integro-differential. An invertible operator in the integro-differential sense is an operator for which there exists a two-sided inverse integro-differential operator. A description of scalar differential operators that are invertible in this sense is obtained. An algorithm for checking the invertibility in the integro-differential sense of a differential operator and for constructing the inverse integro-differential operator is formulated.The results of the work can be used to solve linear equations for matrix differential operators arising in the theory of evolutionary systems with one spatial variable. Such operator equations arise when describing systems that are integrable by the inverse scattering method, when calculating recursion operators, higher symmetries, conservation laws and symplectic operators, and also when solving some other problems. The proposed method for solving operator equations is based on reducing the matrices defining the operator equation to a stepwise or diagonal form and solving the resulting scalar operator equations.

The creation of methods for the study of nonlinear phase systems has a long history, since the 60s of the last century (V.I. Tikhonov, V. Lindsay, M.V. Kapranov, B.I. Shakhtarin, etc.). By now, rigorous and approximate analysis methods of such systems have been developed. However, most methods are limited to the analysis of low order systems. Only in recent years attempts have been made to create methods, which allow to carry out the analysis of high-order phase systems. The material of this article deals with these methods. The article considers the construction of solutions of phase systems on the example of phase-locked frequency of arbitrary dimension with piecewise linear approximation of the nonlinear function. This approximation allowed to use an explicit form of solutions in the linearity and to obtain analytical conditions for the existence of various types of system behavior. The analytical conditions for the existence of solutions leading to the emergence of complex limit sets of the trajectories of phase systems and their bifurcations are obtained. These are homoclinic trajectories in the case of the saddle-focus equilibrium state, which play a decisive role in the occurrence of chaos. It is also shown that it is possible to obtain analytical conditions for the bifurcation of the birth and the existence of multi-pass rotational cycles in a piecewise linear phase system, on the basis of which a criterion for the transition to chaos through bifurcations cascade of doubling the stable cycle period can be obtained; in accordance with the Sharkovsky theorem it ends with the bifurcation of the cycle birth of the period three and the occurrence of developed chaos. It should be noted that the research methods of piecewise linear systems described in the paper were applied by the authors not only to phase systems, but, for example, to the Chua system, allowing various chaotic behavior.

In this paper we introduce a new class of foliations on Rie-mannian 3-manifolds, called B-foliations, generalizing the class of foliations of non-negative curvature. The leaves of B-foliations have bounded total absolute curvature in the induced Riemannian metric. We describe several topological and geometric properties of B-foliations and the structure of closed oriented 3-dimensional manifolds admitting B-foliations with non-positive curvature of leaves.

Let $M$ be a compact two-dimensional manifold and, $f \in C^{\infty}(M,\mathbb{R})$ be a Morse function, and $\Gamma_f$ be its Kronrod-Reeb graph. Denote by $\mathcal{O}_{f}=\{f \circ h \mid h \in \mathcal{D}\}$ the orbit of $f$ with respect to the natural right action of the group of diffeomorphisms $\mathcal{D}$ on $C^{\infty}(M,\mathbb{R})$, and by $\mathcal{S}(f)=\{h\in\mathcal{D} \mid f \circ h = f\}$ the corresponding stabilizer of this function. It is easy to show that each $h\in\mathcal{S}(f)$ induces a homeomorphism of $\Gamma_f$. Let also $\mathcal{D}_{\mathrm{id}}(M)$ be the identity path component of $\mathcal{D}(M)$, $\mathcal{S}'(f)= \mathcal{S}(f) \cap \mathcal{D}_{\mathrm{id}}(M)$ be group of diffeomorphisms of $M$ preserving $f$ and isotopic to identity map, and $G_f$ be the group of homeomorphisms of the graph $\Gamma_f$ induced by diffeomorphisms belonging to $\mathcal{S}'(f)$. This group is one of the key ingredients for calculating the homotopy type of the orbit $\mathcal{O}_{f}$. Recently the authors described the structure of groups $G_f$ for Morse functions on all orientable surfaces distinct from $2$-torus $T^2$ and $2$-sphere $S^2$. The present paper is devoted to the case $M=S^{2}$. In this situation $\Gamma_f$ is always a tree, and therefore all elements of the group $G_f$ have a common fixed subtree $\mathrm{Fix}(G_f)$, which may even consist of a unique vertex. Our main result calculates the groups $G_f$ for all Morse functions $f:S^{2}\to\mathbb{R}$ whose fixed subtree $\mathrm{Fix}(G_f)$ consists of more than one point., Comment: 5 pages

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.

