Краткие сообщения. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
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In this article, the model of forecasting of electricity consumption volume is analyzed on the basis of harmonic analysis of a time series. Previously, we establish the presence of a trend component described by a second-order polynomial. Based on the construction of the autocorrelation function, the periodicity of the time series of electricity consumption volume is proved, with periods equal to 1 year and 1 week, due to decrease in electricity consumption in summer because of increase in daylight hours and decrease in production on weekends. The model is tested on actual hourly data of United Energy System of the wholesale market for electricity and power in Russia. The model is tested for adequacy using Fisher's criterion and the coefficient of determination. The introduction of two harmonic components (annual and weekly ones) instead of the generally accepted one reduces the approximation error for the current model from 2,25 % to 2,08 %, which provides increased energy efficiency for consumers while reducing the fines of the balancing market. The proposed scientific tools are recommended in the entity's operating activity of electric power to forecast the major energy market parameters in order to reduce the fines by improving the accuracy of forecasts.
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A mathematical model of noise in the measuring channels of intelligent systems
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We propose a mathematical model describing quantization frequency selection in the measuring channels of intelligent systems depending on the parameters of noise. We consider internal and external sources of noise, seek the reasons for noise generation, justify the necessity of recording noise in the analog lines of measuring channels of intelligent systems. The parameters of our mathematical model of noise are: stationary white noise with normal distribution, an anti-aliasing bandwidth filter, the variance of noise at the input of an analog-digital converter, the correlation coefficient of noise. We evaluate the dependence of the selection of quantization frequency value on the parameters of noise. We show that if the right sampling frequency is selected in the conditions of broadband noise measuring then the systematic component of error is determined by the bandwidth of the low frequency analog anti-aliasing filter, while the random error is determined by the bandwidth of the low frequency digital averaging filter.
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A modification of Dai-Yuan’s conjugate gradient algorithm for solving unconstrained optimization
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The spectral conjugate gradient method is an essential generalization of the conjugate gradient method, and it is also one of the effective numerical methods to solve large scale unconstrained optimization problems. We propose a new spectral Dai-Yuan (SDY) conjugate gradient method to solve nonlinear unconstrained optimization problems. The proposed method's global convergence was achieved under appropriate conditions, performing numerical testing on 65 benchmark tests to determine the effectiveness of the proposed method in comparison to other methods like the AMDYN algorithm and some other existing ones like Dai-Yuan method.
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A new formula on the conjugate gradient method for removing impulse noise images
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A variety of conjugate gradient algorithms are constructed on the coefficient conjugate. In this paper, a new coefficient conjugate based on the quadratic model for impulse noise removal is proposed. Its global convergence results might be achieved under Wolfe line search circumstances. To demonstrate the performance of the conjugate gradient approach for impulse noise reduction, numerical experiments are provided.
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The article is devoted to the study of the morphology of the phase space of a mathematical model of the nerve impulse spread in a membrane, based on a degenerate Fitz Hugh-Nagumo system, defined on a bounded domain with a smooth boundary. In this mathematical model, the rate of change of one of the components of the system can significantly exceed the other, which leads to a degenerate Fitz Hugh-Nagumo system. The model under inquiry belongs to a wide class of semilinear Sobolev type models. To research the problem of nonuniqueness of solutions of the Showalter-Sidorov problem, the phase space method will be used, which was developed by G.A. Sviridyuk to scrutinize the solvability of Sobolev type equations. We have shown that the phase space of the studied model contains singularity such as the Whitney fold. The conditions of existence, uniqueness or multiplicity of solutions of the Showalter-Sidorov problem depending on the parameters of the system are found.
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About two models of modern chronology. To the problem of history chronology
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The problem of the dispute, arising at the turn of the centuries - when a new century or millennium comes - is viewed in the article from mathematical point. Controversy about starting chronology points is related to Anno Domini chronology invention. The fact that some researchers took "zero" year as a starting point and others - the first year, resulted in occurrence of the dispute about the correct choice of modern chronology. Authors deem that the first group of researchers use continuous model of chronology, and the other - discreet one. Continuous as well as discreet models are used in modern life in different field of activity.
