Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Все статьи: 729
Статья научная
We propose an approach to simulation of information protection processes at informatization objects in the dynamics of realizing threats of its leakage through technical channels using the apparatus of composite Petri–Markov nets. We investigate the interrelationships of the processes of transferring information at the object of informatization, its interception by a violator from outside this object through technical channels and protection against interception. Analytical expressions are obtained for calculating the probabilistic-temporal characteristics of simulated processes in order to assess the possibility of a violator implementing threats of information leakage at objects of informatization; protection of this information through the use of preventive measures aimed at blocking or anticipating the actions of the violator to implement these threats
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Simulation of concurrent games
Статья научная
Concurrent games, in which participants run some distance in real physical time, are investigated. Petri - Markov models of paired and multiple competitions are formed. For paired competition formula for density function of time of waiting by winner the moment of completion of distance by loser is obtained. A concept of distributed forfeit, which amount is defined as a share of sum, which the winner gets from the loser in current moment of time is introduced. With use of concepts of distributed forfeit and waiting time the formula for common forfeit, which winner gets from loser, is obtained. The result, received for a paired competition, was spread out onto multiple concurrent games. Evaluation of common wins and loses in multiple concurrent game is presented as a recursive procedure, in which participants complete the distance one after another, and winners, who had finished the distance get forfeits from participants, who still did not finish it. The formula for evaluation of common winning in concurrent game with given composition of participants is obtained. The result is illustrated with numerical example.
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Статья научная
It is shown that multistage concurrent games, or relay-races, are widely used in practice. It is proposed to model relay-races in the state space, in which discrete co-ordinates are the mathematical analogue of stages, which participants pass in the current time, and basic principle of modelling of residence of participant in space states is the M-parallel semi-Markov process. With use of the proposed formalisms formulae for evaluation of stochastic and time characteristics of relay-races evolution are obtained. For arbitrary realization of switching trajectory the recurrent procedure of evolution with evaluation of stochastic and time characteristics of realization under investigation is worked out. Conception of distributed forfeit, which depends on difference of stages of participants compete in pairs is introduced. Dependence for evaluation of total forfeit for every participant is obtained.
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SkifCh: эффективный коммуникационный интерфейс
Статья научная
В работе описывается SkifCh - низкоуровневый интерфейс передачи сообщений. Данный интерфейс эффективно поддерживается на уровне сетевого оборудования, которое, в свою очередь, может быть реализовано в ПЛИС (как сделано в суперкомпьютере СКИФ-Аврора) или в специализированных микросхемах. Интерфейс SkifCh может быть использован для высокоэффективных сетевых обменов непосредственно из прикладных программ, а также для реализации коммуникационных библиотек более высокого уровня. На данный момент поверх интерфейса SkifCh реализованы системы MPI, SHMEM, GASNet и ARMCI. В работе также приведено сравнение эффективности использования SkifCh и MPI на суперкомпьютере СКИФ-Аврора.
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Smooth approximation of the quantile function derivatives
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In this paper, a smooth approximation of the second-order derivatives of quantile function is provided. The convergence of approximations of the first and second order derivatives of quantile function is studied in cases when there exists a deterministic equivalent for the corresponding stochastic programming problem. The quantile function is one of common criteria in stochastic programming problems. The first-order derivative of quantile function can be represented as a ratio of partial derivatives of probability function. Using smooth approximation of probability function and its derivatives we obtain approximations of these derivatives in the form of volume integrals. Approximation of the second-order derivative is obtained directly as derivative of the first-order derivative. A numerical example is provided to evaluate the accuracy of the presented approximations.