В работе демонстрируется, как метод апостериорной оценки порядка точности разностной схемы по Ричардсону позволяет сделать вывод о некорректности постановки (в смысле отсутствия решения) решаемой численно начально-краевой задачи для уравнения в частных производных. Это актуально в ситуации, когда аналитическое доказательство некорректности постановки ещё не получено или принципиально невозможно., The paper demonstrates how the method of a posteriori estimation of the order of accuracy for the difference scheme according to the Richardson extrapolation method allows one to conclude that the formulation of the numerically solved initial-boundary value problem for a partial differential equation is ill-posed (in the sense of the absence of a solution). This is important in a situation when the ill-posedness of the formulation is not analytically proved yet or cannot be proved in principle., ВЫЧИСЛИТЕЛЬНЫЕ МЕТОДЫ И ПРОГРАММИРОВАНИЕ: НОВЫЕ ВЫЧИСЛИТЕЛЬНЫЕ ТЕХНОЛОГИИ, Выпуск 1 2021

При решении задач динамики жидкостей и газов в области малых скоростей потока и при изотермических условиях с успехом применяется метод решеточных уравнений Больцмана (LBM). Для решения дискретного уравнения Больцмана может быть использован новый метод Particles-on-Demand (PonD), в котором в каждой точке сетки дискретизация функции распределения в пространстве скоростей центрирована относительно текущей скорости потока. В отличие от классического LBM, метод PonD применим не только для задач с малыми скоростями потока и при изотермических условиях. В данной работе реализован метод PonD D1Q5 с итерационным расчетом скорости переноса и явным расчетом первых трех моментов, включая скорости переноса. Показано, что рассмотренная модификация метода PonD хоть и накладывает ограничения на параметры, позволяет проводить расчеты в большем диапазоне допустимых скоростей., The purpose of the paper is to demonstrate applicability of the Particle on Demand (PonD) D1Q5 method with the explicit calculation of the first three moments to problem with high speed of the flow. The standard LBM is applicable for small flow velocities. Thus to overcome this limitation we use PonD. In this work, we use conservative version of PonD - the D1Q5 method with the explicit calculation of the first three moments. Methodology. The Pond over LBM was applied to the Riemann problem in order to demonstrate the advantage of the method. In this work, we choose the case when contact discontinuities could propagate at variable speed. Findings. If the interpolation pattern is fixed relative to the point at which there is a current update of the discrete distribution function, then the transfer step can be written explicitly, thus the scheme is conservative. On the other hand, this imposes additional restrictions on the temperature and the flow rate. But even if the PonD scheme is limited to a fixed interpolation pattern, it is possible to simulate flows with larger values of the Mach number than in the case when the classical method of lattice Boltzmann equations is used. Originality/value. In the described particular case of the PonD method, it is possible to avoid iterations by calculating the temperature and velocity values directly at a new time layer. In this work, we have investigated the properties and the range of applicability (admissible values of temperature and velocity) of such modification of PonD., Вычислительные технологии, Выпуск 1 (26) 2021, Pages 62-71

The paper analyzes again the unique Greek inscription – the votive poem honouring Achillesgod and his island founded in 1987 on the Berezan.The purpose of the studyis to analyze the reading options of the famous Berezan hymn to the island and Achilles and establish which of them most closely matches the idea of the ancient poet and the current state of the work he created.Scientific novelty. For the first time, a comprehensive study of all the versions of the Berezan votive reconstruction proposed in world science is undertaken.The research methodologyis based on the principle of scientific objectivity and the generally accepted methods, and the technique of historical and epigraphic discipline.Conclusions. The criteria for the restoration of this epigraphic document have been determined. The boundaries of the permissible in its reconstruction have been established. The meaning, structure and condition of the inscription correspond to the version published by the author in 1990 the most closely: [Κυκλο]τερὲς κτεάτισμα θεῶν| [Ἀχιλῆο]ς ὄχημα, νῆσσε περικλυ|[τή. ἥ], κύμασι γηθομένη, σὸν πέ|[ρι εἴ]ληχεν Θέτιδος γόνος αἷμα|[ἔνερ]θε Αἰακίδης Ἀχιλλεὺς ἀθα|[νατοῖ]σιν ἵσος. Ἀλλ’ Ἀχιλλεῦ δέ|[ξ’ εὔθυ]σίην καὶ εἵλλαος ἴσθι ἡμετέ|ρ[αν ὁσιὴ]ν μοῦσαν ἀπὸ γραφίδος. “Oh, carved around the gods’ property, Achilles’ chariot, glorious, wave-rejoiced island. You, who before all received your blood as an inheritance, the child of Thetis, equal to the immortals below (= in hell) Achilles Eakid. You, O Achilles, receive a good sacrifice and, merciful, acknowledge (= accept) our (outgoing) pious muse”.