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Adaptive estimation of a moving object trajectory using sequential hypothesis testing
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The present paper addresses the problem of adaptive estimation of a moving object trajectory and detection of changes in the motion mode. It is supposed that an object moves along a complex trajectory and at known discrete-time instants it may change its motion to one of three possible modes: a uniform straight line motion or a uniform anticlockwise/clockwise circular motion. We propose a new algorithm for adaptive trajectory estimation that combines a hybrid linear stochastic model of an object trajectory with a bank of competitive Kalman filters and a decision rule based on a sequential hypothesis testing. A detailed description of the decision rule and pseudocode of the proposed algorithm are given. The software implementation of the algorithm is made in Matlab. A numerical example of adaptive estimation of the motion of an object along a complex trajectory consisting of nine different pieces is considered. We have conducted computational experiments with different levels of noise in the measurements. The results confirm the effectiveness of the proposed algorithm.
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Algorithm for verifying the measurements
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This paper describes the Kramers-Kronig relation for verifying the obtained values of S-parameters for different operation conditions of a transmission line. We obtain and prove lemmas for S-parameters for operation conditions of the line under short-circuit, open-circuit, and matched load. We give a comparison of theoretical and experimental values, which confirm the correctness of the obtained relations and conclusions.
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The Avalos-Triggiani problem for a system of wave equations and a linear Oskolkov system of non-zero order is investigated. The mathematical model contains a linear Oskolkov system describing the flow of an incompressible viscoelastic Kelvin-Voigt fluid of non-zero order, and a wave vector equation corresponding to some structure immersed in the fluid. Based on the method proposed by the authors of this problem, the existence of a unique solution to the Avalos-Triggiani problem for the indicated systems is proved.
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The paper considers a recurrent regularizing algorithm for joint estimation of distortions of a M-ary quadrature amplitude modulation (M-QAM) signal obtained in a direct conversion receiver path. The algorithm is synthesized using a modified least squares method in the form of Tikhonov's functional under conditions of a priori uncertainty about the laws of noise distribution. The resulting procedure can work both on the test sequence and on information symbols after the detection procedure. We analyze the influence of the Lagrange multiplier on the accuracy of the estimation procedure and on the complexity of the algorithm. It is shown that, with the same accuracy, the regularizing algorithm requires significantly fewer iterations than the procedure without the Lagrange multiplier, and therefore has a lower computational complexity.
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Anisotropic diffusion in anisotropic Stepanov spaces
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We consider a problem on the image processing and computer vision. A wide range of methods allows to solve problems of this type. The methods of partial differential equations are the most useful and interesting ones. A non-linear diffusion takes special place in these studies. In this context, fundamental theoretical foundation is a central part of this approach. Therefore, we introduce a new functional class of spaces, formulate and prove the lemma on the equivalent norms in anisotropic Stepanov spaces. Another important result of this study is the lemma that the anisotropic Stepanov spaces are Banach. In addition, we obtain the theorem on the solvability of the equation of anisotropic diffusion in anisotropic Stepanov spaces. The results can be applied to the image processing and computer vision. Also, the obtained results open the new view to this problem.
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Applicability of forecasted bankruptcy models to Russian industrial companies
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The application of effective methods for forecasting of the bankruptcy of industrial companies is always an urgent task for businesses, especially at the present stage which is characterized by an extremely high uncertainty. The paper presents the main techniques of bankruptcy modelling used in the world's practice: logit, probit and MDA-models, as well as the special private methods developed on their basis. These tools constitute the methodological foundation of our research. To assess the practical applicability of these methods to the contemporary Russian market, two sectorial companies (bankrupt and non-bankrupt) are selected as the object of study. A feature of the research is the use of financial statements of companies developed according to Russian and international standards. In the course of the calculations, we apply external and internal restrictions related to the key rate, credit history characteristics, age and regional affiliation of companies. Based on the dynamic assessment, we draw conclusions about the practical applicability and inapplicability of certain forecast models for the Russian economy. We investigate the relationship between the assessment results and the type of source data used. Research veracity is confirmed using generally recognized models and methods, as well as the practical implementation of the results obtained. We can recommend to use these results for improving the existing models for predicting bankruptcy and developing new ones, as well as for owners and investors of companies who need to make strategic decisions.
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Approximation of solutions to the boundary value problems for the generalized Boussinesq equation
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The paper is devoted to one of the Sobolev type mathematical models of fluid filtration in a porous layer. Results that allow to obtain numerical solutions are significant for applied problems. We propose the following algorithm to solve the initial-boundary value problems describing the motion of a free surface filtered in a fluid layer having finite depth. First, the boundary value problems are reduced to the Cauchy problems for integro-differential equations, and then the problems are numerically integrated. However, numerous computational experiments show that the algorithm can be simplified by replacing the integro-differential equations with the corresponding approximating Riccati differential equations, whose solutions can also be found explicitly. In this case, the numerical values of the solution to the integro-differential equation are concluded between successive values of approximating solutions. Therefore, we can pointwise estimate the approximation errors. Examples of results of numerical integration and corresponding approximations are given.
