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The article discusses the main approaches to the design and development of an automated information system for a secure cloud repository of paperworks of students and employees of educational and research organisations (abstracts, essays, term papers and theses, master’s theses, deposited articles), providing secure storage and secure mobile access to stored data. The research and development of architectural solutions to ensure reliable and secure data storage using cloud technologies are described. The main problems of designing a secure repository and ways to solve them are considered. The cloud repository of written works is built on the basis of the Ceph distributed file system, which uses the NextCloud system and the virtual computing resources of the virtual network infrastructure of the Belarusian State University as a platform for building a cloud interface.

A scheme is constructed for the numerical solution of a singular integral equation with a logarithmic kernel by the method of orthogonal polynomials. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

Asymptotic properties of random walks on minimal Cayley graphs of complex reflection groups are investigated. The main result of the paper is theorem on fast mixing for random walks on Cayley graphs of complex reflection groups. Particularly, bounds of diameters and isoperimetric constants, a known result on fast fixing property for expander graphs play a crucial role to obtain the main result. A constructive way to prove a special case of Babai’s conjecture on logarithmic order of diameters for complex reflection groups is proposed. Basing on estimates of diameters and Cheeger inequality, there is obtained a non-trivial lower bound for spectral gaps of minimal Cayley graphs on complex reflection groups.

An important mathematical problem of computer data analysis – the problem of statistical sequential testing of simple hypotheses on parameters of probability distributions of observed binary data – is considered in the paper. This problem is being solved for two models of observation: for independent observations and for homogeneous Markov chains. Explicit expressions of the sequential tests statistics are derived, transparent for interpretation and convenient for computer realisation. An approach is developed to calculate the performance characteristics – error probabilities and mathematical expectations of the random number of observations required to guarantee the requested accuracy for decision rules. Asymptotic expansions for the mentioned performance characteristics are constructed under «contamination» of the probability distributions of observed data.

The solution of the stationary heat conduction problem for profiled polar-orthotropic annular plates with heat-insulated bases from N extended heat sources at their external border is presented. The temperature distribution in such plates will be non-axisymmetric. The solution of the stationary heat conduction problem for anisotropic annular plates of an random profile is resolved through the solution of the corresponding Volterra integral equation of the second kind. The formula of a temperature calculations in anisotropic annular plates of an random profile is given. The exact solution of stationary heat conduction problem for polar-orthotropic annular plate of an exponential profile is recorded. The temperature distribution in such anisotropic plate from N extended heat sources at its outer border is more complex than in the case of temperature distribution from N point heat sources at their external border.

The article describes an analytical model of mine skip dynamics taking into account the presence of the head and balancing ropes and the existing curvilinearity of the guides. Expressions for the forces acting on the skip from the side of the guides have been constructed. It is shown, that the frequencies of natural vibrations of skip depend on the vertical acceleration and the distance traveled during its lifting. A graph (diagram) of skips vertical speed which observance does not lead to the appearance of skips vertical vibrations due to elasticity of the ropes is developed. An algorithm for finding the forces principal vector and the forces principal moment acting on the skip based on the reading of three accelerometers recording horizontal accelerations of skip during its movement is presented.

The paper proposes an identification technique of objects on the Earth’s surface images based on combination of machine learning methods. Different variants of multi-layer convolutional neural networks and support vector machines are considered as original models. A hybrid convolutional neural network that combines features extracted by the neural network and experts is proposed. Optimal values of hyperparameters of the models are calculated by grid search methods using k-fold cross-validation. The possibility of improving the accuracy of identification based on the ensembles of these models is shown. Effectiveness of the proposed technique is demonstrated by the example of images obtained by synthetic aperture radar.

This paper deals with an optimal control problem for a linear discrete system subject to unknown bounded disturbances, where the control goal is to steer the system with guarantees into a given terminal set while minimising the terminal cost function. We define an optimal control strategy which takes into account the state of the system at one future time instant and propose an efficient numerical method for its construction. The results of numerical experiments show an improvement in performance under the optimal control strategy in comparison to the optimal open-loop worst-case control while maintaining comparable computation times.