In $L_2(\mathbb{R}^d;\mathbb{C}^n)$, consider a self-adjoint matrix second order elliptic differential operator $\mathcal{B}_\varepsilon$, $0, Comment: 41 pages. This is an English translation of the paper originally published in Russian. Translated version: St. Petersburg Math. J. 31 (2020), to appear. Minor revision of the first version. Some typos removed

Рассмотрена задача прогнозирования временных рядов в ее одно- и многомерной постановках. Описан метод прогнозирования, в основе которого лежат алгоритмы сжатия данных и искусственного интеллекта. В качестве примера практического применения этого метода приведены вычисления для некоторых социальных, демографических и экономических показателей Новосибирской области. Наряду с широко используемыми на практике программами для сжатия данных, использованы реализации относительно менее известных моделей на основе формальных грамматик, а также метода, основанного на конечных автоматах. Для прогнозных значений построены доверительные интервалы., This paper describes and experimentally investigates a time series forecasting method based on data compression and artificial intelligence techniques. Its basic idea is to combine various algorithms that can estimate the compressed size of a sequence of discrete values into a single method of forecasting. Generalizations of the method to continuous and multivariate cases are described. We use several popular data compression libraries (zlib, bzip2, ppmd), as well as applications of relatively lesser-known algorithms based on formal grammars (re-pair) along with our implementation of an algorithm based on finite automata with multiple heads. All these are employed to make forecasts for some social, economic and demographic indicators of the Novosibirsk region. The article elaborates both our methodology and the methods used for data preprocessing. Confidence intervals with a confidence level of 0.95 are plotted for all predicted values; the average relative errors calculated from the results of predicting the already fixed values are given. Cases in which multivariate forecasting turned out to be more accurate than univariate forecasting are: 1. Average annual population and natural population growth in the Novosibirsk region. Figure 1 shows their graphs along with multivariate predictions, while confidence intervals and mean relative errors are presented in tables 1, 2. 2. Life expectancy and number of deaths in the Novosibirsk region. Figure 2 shows their graphs along with multivariate predictions, confidence intervals and mean relative errors are presented in tables 3, 4. Table 5 gives univariate forecasts of some other indicators of the Novosibirsk region. We think that the results of computations show that the proposed method is capable of finding non-trivial patterns in the data and can be used in practice., Вычислительные технологии, Выпуск 5 (25) 2020, Pages 80-90

Отмечена роль ведущих отечественных ученых В.И. Денисова и В.Г. Горского в становлении и развитии теории планирования эксперимента и прикладных статистических методов анализа данных. На основе математической теории планирования эксперимента исследовано влияние параметров воздушной среды рабочих помещений на интенсивность испарения опасных химических веществ с поверхности пролива. Создана математическая модель, описывающая этот процесс. Данные исследования необходимы для оценки профессионального риска на химических производствах, The article addresses the important problem of assessing the professional risk of workers in industries where the chemical factor is possible, as well as the role of the mathematical theory of experimental design in conducting of experimental research. The role of leading Russian scientists Denisov Vladimir Ivanovich and Gorsky Vladimir Grigoryevich in the establishment and development of experimental design theory and applied statistical data analysis methods is noted in the work. It is noted that in order to successfully solve scientific, industrial, and technological problems, modern research engineer must have competencies in the field of the theory of mathematical statistics, practical skills in planning an experiment, and methods for statistical processing of experimental data. The aim of the studies presented in this paper is to assess the influence of the air environment of the working room (temperature, humidity) on the rate of evaporation of a hazardous chemical from the spill. The experiment was carried out on a facility that allowed varying over a wide range the temperature and humidity. The experiment was carried out in accordance with the plan developed on the basis of the mathematical theory of experimental design. The obtained mathematical model allows estimating the level of air pollution by hazardous chemicals in the temperature and humidity range typical of workrooms. A comparison of the concentrations calculated according to the model with the permissible level of air pollution (MPC of the working zone air) will allow assessing the risk of inhalation exposure to hazardous chemicals on working personnel in local emergency situations. Conducted research is necessary to assess occupational risk in chemical industries, Вычислительные технологии, Выпуск 3 2020

