Краткие сообщения. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Models and methods of computer-aided design of the user interface of software systems
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The article deals with the design, mathematical modeling and development of the prototype of the interface of the information system of the audit fund. In this paper, the study of the design of the user interface, methods for creating interfaces, user interaction types, a list of the functionality of the system, a mathematical model, nondeterministic finite automaton (NFA) was built user interaction with system elements and a prototype of the system have been created. According to the authors, this system will automate and simplify the process of managing the audit fund, joint use of the NFA and one of the methods of creating interfaces, namely, ontology will allow describing domain concepts, edit domain concepts, generate databases and knowledge bases, broadcast from one language to another, get rid of information redundancy, reduce time for database development as well as simplify the process of scheduling the university.
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Multipoint initial-final value problem for the model of Devis with additive white noise
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The evolution of the free surface of the filtering fluid in a reservoir of limited power is modeled by the Davis equation with homogeneous Dirichlet conditions. Depending on the nature of the free term describing the internal source of the liquid, the model will be deterministic or stochastic. The deterministic model has been studied in various aspects by many researchers with different initial (initial-final value conditions). The stochastic model is studied for the first time. The main result is the proof of the unique solvability of the evolutionary model with an additive white noise and a multipoint initial-final value condition.
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Neural net decoders for linear block codes
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The work is devoted to neural network decoders of linear block codes. Analytical methods for calculating synaptic weights based on a generator and parity-check matrices are considered. It is shown that to build a neural net decoder based on a parity-check matrix was sufficiently four layers feedforward neural net. The activation functions and weight matrices for each layer are determined, as well as the number of weights for the neural net decoder. An example of error correction with uses of the BCH neural net decoder is considered. As a special case of a neural network decoder built on the basis of a parity-check matrix, a model for decoding Hamming codes has been proposed. This is the two-layer feedforward neural net for with a neuron number equal to the length of the codeword and a number of weight coefficients equal to the square of the codeword length. The graphs of the number of a synaptic weight of neural net decoders based on the generator and parity-check matrices, on the number of bits and the number of corrected errors, are shown.
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Non-Uniqueness of Solutions to Boundary Value Problems with Wentzell Condition
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Recently, in the mathematical literature, theWentzell boundary condition is considered from two points of view. In the first case, let us call it classical one, this condition is an equation containing a linear combination of the values of the function and its derivatives on the boundary of the domain. Moreover, the function itself also satisfies the equation with an elliptic operator defined in the domain. In the second case, which we call neoclassical one, the Wentzell condition is an equation with the Laplace–Beltrami operator defined on the boundary of the domain understood as a smooth compact Riemannian manifold without boundary, and the external action is represented by the normal derivative of a function defined in the domain. The paper shows the non-uniqueness of solutions to boundary value problems with the Wentzell condition in the neoclassical sense both for the equation with the Laplacian and for the equation with the Bi-Laplacian given in the domain.
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Numerical investigation of the Boussinesq - Love mathematical models on geometrical graphs
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The article is devoted to the numerical investigation of the Boussinesq - Love mathematical models on geometrical graphs representing constructions made of thin elastic rods. The first paragraph describes the developed algorithm for numerical solution of the Boussinesq - Love equation with initial conditions and boundary conditions in the vertices. The block diagram of the algorithm is given and described. The result of computation experiment is given in the second paragraph.
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The process of unsteady flow of incompressible viscoelastic fluid in a cylindrical tube of constant cross-section is considered. To describe the rheological properties of a viscoelastic fluid, the Kelvin-Voigt model is used and the mathematical model of this process is presented as an integro-differential partial differential equation. Within the framework of this model, the problem is to determine the pressure drop along the length of the pipe, which ensures the passage of a given flow rate of viscoelastic fluid through the pipe. This problem belongs to the class of inverse problems related to the recovery of the right parts of integro-differential equations. By replacing variables, the integro-differential equation is transformed into a third-order partial differential equation. First, a discrete analog of the problem is constructed using finite-difference approximations. To solve the resulting difference problem, we propose a special representation that allows splitting the problems into two mutually independent second-order difference problems. As a result, an explicit formula is obtained for determining the approximate value of the pressure drop along the length of the pipeline for each discrete value of the time variable. Based on the proposed computational algorithm, numerical experiments were performed for model problems.