In the paper, the linear differential-difference dynamic systems with delayed arguments are considered. Such systems have a lot of application areas, in particular, processes with repetitive and learning structure. We apply the method of the separation hyperplane theorem for convex sets to establish optimality conditions for the control function to drive the trajectory to zero equilibrium state in the fastest possible way. For the special case of the integral control constraints, the proposed method is detailed to establish an analytical form of the optimal control function. The illustrative example is given to demonstrate the obtained results with the step-by-step calculation of the basic elements of the optimal control.

Throughout this paper, all groups are finite. Let $σ=\{σ_i{}|i\in I\}$ be some partition of the set of all primes $\Bbb{P}$. If $n$ is an integer, $G$ is a group, and $\mathfrak{F}$ is a class of groups, then $σ(n)=\{σ_i{}|σ_i{}\cap \pi(n)\ne \emptyset\}$, $σ(G)=σ(|G|)$, and $σ(\mathfrak{F})=\cup _G{}_\in{}_\mathfrak{F}σ(G)$. A function $f$ of the form $f\colon σ\to$ {formations of groups} is called a formation σ-function. For any formation $σ$-function $f$ the class $LF_σ(f)$ is defined as follows: $LF_{\sigma}(f)=(G$ is a group $|G=1$ или $G\ne1$ and $G/O_σ{}'_i{}_,{}_σ{}_i{}(G)\in f(σ_i{})$ for all $σ_i{}\in σ(G))$. If for some formation $σ$-function $f$ we have $\mathfrak{F}=LF_{\sigma}(f)$, then the class $\mathfrak{F}$ is called $σ$-local definition of $\mathfrak{F}$. Every formation is called 0-multiply $σ$-local. For $n$ > 0, a formation $\mathfrak{F}$ is called $n$-multiply $σ$-local provided either $\mathfrak{F}=(1)$ is the class of all identity groups or $\mathfrak{F}=LF_{\sigma}(f)$, where $f(σ_i{})$ is $(n – 1)$-multiply $σ$-local for all $σ_i{}\in σ(\mathfrak{F})$. A formation is called totally $σ$-local if it is $n$-multiply $σ$-local for all non-negative integer $n$. The aim of this paper is to study properties of the lattice of totally $σ$-local formations. In particular, we prove that the lattice of all totally $σ$-local formations is algebraic and distributive.

Регуляризация задачи оптимизации состоит в ее сведении к эквивалентной задаче, удовлетворяющей условиям регулярности, которые гарантируют выполнение соотношений двойственности в строгой форме. В настоящей статье для линейных задач полуопределенного программирования предлагается процедура регуляризации, основанная на понятии неподвижных индексов и их свойствах. Эта процедура описана в виде алгоритма, который за конечное число шагов преобразует любую задачу линейного полубесконечного программирования в эквивалентную задачу, удовлетворяющую условию Слейтера. В результате использования свойств неподвижных индексов и предложенной процедуры регуляризации получены новые двойственные задачи полубесконечного программирования в явной и неявной формах. Доказано, что для этих двойственных задач и исходной задачи соотношения двойственности выполняются в строгой форме.

The problem of malicious host detection in a computer network is reviewed. Activity of computer network hosts is tracking by a noisy detector. The paper suggests method for detection malicious hosts using activity timeseries classification. The approach is based on hidden Markov chain model that analyses timeseries and consecutive search of the most probable final state of the model. Efficiency of the approach is based on assumption that advanced persisted threats are localised in time, therefore malicious hosts in a computer network can be detected by virtue of activity comparison with reliable safe hosts.

We study the influence of N extended heat sources at external boundaries on the nonaxisymmetric temperature distribution on profiled polar-orthotropic ring plates and take into account heat exchange with the external environment. The solution of the stationary heat conduction problem for anisotropic annular plates of a random profile is resolved through the solution of the corresponding Volterra integral equation of the second kind. The formula of a temperature calculations in anisotropic annular plates of an random profile is given. The exact solution of stationary heat conductivity problem for a reverse conical polar-orthotropic ring plate is recorded. The temperature distribution in such anisotropic plate from N extended heat sources at its outer border is more complex than in the case of temperature distribution from N point heat sources at their external border.