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Sobolev type equations in spaces of differential forms on Riemannian manifolds without boundary
Статья научная
The article contains a review of the results obtained by the author both independently and in collaboration with other members of the Chelyabinsk scientific school founded by G.A. Sviridyuk and devoted to Sobolev-type equations in specific spaces, namely the spaces of differential forms defined on some Riemannian manifold without boundary. Sobolev type equations are nonclassical equations of mathematical physics and are characterized by an irreversible operator at the highest derivative. In our spaces, we need to use special generalizations of operators to the space of differential forms, in particular, the Laplace operator is replaced by its generalization, the Laplace-Beltrami operator. We consider specific interpretations of equations with the relatively bounded operators: linear Barenblatt-Zheltov-Kochina, linear and semilinear Hoff, linear Oskolkov ones. For these equations, we investigate the solvability of the Cauchy, Showalter-Sidorov and initial-final value problems in different cases. Depending on the choice of the type of equation (linear or semi-linear), we use the corresponding modification of the phase space method. In the spaces of differential forms, in order to use this method based on domain splitting and the actions of the corresponding operators, the basis is the Hodge-Kodaira theorem on the splitting of the domain of the Laplace-Beltrami operator.
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Software for the mobile robot spatial orientation system
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Ensuring the orientation of the mobile robot in space requires solving the problem of its precise positioning. For this purpose, it is proposed to include an additional measuring complex in the control system of the mobile robot. The complex is designed to determine the orientation of the robot in space by the values of the angles of course, roll and pitch. The analysis of the mobile robot positioning solutions allows to justify the feasibility of using inertial navigation systems based on microelectromechanical sensors to obtain navigation information about the orientation of the mobile robot in space. The key element of the developed functional scheme of the mobile robot control system is the software of the mobile robot spatial orientation system. The software implements separate sections of the code that determine the orientation in space using inertial sensors in parallel to the main algorithm of the mobile robot. The result of the developed software is a string containing up-to-date information about the three orientation angles of the robot: the string is sent to the server to form control actions to correct the spatial orientation of the mobile robot. To improve the accuracy of determining the robot orientation in space based on the values of the angles of course, roll and pitch, the developed software eliminates the systematic error of microelectromechanical sensors and corrects the magnetometer readings taking into account the displacement of the magnetic field along its three axes. The developed software of the mobile robot spatial orientation system provides a significant increase in the positioning accuracy of the mobile robot designed for use in a room with an area of 30 to 200 m with a known layout and a priori set starting point.
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Software set of intellectual support and security of LMS MAI Class.Net
Статья научная
The article discusses integration of mathematical methods and security components in a learning management system (LMS). The discussed methods provide a statistical analysis of user data in process of online training in mathematical disciplines and adapting the content of the system for different users. The software package allows automatic calculation of questions complexity and user ratings by using statistical data. This helps the system administrator to detect users which use illegal hints or help from others. A procedure is designed for content selection for a variety of tests and control activities, with restrictions and without a time limit for the test. Two probabilistic models were used during development of mathematical methods: the Rasch model to describe the probability of users' answer correctness and Van der Linden model to describe the time it took for a user to respond to the question. The software package contains special optimization procedures that estimate the parameters of these models based on the accumulated statistics across all users. Apart from discussing efficiency of LMS usage with the above mentioned methods, the article also discusses general security architecture of the LMS, a set of technologies used for developing system security with specific implementation examples.
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Статья научная
The linear system of partial differential equations is considered. It is assumed that there is an irreversible linear operator in the main part of the system. The operator is assumed to enjoy the skeletal decomposition. The differential operators of such system are assumed to have sufficiently smooth coefficients. In the concrete situations the domains of such differential operators are linear manifolds of smooth enough functions with values in Banach space. Such functions are assumed to satisfy additional boundary conditions. The concept of a skeleton chain of linear operator is introduced. It is assumed that the operator generates a skeleton chain of the finite length. In this case, the problem of solution of a given system is reduced to a regular split system of equations. The system is resolved with respect to the highest differential expressions taking into account certain initial and boundary conditions. The proposed approach can be generalized and applied to the boundary value problems in the nonlinear case. Presented results develop the theory of degenerate differential equations summarized in the monographs MR 87a:58036, Zbl 1027.47001.