Математические модели многих технических систем имеют вид интегрального уравнения Вольтерра I рода с разностным ядром. Для таких систем задача идентификации заключается в построении оценки для импульсной переходной функции системы по измеренным (с шумами) значениям входного и выходного сигналов и является некорректно поставленной. В недавней работе авторов предложен устойчивый алгоритм идентификации, использующий аппарат сглаживающих кубических сплайнов для вычисления первых производных входного и выходного сигналов. К сожалению, сглаживающие кубические сплайны неудовлетворительно фильтруют аномальные измерения. Поэтому предложен двухшаговый алгоритм идентификации, на первом шаге которого аномальные измерения удаляются с использованием пространственно-локального фильтра, а затем строятся сглаживающие сплайны, Volterra integral equation of the first kind often represents stationary dynamic systems. For such a model, the non-parametric identification problem reduces to the estimation of pulse transition characteristics (that is the kernel of integral equation) from the registered noise-contaminated values of input and output signals. To formulate stable solution for identification problem authors propose algorithm that estimates pulse transition characteristics by solving Volterra integral equation of the second kind and involving first derivatives of input and output signals application that corresponds to non-stable problem. Smoothing cubic splines employed in robust calculation of first derivatives allow finding a stable solution of identification problem even when input and output signals of system identified are essentially noise-contaminated. Unfortunately, measured values of input and output signals also contain anomalous measurements such as pulse noises, glitches, etc. Such measurements are poorly smoothable by splines that cause high levels of first derivatives errors and, conversely, significant pulse transition characteristics identification errors of dynamic system. For all the reasons aforementioned, in this paper authors present the new stable two-step identification algorithm in case of anomalous measurements. The first step of the algorithm is for non-linear local-spatial combined filtration procedure of input and output signals that helps to effectively remove anomalous measurements. At the second step, smoothing cubic splines are used to calculate stable first derivatives of previously filtered signals. An extensive computational experiment showed the effectiveness of the proposed algorithm, which allows solving the identification problem with acceptable accuracy in practice even at high intensity of anomalous measurements. The experimental results give reason to recommend this algorithm for solving practical problems of identifying stationary systems, the mathematical model of which is the Voltaire integral equation of the first kind with a difference kernel, Вычислительные технологии, Выпуск 3 2020

Рассматривается проблема оценки относительной активности мономеров на основе дифференциального уравнения сополимеризации. Обосновано включение в модель погрешности измерения входного признака в виде ошибки Берксона. Предложен алгоритм одновременного оценивания констант сополимеризации и дисперсий ошибок с помощью метода максимального правдоподобия. На примере сополимеризации виниловых эфиров произведено сравнение разных методов оценивания констант сополимеризации. Показано, что метод на основе симметричных уравнений дает некорректные результаты. Результаты оценивания с помощью предложенного алгоритма наиболее близки к оценкам, полученным по нелинейному методу наименьших квадратов, Purpose. The purpose of this paper is to study methods for estimating copolymerization reactivity ratios based on the differential composition equation. Methodology. Most estimation methods reduce the differential composition equation to a linear form. They are based on the least squares method and do not take into account the measurement error in the input variable. Therefore they lead to statistically incorrect results. When analyzing the problem on the basis of the error-in-variables model in the classical case, additional information is required to determine the magnitude of the errors in measuring the concentration of monomers in the mixture and in the copolymer. Inclusion of the measurement error in the input variable into the model as the Berkson error is more consistent with the actual conditions of the experiments. It allows simultaneous estimating both the reactivity ratios and the variances of measurement errors using the maximum likelihood method. Results. The algorithm have been developed for estimating reactivity ratios with no additional information. The empirical study of estimation methods has been carried out using the example of copolymerization of vinyl esters. Findings. It is shown that the method based on symmetric equations gives incorrect results. Estimation results using the proposed algorithm are closest to the estimates obtained by the nonlinear least squares method, Вычислительные технологии, Выпуск 3 2020

The paper deals with the problem of optimal control with a Boltz–type quality index over a finite time interval for a linear steady–state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A. B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady–state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time–optimal problems. © 2020 Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. All rights reserved.