This article is devoted to the investigation of stability of equilibrium of ordinary differential equations using the method of semi-definite Lyapunov’s functions. Types of scalar nonlinear sixth-order differential equations for which regular constant auxiliary functions are used are emphasized. Sufficient conditions of global asymptotic stability and instability of the zero solution have been obtained and it has been established that the Aizerman problem has a positive solution concerning the roots of the corresponding linear differential equation. Studies highlight the advantages of using semi-definite functions compared to definitely positive Lyapunov’s functions.

Hausdorff operators on the real line and multidimensional Euclidean spaces originated from some classical summation methods. Now it is an active research area. Hausdorff operators on general groups were defined and studied by the author since 2019. The purpose of this paper is to define and study Hausdorff operators on Lebesgue and real Hardy spaces over homogeneous spaces of locally compact groups. We introduce in particular an atomic Hardy space over homogeneous spaces of locally compact groups and obtain conditions for boundedness of Hausdorff operators on such spaces. Several corollaries are considered and unsolved problems are formulated.

Two computational schemes for solving boundary value problems for a singular integro-differential equation, which describes the scattering of H-polarized electromagnetic waves by a screen with a curved boundary, are constructed. This equation contains three types of integrals: a singular integral with the Cauchy kernel, integrals with a logarithmic singularity and with the Helder type kernel. The integrands, along with the solution function, contain its first derivative. The proposed schemes for an approximate solution of the problem are based on the representation of the solution function in the form of a linear combination of the Chebyshev orthogonal polynomials and spectral relations that allows to obtain simple analytical expressions for the singular component of the equation. The expansion coefficients of the solution in terms of the Chebyshev polynomial basis are calculated by solving a system of linear algebraic equations. The results of numerical experiments show that on a grid of 20 –30 points, the error of the approximate solution reaches the minimum limit due to the error in representing real floating-point numbers.

The problem of construction of «continuous» (number of observations is not fixed) and «exact» (number of observations is fixed) D-optimal experimental designs for linear multiple regression in the case when variance of errors of observations depends on regressor value is studied in this paper. Families of functions that determine heteroscedastic observations are found for which it is possible to construct «continuous» and «exact» D-optimal experimental designs. «Continuous» D-optimal experimental designs under four different types of heteroscedasticity are constructed for linear multiple regression with three regressors.

An algorithm is developed for finding the structure of the optimal subset in the knapsack problem based on the proposed multicriteria optimization model. A two-criteria relation of preference between elements of the set of initial data is introduced. This set has been split into separate Pareto layers. The depth concept of the elements dominance of an individual Pareto layer is formulated. Based on it, conditions are determined under which the solution to the knapsack problem includes the first Pareto layers. They are defined on a given set of initial data. The structure of the optimal subset is presented, which includes individual Pareto layers. Pareto layers are built in the introduced preference space. This does not require algorithms for enumerating the elements of the initial set. Such algorithms are used when finding only some part of the optimal subset. This reduces the number of operations required to solve the considered combinatorial problem. The method for determining the found Pareto layers shows that the number of operations depends on the volume of the knapsack and the structure of the Pareto layers, into which the set of initial data in the entered two-criteria space is divided.

The goal of this work is to study the effect of bone ingrowth into open pores of the implant and estimate of the mechanical characteristics for obtained biocomposite. Reconstruction of the isotropic model based on data acquired from computed tomography allows us to study the metallic and bone components integration under compressive load. Results are compared to performed mechanical tests of the porous specimen. The finite element modeling allows obtaining a stress-stain curve for the bone – titanium biocomposite. Young’s modulus of the metallic specimen is increased by 29 % after pores is filled with bone tissues. The conditional yield strength of the bone – titanium biocomposite is 2 times higher than that of porous open-pore titanium.

In this paper, we investigate the urgent problem associated with bone age determination using hand X-rays. A combined algorithm for the recognition of radiographs is proposed, which uses simultaneous two neural network models, based on Xception and DenseNet169. The method allows to generalize the knowledge of different medical experts and increases the accuracy of bone age prediction in general.