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Статья научная
A new method for solving the Cauchy problem for an ordinary differential equation is proposed and implemented using the collocation and least squares method of increased accuracy. It is based on the derivation of an approximate nonlinear equation by a multipoint approximation of the problem under consideration. An approximate solution of the problem in the form of the Pade approximation is reduced to an iterative solution of the linear least squares problem with respect to the coefficients of the desired rational function. In the case of nonlinear differential equations, their preliminary linearization is applied. A significant superiority in accuracy of the method proposed in the paper for solving the problem over the accuracy of the NDSolve procedure in the Mathematica system is shown. The solution of a specific example shows the superiority in accuracy of the proposed method over the fourth-order Runge-Kutta method. Examples of solving the Cauchy problem for linear and non-linear equations with an accuracy close to the value of rounding errors during operations on a computer with numbers in the double format are given. It is shown that the accuracy of solving the problem essentially depends on the complexity of the behavior of the values of the right-hand side of the equation on a given interval. An example of constructing a spline from pieces of Pade approximants on partial segments into which a given segment is divided is given in the case when it is necessary to improve the accuracy of the solution.
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Краткое сообщение
The existence theorem for systems of nonlinear Volterra integral equations kernels of the first kind with piecewise continuous is proved. Such equations model evolving dynamical systems. A numerical method for solving nonlinear Volterra integral equations of the first kind with piecewise continuous kernels is proposed using midpoint quadrature rule. Also numerical method for solution of systems of linear Volterra equations of the first kind is described. The examples demonstrate efficiency of proposed algorithms. The accuracy of proposed numerical methods is O(N-1).
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Solving a routing problem with the aid of an independent computations scheme
Статья научная
This paper is devoted to the issues in development and implementation of parallel algorithms for solving practical problems. We consider a routing problem with constraints and complicated cost functions. The visited objects are assumed to be clusters, or megalopolises (nonempty finite sets), and the visit to each one entails certain tasks, which we call interior jobs. The order of visits is subject to precedence constraints. The costs of movements depend on the set of pending tasks (not yet complete at the time of the movement), which is also referred to as «sequence dependence», «position dependence», and «state dependence». Such dependence arises, in particular, in routing problems concerning emergencies at nuclear power plants, similar to the Chernobyl and Fukushima Daiichi incidents. For example, one could consider a disaster recovery problem concerned with sequential dismantlement of radiation sources; in this case, the crew conducting the dismantlement is exposed to the radiation from the sources that have not yet been dealt with. Hence the dependence on pending tasks in the cost functions that measure the crew's radiation exposure. The latter dependence reflects the «shutdown» operations for the corresponding radiation sources. This paper sets forth an approach to a parallel solution for this problem, which was implemented and run on the URAN supercomputer. The results of the computational experiment are presented.
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Solving of a minimal realization problem in Maple
Статья научная
In the computer algebra system Maple, we have created a package MinimalRealization to solve the minimal realization problem for a discrete-time linear time-invariant system. The package enables to construct the minimal realization of a system starting with either a finite sequence of Markov parameters of a system, or a transfer function, or any non-minimal realization. It is designed as a user library and consists of 11 procedures: ApproxEssPoly, ApproxSpace, Approxrank, ExactEssPoly, FractionalFactorizationG, FractionalFactorizationMP, MarkovParameters, MinimalityTest, MinimalRealizationG, MinimalRealizationMP, Realization2MinimalRealization. The realization algorithm is based on solving of sequential problems: (1) determination of indices and essential polynimials (procedures ExactEssPoly, ApproxEssPoly), (2) construction of a right fractional factorization of the transfer function (FractionalFactorizationG, FractionalFactorizationMP), (3) construction of the minimal realization by the given fractional factorization (MinimalRealizationG, MinimalRealizationMP, Realization2MinimalRealization). We can solve the problem both in the case of exact calculations (in rational arithmetic) and in the presence of rounding errors, or for input data which are disturbed by noise. In the latter case the problem is ill-posed because it requires finding the rank and the space of a matrix. We use the singular value decomposition as the most accurate method for calculation of the numerical rank (Approxrank) and the numerical space (ApproxSpace). Numerical experiments with the package MinimalRealization demonstrate good agreement between the exact and approximate solutions of the problem.
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Some inverse problems for convection-diffusion equations
Статья научная
We examine the well-posedness questions for some inverse problems in the mathematical models of heat-and-mass transfer and convection-diffusion processes. The coefficients and right-hand side of the system are recovered under certain additional overdetermination conditions, which are the integrals of a solution with weights over some collection of domains. We prove an existence and uniqueness theorem, as well as stability estimates. The results are local in time. The main functional spaces used are Sobolev spaces. These results serve as the base for justifying of the convergence of numerical algorithms for inverse problems with pointwise overdetermination, which arise, in particular, in the heat-and-mass transfer problems on determining the source function or the parameters of a medium.