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Numerical research of the Barenblatt - Zheltov - Kochina stochastic model
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At present, investigations of Sobolev-type models are actively developing. In the solution of applied problems the results allowing to get their numerical solutions are very significant. In the article the algorithm for numerical solving of the initial boundary value problem is developed. The problem describes the pressure distribution of the homogeneous fluid in the horizontal layer in the circle. The layer is opened by a vertical well of a small radius. In our research we suppose that random disturbing loads have an influence on the fluid. The problem was solved under two assumptions. Firstly, we suppose that an unstable fluid flow is axially symmetric, and secondly, that in initial moment the pressure in the layer is constant. After the process of the discretization we modify the original model to the Cauchy problem for the system of ordinary differential equations. For the numerical solution we use algorithms based on explicit one-step formulas of the Runge - Kutta type with the seventh-order accuracy and with the selection of the integration step. We also use the scheme of the eighth-order accuracy to evaluate the calculation accuracy on each steps of time. According to the results of this control, we choose the time-step. A lot of numerical experiments have shown high numerical efficiency of the algorithm that we use to solve the investigated initial-boundary problem.
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Numerical research of the mathematical model for traffic flow
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The problems of distribution of transport flows are currently relevant in connection with the increase in vehicles. In the 50s of the last century, the first macroscopic (hydrodynamic) models appeared, where the transport flow resembles the flow "motivated" compressible liquid. The scientific approach based on the Navier - Stokes system. The main idea of the scholars is consideration the hydrodynamic models on the grounds of interrelation between the transport flow and incompressible fluid. For modelling traffic flows we examine Oskolkov equation on the geometric graph, where the edge has two positive values corresponding to it "length" and "width". Certainly, in the context of mathematical model the values lk and bk are dimensionless, but for clarity it is convenient to imagine that lk is measured in linear metric units, for example, kilometers or miles, and bk is equal to the number of traffic lanes on the roadway in one direction. In terms of the Oskolkov model, we obtained a non-classical multipoint initial-final value condition. We will study such a model using the idea and methods of the Sobolean equation theory. These notes describe a numerical experiment based on the Galerkin method for the Oskolkov equation with a multipoint initial-final condition on the graph.
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On a heat and mass transfer model for the locally inhomogeneous initial data
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We consider a model case of the problem of heat diffusion in a homogeneous body with a special initial state. The peculiarity of this initial state is its local inhomogeneity. That is, there is a closed domain Ω inside a body, the initial state is constant out of the domain. Mathematical modelling leads to the problem for a homogeneous multi-dimensional diffusion equation. We construct the boundary conditions on the boundary of the domain Ω, which can be characterized as "transparent" boundary conditions. We separately consider a special case - a model of redistribution of heat in a uniform linear rod, the side surface of which is insulated in the absence of (internal and external) sources of heat and of locally inhomogeneous initial state.
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On a model of oscillations of a thin flat plate with a variety of mounts on opposite sides
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We consider a model case of stationary vibrations of a thin flat plate, one side of which is embedded, the opposite side is free, and the sides are freely leaned. In mathematical modeling there is a local boundary value problem for the biharmonic equation in a rectangular domain. Boundary conditions are given on all boundary of the domain. We show that the considered problem is self-adjoint. Herewith the problem is ill-posed. We show that the stability of solution to the problem is disturbed. Necessary and sufficient conditions of existence of the problem solution are found. Spaces of the ill-posedness of the considered problem are constructed.