Some new upper bounds for noncentral chi-square cumulative density function are derived from the basic symmetries of the multidimensional standard Gaussian distribution: unitary invariance, components independence in both polar and Cartesian coordinate systems. The proposed new bounds have analytically simple form compared to analogues available in the literature: they are based on combination of exponents, direct and inverse trigonometric functions, including hyperbolic ones, and the cdf of the one dimensional standard Gaussian law. These new bounds may be useful both in theory and in applications: for proving inequalities related to noncentral chi-square cumulative density function, and for bounding powers of Pearson’s chi-squared tests. Некоторые новые верхние границы для функции нецентрального хи-квадрат распределения выводятся из базовых симметрий плотности многомерного стандартного нормального закона: унитарной инвариантности, независимости компонент как в полярной, так и в декартовой системе координат. В сравнении с известными в литературе аналогами предложенные новые верхние оценки имеют простой аналитический вид: они представляют собой комбинации из экспонент, прямых и обратных тригонометрических функций, в том числе гиперболических, а также функции распределения одномерного стандартного нормального закона. Данные оценки могут быть полезны как в теории, так и в приложениях для доказательства неравенств, связанных с функцией нецентрального хи-квадрат распределения, и построения нижних оценок мощности хи-квадрат критерия Пирсона.

The solution of the nonaxisymmetric stationary problem of the heat conductivity for profiled polar-orthotropic annular plates considering the heat exchange with external environment through the bases is presented. Thermophysical characteristics of the material of the plate are assumed to be temperature-independent. A constant temperature T [1] ∗ is maintained on the inner contour of the ring plate and on the outer contour N equidistant point sources of heat with the same tempe­ rature T [2] ∗ each are applied. Plate temperature is higher than ambient temperature T [0] ( T [0] < T [1] ∗ < T [2] ∗ ). It is assumed that the temperature does not vary in thickness of a thin ring plate. The temperature values on the contours of the annular plate are given. There are no internal heat sources in the plate. The temperature distribution in such plates will be nonaxisymmetric. Analytical solutions of the stationary heat conductivity problem for the following anisotropic annular plates are presented: the plate of constant thickness, the back conical and the conical plate. The Volterra integral equation of the second kind corresponding to the given differential equation of the stationary heat conductivity for profiled anisotropic annular plates is written to obtain the solution in the general case. The kernels of the integral equation for anisotropic annular plates of power and exponential profiles are given explicitly. The solution of the integral equation is written by using the resolvent. It is indicated that due to the presence of irrational functions in the kernels of the integral equation it is necessary to apply numerical methods in the calculation of iterated kernels or numerically solve the Volterra integral equation of the second kind. A formula for the calculation of temperatures in anisotropic annular plates of an arbitrary profile is given.

Approximation properties of Fejer means of Fourier series by Chebyshev – Markov system of algebraic fractions and approximation by Fejer means of function |x|s, 0 < s < 2, on the interval [−1,1], are studied. One orthogonal system of Chebyshev – Markov algebraic fractions is considers, and Fejer means of the corresponding rational Fourier – Chebyshev series is introduce. The order of approximations of the sequence of Fejer means of continuous functions on a segment in terms of the continuity module and sufficient conditions on the parameter providing uniform convergence are established. A estimates of the pointwise and uniform approximation of the function |x|s, 0 < s < 2, on the interval [−1,1], the asymptotic expressions under n→∞ of majorant of uniform approximations, and the optimal value of the parameter, which provides the highest rate of approximation of the studied functions are sums of rational use of Fourier – Chebyshev are found.