Представлены две адаптивные модификации сигма-точечного фильтра Калмана с рекуррентным оцениванием ковариационных матриц шумов системы и измерений, на основе которых выполняется процедура параметрической идентификации нелинейных непрерывно-дискретных систем. Применение процедуры адаптивной параметрической идентификации позволило вычислить с достаточной точностью оценки параметров нескольких моделей радиационного давления солнечного излучения. Полученные результаты повысили качество прогнозирования траектории движения навигационного спутника, Purpose. The paper considers the problem of estimation of unknown parameters for various models of solar radiation based on adaptive modifications of the unscented Kalman filter. The estimations of the obtained parameters are used both in solar radiation models and in construction of trajectory of a navigation satellite. Methodology. To solve the problem of parametric identification of stochastic nonlinear continuous-discrete systems, several adaptive modifications of the unscented Kalman filter are considered. The algorithms assume recurrent estimation of covariance matrices of system noise and measurements. The maximum likelihood method is used for parametric identification of stochastic nonlinear continuous-discrete systems. Adaptive modifications of the unscented Kalman filter are used in the construction of the identification criterion. Estimates of unknown parameters of various solar radiation models are found for the movement for the navigation satellite model as an example. The satellite orbital movement forecast is made. Finding and value. The application of the adaptive parametric identification procedure allows calculating the estimates for the parameters of several models of the solar radiation with sufficient accuracy. The obtained results lead to significant improvement of quality of the prediction for satellite trajectory, Вычислительные технологии, Выпуск 3 2020

Исследуется зондирование снежного покрова методом радарной интерферометрии. Предложена приближенная модель интерферометрического зондирования, основанная на методе малых возмущений. Поле обратного рассеяния рассматривается как когерентная сумма волн, рассеянных на неровностях границ раздела “воздух - снег” и “снег - почва”. Представлены результаты определения водного эквивалента снега с помощью радарной интерферометрии на ровной земной поверхности (равнина) и поверхности с уклонами рельефа (холмисто-гористая местность). Проведена оценка вклада рассеяния от поверхности снега в значения амплитуды и интерферометрической фазы. Выполнен анализ влияния уклонов рельефа на оценку параметров снежного покрова. Приведены результаты численных оценок. Показано, что для местности с уклоном рельефа около 45изменения в интерферометрической фазе относительно ровной поверхности достигают 40%. Однако если склоны сравнительно ровные (менее 10), эти изменения не превышают 10%, The main characteristics of the snow cover determining its impact on the environment are its thickness and the snow water equivalent (SWE). SWE assesses the water content in the snow cover. Radar interferometry is one of the methods for determining SWE. The paper presents the results of snow cover sensing by radar interferometry on both flat Earth’s surface and terrain with relief. A backscattering model taking into account backscattering from the snow surface is proposed in contrast to the existing methods. The backscattering field is considered as a coherent sum of waves scattered on the irregularities of the air - snow and snow - soil interfaces. These interfaces are statistically rough surfaces with random irregularities, whose heights are described by uncorrelated stationary random functions with their mean values, standard deviations, and correlation radii. It is assumed that the irregularities are small compared to the wavelength, their slopes are small, and the conditions for the applicability of the method of small perturbations are satisfied. It is also supposed that roughness does not affect the coherent field according to the Born approximation. The incident and scattered waves are assumed to follow Snell’s law. The coherent waves reflection and transmission coefficients are determined by Fresnel formulas for a flat interface. The contribution of backscattering from the snow surface to the values of the amplitude and interferometric phase is estimated using small perturbation method. It is shown that the relative error of interferometric phase determination due to the influence of the wave scattered by the air - snow boundary does not exceed 8% for the angles of incidence of 20-45and the density of snow 0.2-0.3 g/cm. The approximate relations show the linear dependence between the interferometric phase and SWE. The model is extended to the general case of backscattering from snow cover on the Earth’s surface with relief. The influence of terrain slopes on the interferometric phase is estimated. It is shown that for hilly terrain with slopes of about 45, the relative changes in the interferometric phase could reach 40%. However, if the slopes are relatively flat (less than 10), these changes do not exceed 10%., Вычислительные технологии, Выпуск 4 (25) 2020