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On existence of solutions to stochastic differential equations with current velocities
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The notion of mean derivatives was introduced by E. Nelson in 60-th years of XX century and at the moment there are a lot of mathematical models of physical processes constructed in terms of those derivatives. The paper is devoted to investigation of stochastic differential equations with current velocities, i.e. with Nelson's symmetric mean derivatives. Since the current velocities of stochastic processes are natural analogues of ordinary physical velocities of deterministic processes, such a research is important for investigation of models of physical processes that take into account stochastic properties. An existence of solution theorem for those equations is obtained.
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On one Sobolev type mathematical model in quasi-Banach spaces
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The theory of Sobolev type equations experiences an epoch of blossoming. In this article the theory of higher order Sobolev type equations with relatively spectrally bounded operator pencils, previously developed in Banach spaces, is transferred to quasi-Banach spaces. We use already well proved for solving Sobolev type equations phase space method, consisting in reduction of singular equation to regular one defined on some subspace of initial space. The propagators and the phase space of complete higher order Sobolev type equations are constructed. Abstract results are illustrated by specific examples. The Boussinesq-Love equation in quasi-Banach space is considered as application.
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Interest in Sobolev type equations has recently increased significantly, moreover, there arose a necessity for their consideration in quasi-Banach spaces. The need is dictated not so much by the desire to fill up the theory but by the aspiration to comprehend non-classical models of mathematical physics in quasi-Banach spaces. Notice that the Sobolev type equations are called evolutionary if solutions exist only on R +. The theory of holomorphic degenerate semigroups of operators constructed earlier in Banach spaces and Frechet spaces is transferred to quasi-Sobolev spaces of sequences. This article contains results about existence of the exponential dichotomies of solutions to evolution Sobolev type equation in quasi-Sobolev spaces. To obtain this result we proved the relatively spectral theorem and the existence of invariant spaces of solutions. The article besides the introduction and references contains two paragraphs. In the first one, quasi-Banach spaces, quasi-Sobolev spaces and polynomials of Laplace quasi-operator are defined. Moreover the conditions for existence of degenerate holomorphic operator semigroups in quasi-Banach spaces of sequences are obtained. In other words, we prove the first part of the generalization of the Solomyak - Iosida theorem to quasi-Banach spaces of sequences. In the second paragraph the phase space of the homogeneous equation is constructed. Here we show the existence of invariant spaces of equation and get the conditions for exponential dichotomies of solutions.
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On the existence of an integer solution of the relaxed Weber problem for a tree network
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The problem of finding the optimal arrangement of vertices of a tree network in the installation space representing a finite set is considered. The criterion of optimality is the minimization of the total cost of deployment and the cost of communications. Placement of different tree vertices in one point of the installation space is allowed. This problem is known as Weber problem for a tree network. The statement of Weber problem as an integer linear programming problem is given in this research. It's proved that a set of optimal solutions of corresponding relaxed Weber problem for a tree-network contains the integer solution. This fact allows to prove the existence a saddle point while proving the performance of decomposition algorithms for problems different from problems because of additional restrictions.
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On the uniqueness of a nonlocal solution in the Barenblatt - Gilman model
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This article deals with the question of uniqueness of a generalized solution to the Dirichlet-Cauchy problem for the Barenblatt-Gilman equation, which describes nonequilibrium countercurrent capillary impregnation. The unknown function corresponds to effective saturation. The main equation of this model is nonlinear and implicit with respect to the time derivative, which makes it quite hard to study. In a suitable functional space, the Dirichlet-Cauchy problem for the Barenblatt-Gilman equation reduces to the Cauchy problem for a quasilinear Sobolev-type equation. Sobolev-type equations constitute a large area of nonclassical equations of mathematical physics. The techniques used in this article originated in the theory of semilinear Sobolev-type equations. For the Cauchy problem we obtain a sufficient condition for the existence of a unique generalized solution. We establish the existence of a unique nonlocal generalized solution to the Dirichlet-Cauchy problem for the Barenblatt-Gilman equation.