In this work, a problem of partitioning a complete weighted graph into cliques in such a way that sum of edge weights between vertices belonging to the same clique is maximal is considered. This problem is known as a clique partitioning problem. It arises in many applications and is a varian of classical clustering problem. However, since the problem, as well as many other combinatorial optimization problems, is NP-hard, finding its exact solution often appears hard. In this work, a new method for constructing upper bounds of partition quality function values is proposed, and it is shown how to use these upper bounds in branch and bound technique for finding an exact solution. Proposed method is based on the usage of triangles constraining maximal possible quality of partition. Novelty of the method lies in possibility of using triangles overlapping by edges, which allows to find much tighter bounds than when using only non-overlapping subgraphs. Apart from constructing initial estimate, a method of its recalculation, when fixing edges on each step of branch and bound method, is described. Test results of proposed algorithm on generated sets of random graphs are provided. It is shown, that version that uses new bounds works several times faster than previously known methods.

This article offers a mathematical model for distributed object­oriented stochastic hybrid systems (DOBSHS). DOBSHS are composite objects communicating with other objects through the exchange of messages through an asynchronous medium such as a network. An important component of the model is the probabilistic nature of the DOBSHS, in which the state of the system is described by stochastic differential equations with instantaneous probabilistic state changes when certain conditions are met. Also probabilistic is the nature of the messaging environment, in which the model of message delivery time is a random variable. Such problems are often encountered in practice in various areas and issues of formal modeling and verification of their properties are very important. The article presents a mathematical model of DOBSHS and proved that it has a Markov property.

The article proposes a new method for determining the number of splitting intervals and the number of observations in them when building estimates of the spectral densities of stationary random processes with a given accuracy over intersecting observation intervals based on asymptotic results, obtained for the first moment of convergence rate under the assumption that the spectral density satisfies the Lipschitz condition. Two cases are considered: with a single and arbitrary data taper. As a result, an algorithm is proposed for constructing estimates for intersecting intervals of observations with a given accuracy. This algorithm was tested on model examples for random AR(4) processes, using data taper of Riesz, Bochner, Parzen. The proposed method will be useful to the researcher in analyzing data in the form of stationary random processes using non­parametric methods of spectral analysis in an automated mode.

Let P be the set of all primes, Zn a cyclic group of order n and X wr Zn the regular wreath product of the group X with Zn. A Fitting class F is said to be X-quasinormal (or quasinormal in a class of groups X ) if F ⊆ X, p is a prime, groups G ∈ F and G wr Zp ∈ X, then there exists a natural number m such that G m wr Zp ∈ F. If X is the class of all soluble groups, then F is normal Fitting class. In this paper we generalize the well-known theorem of Blessenohl and Gaschütz in the theory of normal Fitting classes. It is proved, that the intersection of any set of nontrivial X-quasinormal Fitting classes is a nontrivial X-quasinormal Fitting class. In particular, there exists the smallest nontrivial X-quasinormal Fitting class. We confirm a generalized version of the Lockett conjecture (in particular, the Lockett conjecture) about the structure of a Fitting class for the case of X-quasinormal classes, where X is a local Fitting class of partially soluble groups.

The problem of sequential test for many simple hypotheses on parameters of time series with trend is considered. Two approaches, including M-ary sequential probability ratio test and matrix sequential probability ratio test are used for constructing the sequential test. The sufficient conditions of finite terminations of the test and the existence of finite moments of their stopping times are given. The upper bounds for the average numbers of observations are obtained. With the thresholds chosen suitably, these tests can belong to some specified classes of statistical tests. Numerical examples are presented. Рассмотрена проблема последовательного тестирования многих простых гипотез о параметрах временных рядов с трендом. Для построения последовательного теста использованы два подхода, в том числе M-нарный последовательный критерий отношения вероятностей и матричный последовательный критерий отношения вероятностей. Даны достаточные условия конечных завершений теста и существования конечных моментов их времени остановки. Получены верхние оценки для среднего числа наблюдений. При подходящих порогах эти тесты могут принадлежать некоторым определенным классам статистических тестов. Приводятся результаты вычислительных экспериментов.

A finite non-nilpotent group G is called a B-group if every proper subgroup of the quotient group G/Φ(G) is nilpotent. We establish the r-solvability of the group in which some Sylow r-subgroup permutes with the derived subgroups of 2-nilpotent (or 2-closed) B-subgroups of even order and the solvability of the group in which the derived subgroups of 2-closed and 2-nilpotent B-subgroups of even order are permutable.