Системы уравнений с запаздываниями широко применяются в различных областях современного математического моделирования. В ходе разработки структуры математической модели и идентификации ее параметров приходится многократно решать задачу Коши для подобных систем. В случае высокой размерности системы, а также при условии жесткости задачи численное решение уравнений с запаздываниями может требовать значительных вычислительных и временных затрат. Таким образом, разработка и реализация эффективных алгоритмов численного решения различных классов уравнений с запаздывающими аргументами является актуальной задачей. В настоящей статье представлена модифицированная версия программного комплекса DIFSUBDEL, в которой реализованы методы численного решения дифференциальных уравнений с запаздываниями на основе линейных многошаговых методов. Переработанная версия разработана с применением принципов структурного программирования и является значительно более удобной в эксплуатации, чем исходная, а также обладает свойством потокобезопасности, что позволяет использовать комплекс в качестве блока в системах, основанных на технологиях параллельного программирования с общей памятью. Был проведен сравнительный анализ производительности переработанной системы DIFSUBDEL c другими существующими программными реализациями численных методов решения дифференциальных уравнений с запаздыванием и показана ее эффективность., The systems of delay differential equation are widely used in modern mathematical modeling. The process of mathematical model development and identification requires the repeated solution of initial value problems for such systems. The numerical solution of delay differential equations may be computationally expensive, especially when the problem is stiff and high-dimensional. Therefore, it is important to develop and implement the efficient algorithms for the numerical solution of different classes of delay differential equations. In this paper, a new implementation of DIFSUBDEL program package for the numerical solution of delay differential equations based on linear multistep methods is discussed. The modified version is based on structured programming principles to make the program thread safe and user-friendly. The modified program performance is compared with the existing implementations of numerical methods for the solution of delay differential equations., ВЫЧИСЛИТЕЛЬНЫЕ МЕТОДЫ И ПРОГРАММИРОВАНИЕ: НОВЫЕ ВЫЧИСЛИТЕЛЬНЫЕ ТЕХНОЛОГИИ, Выпуск 1 2020

Проведено сравнение качества решений модельного уравнения второго порядка с малым параметром, полученных по трем различным разностным схемам на специальных адаптивных сетках, явно задаваемых координатным преобразованием, а также на равномерных сетках в новых переменных, соответствующих этому преобразованию. Исследуются схемы второго порядка точности с диагональным преобладанием и без него и простейшая противопотоковая схема. На основе оценок погрешностей сделаны прогнозы относительно свойств решений, подтвержденные анализом и численными экспериментами. Показано, что схема второго порядка аппроксимации с диагональным преобладанием сходится равномерно по малому параметру со вторым порядком лишь в частном случае, когда коэффициент при старшей производной мал только в слое; если же он мал также и вне слоя, порядок сходимости первый. Установлено также, что схема без диагонального преобладания имеет существенно более качественные решения без осцилляций в новых переменных на равномерной сетке, чем в соответствующих им исходных физических координатах. В противоположность ей схемы с диагональным преобладанием не чувствительны к выбору системы координат., The paper compares solution quality to some model second- order equation with a small parameter obtained through three different schemes both on special adaptive grids specified explicitly by coordinate transformations eliminating layers and on uniform grids in a new coordinate related to the transformations. The schemes up to second order in physical and transformation variables both with a diagonal and not diagonal dominance and the simplest counter-flow scheme are analyzed. Predictions of a solution behavior based on estimates of solution errors are described, which are confirmed by numerical experiments and proofs. It is established, in particular, that the scheme of the second order with a diagonal dominance converges uniformly if the coefficient before the second derivative is small at the points of the boundary layer only. It was also demonstrated for the schemes without a diagonal dominance, mach better solutions without oscillations are obtained on uniform grids in new variables than on corresponding adaptive grids in the original physical coordinates., Вычислительные технологии, Выпуск 5 (25) 2020, Pages 66-79