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Статья научная
The article is devoted to the research of the class of stochastic models in mathematical physics on the basis of an abstract Sobolev type equation in Banach spaces of sequences, which are the analogues of Sobolev spaces. As operators we take polynomials with real coefficients from the analogue of the Laplace operator, and carry over the theory of linear stochastic equations of Sobolev type on the Banach spaces of sequences. The spaces of sequences of differentiable "noises" are denoted, and the existence and the uniqueness of the classical solution of Showalter - Sidorov problem for the stochastic equation of Sobolev type with a relatively bounded operator are proved. The constructed abstract scheme can be applied to the study of a wide class of stochastic models in mathematical physics, such as, for example, the Barenblatt - Zheltov - Kochina model and the Hoff model.
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Статья научная
This paper investigates the possibility of effective implementation of calculations in low-precision neural network models on the Elbrus platform with the VLIW architecture. Such models are widely used in practice to increase the computational efficiency of recognition and well suit computers with the x86 and ARM architectures. In this paper, we consider an 8-bit neural network model, in which matrix multiplication is the most resource-intensive part of the implementation. This paper presents an effective implementation of matrix multiplication that takes into account the features of the Elbrus architecture: the presence of several computational channels with various arithmetic and logic devices, an array prefetch buffer, and its own SIMD extension. We carry out theoretical and experimental comparisons of the computational efficiency of low-precision and classical neural network models, which show that Elbrus processors have much more capabilities for performing fast floating point calculations and require the development of new approaches to increase the computational efficiency of neural network models.
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Spectral problems on compact graphs
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The method of finding the eigenvalues and eigenfunctions of abstract discrete semi-bounded operators on compact graphs is developed. Linear formulas allowing to calculate the eigenvalues of these operators are obtained. The eigenvalues can be calculates starting from any of their numbers, regardless of whether the eigenvalues with previous numbers are known. Formulas allow us to solve the problem of computing all the necessary points of the spectrum of discrete semibounded operators defined on geometric graphs. The method for finding the eigenfunctions is based on the Galerkin method. The problem of choosing the basis functions underlying the construction of the solution of spectral problems generated by discrete semibounded operators is considered. An algorithm to construct the basis functions is developed. A computational experiment to find the eigenvalues and eigenfunctions of the Sturm - Liouville operator defined on a two-ribbed compact graph with standard gluing conditions is performed. The results of the computational experiment showed the high efficiency of the developed methods.
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Stability of a stationary solution to one class of non-autonomous Sobolev type equations
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The article is devoted to the study of the stability of a stationary solution to the Cauchy problem for a non-autonomous linear Sobolev type equation in a relatively bounded case. Namely, we consider the case when the relative spectrum of the equation operator can intersect with the imaginary axis. In this case, there exist no exponential dichotomies and the second Lyapunov method is used to study stability. The stability of stationary solutions makes it possible to evaluate the qualitative behavior of systems described using such equations. In addition to introduction, conclusion and list of references, the article contains two sections. Section 1 describes the construction of solutions to non-autonomous equations of the class under consideration, and Section 2 examines the stability of a stationary solution to such equations.
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Статья научная
Least Absolute Deviations (LAD) method is a method alternative to the Ordinary Least Squares OLS method. It allows to obtain robust errors in case of violation of OLS assumptions. We present two types of LAD: Weighted LAD method and Generalized LAD method. The established interrelation of methods made it possible to reduce the problem of determining the GLAD estimates to an iterative procedure with WLAD estimates. The latter is calculated by solving the corresponding linear programming problem. The sufficient condition imposed on the loss function is found to ensure the stability of the GLAD estimators of the autoregressive models coefficients under emission conditions. It ensures the stability of GLAD-estimates of autoregressive models in terms of outliers. Special features of the GLAD method application for the construction of the regression equation and autoregressive equation without exogenous variables are considered early. This paper is devoted to extension of the previously discussed methods to the problem of estimating the parameters of autoregressive models with exogenous variables.
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