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We study the problem of optimal control of solutions to an operator-differential equation, which is not solved with respect to the time derivative, together with a multipoint initial-final condition. In this case, one of the operators in the equation is multiplied by a scalar function of time. By the properties of the operators involved, the stationary equation has analytical resolving group. We construct a solution to the multipoint initial-final problem for the nonstationary equation. We show that a unique optimal control of solutions to this problem exists. Apart from the introduction and bibliography, the article consists of three sections. The first section provides the essentials of the theory of relatively p-bounded operators. In the second section we construct a strong solution to the multipoint initial-final problem for nonstationary Sobolev-type equations. The third section contains our proof that there exists a unique optimal control of solutions to the multipoint initial-final problem.
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Parallel delivery operations modelling
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Some delivery organization scheme is considered. The key point is the principle of routes parallelization using several carriers at the same time and these auxiliary carriers can be based on the main carrier. An example of such a delivery system is a van carrying several autonomous carriers, which in turn can carry out simultaneous so-called parallel deliveries. Delivery routes are determined based on the coordinates of customers, the determination of acceptable starting points for auxiliary carriers, the technical and energy limitations of the main and auxiliary carriers, and the minimization of the amount of time spent on delivery operations. The developed algorithm for solving the problem on routing of delivery using primary and secondary carriers allows to reduce delivery time and resources. The algorithm is implemented in Python using the libraries for processing and visualization of trajectories and other space-time data, packages for extracting, modelling, analyzing and visualizing street networks on the example of the Yekaterinburg city.
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Parameters identification algorithm for the SUSUPLUME air pollution propagation model
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The article presents the method of identifying the parameters of a dynamic dispersion calculation model SUSUPLUME. It is supposed that the model parameters contain not only the characteristics of the atmosphere and the pollutant, but also information about the influence of other particular conditions such as terrain, building, background, etc. The model parameters are configured based on instrumental measurements of concentrations of pollutants in the atmospheric air in the surface layer (2 meters above ground level). Three identification strategies are considered: identification of parameters by all measurements, identification of parameters by measurements of a given source and identification of parameters using another approved model. A method for weighing these strategies is proposed in the issue. The article also provides objective functions for optimization criteria, an acceptable set of parameters, an algorithm for solving an optimization problem, a decision tree of a feasible set and a global optimization algorithm.
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Research of default risk level of Russian energy
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The liberalization of the electricity market intensifies the competition for investors in the Russian energy market, for which the financial stability of energy companies is an important criterion. The paper presents the mathematical modelling of integrated assessment of the default risk level of Russian energy companies, taking into account their specific characteristics: type of the energy business, form of ownership, and regional specific. The research is based on the industry approach to the diagnostics of default risk level of energy companies. The approach includes the logit-model and assessment of the coefficients significance. The complexity of this model is conditioned by the study of external and internal financial and economic indicators, as well as qualitative criteria based on the introduction of dummy-variables. Four groups of default risk level of Russian energy companies are proposed. The use of mathematical modelling tools increases the accuracy of assessing the financial insolvency of energy companies in comparison with the traditional methods. Therefore, the proposed approach is novel, relevant, and practically significant. Research veracity is confirmed by the practical implementation. We recommend the use of the proposed methodology in assessment of the current state and development strategy of Russian energy companies, as well as by investors and analysts to make financial decisions.
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We propose development of examination methodology based on a sequential application of the MAI method (i.e., the hierarchy analysis method) and associative training of neural networks. The proposed method is an alternative to the usual methods to solve a direct examination problem. We present a methodological approach to the examination problem. The approach allows to save information about all objects and consider their indicators in total. Therefore, there is the soft maximum principle (softmax), based on the model of expert evaluations mixing. This approach allows different interpretations of the examination results, which save quality unchanged overall picture of the examination object indicators ratio, and to get more reliable examination results, especially in cases where the objects characteristics are very different.
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