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The paper is focused on the modelling of a one-dimensional chaotic process which dynamics is described by a one-parameter nonlinear map. The problem is to estimate the initial condition and model parameter from measurements corrupted by additive errors. The considered guaranteed (set-membership) approach assumes that the prior information about the unknown variables (initial condition, model parameter and measurement errors) is presented as interval estimates. In this context, the estimation problem can be stated as a problem of solving a system of nonlinear inequalities. Due to the nonlinearity, it is not possible to obtain an exact characterization of the solution set. The developed algorithm computes an outer approximation as a union of non-overlapping boxes.
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Assessment of economic sustainability of an enterprise
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In a turbulent economy and increasing economic crises, the question of assessing the economic sustainability of a corporation is quite relevant and in demand in scientific research. At the same time, with a certain degree of conditionality, all works in this direction can be divided into two clusters: works based on traditional indicators of financial stability and ones using original scientific approaches to determining the economic condition of an enterprise. This article belongs to the works of the second cluster. The basis for assessing the economic sustainability of an enterprise is the economic and mathematical modelling of the operating activities of the enterprise based on the well-known Cobb - Douglas production function. A methodology is developed for constructing a mathematical model of the enterprise's activity, for determining elasticity indicators of output according to the resources used in the production process and elasticity of production, on the basis of the dynamics of which an approach to assessing economic stability is proposed. The methodology was tested on the data of PJSC Chelyabinsk Tube Rolling Plant during the implementation of the innovation project "Height 239".
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Balance on Nash in "soft" control problems for systems of the hierarchal type
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This paper discusses the hierarchical system, which includes the Center, symbolizing the global interests of the system and control objects (CO), perform guidance Center, but at the same time pursuing its own local interests. In the first phase, the Centre produces a decision, taking into account global interests, and in the second stage CO chooses a specific version of the decision, taking into account their own administration and their local interests. It is anticipated that decision is consistent if the interests of each of the parties are counted so that a departure from it cannot improve the performance of any of the parties, i.e. it is balance on Nash. «Soft» control suggests that for each pair of «Center-facility control» global and local interests are taken into account in varying degrees. The method of finding equilibrium solutions depending on the extent of the accounting of interests and levels of reflection is found.
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Calculation of aircraft area on satellite images by genetic algorithm
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Classical genetic algorithm is used in a computational experiment, which is described in this article, to measure an area of the aircraft Boeing 737-300. Satellite images containing objects of interest are selected as initial data. The computational experiment consists of two steps. The first step presents the generalized formulas of values. These formulas are needed to select initial images, using the theory of modified descriptive algebra of images. After that, the initial data is formed according to the calculated values, i.e. the satellite image is chosen in the needed scale and the shooting angle. At the second step, a model of technical vision system (computer vision system) in the modified descriptive images algebra and a fitness function for the genetic algorithm are developed. Then varying parameters of the model are chosen and their optimization is carried out in MATLAB. The article demonstrates development of the model due to its complexity by additional imaging techniques. Experimentally it was found that the evolution of the model improves optimization results.
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Choosing average values when determining characteristics of the unsteady boiling of liquid
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This paper presents an analysis of the issues associated with constructing mathematical models for processes of intense phase transformations and, in particular, focuses on the aspect of using closing relations of empirical origin. The main trend in the implementation of modern numerical algorithms for practical problems is aimed at improving the accuracy of calculation results. The latter is usually achieved by refining a certain set of coefficients in mathematical models. These refinements are carried out both on the basis of the modernization of existing approaches, and with the involvement of new empirical information obtained for a limited number of regime conditions. Predictive models for describing the dynamics of phase transformations, as one of the most difficult in the mathematical formulations, refer to a particularly striking manifestation of the problem under study. In this research, we discuss the existing and widely used experimental work devoted to the extraction of primary information about the dynamics of vapor bubbles on the surface of metal heaters. Their example reveals the presence of a simplified approach in the existing development methodology, and shows a way to determine the correct generalization of empirical information that has a pseudo-stochastic nature.
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Complex approach to assessment of investment attractiveness of power generating company
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Present approaches based on the qualitative analysis methods are not effective enough for a comprehensive evaluation of the investment attractiveness of the power generating company (PGC). It resolves the urgency of the complex deterministic method of accounting for aggregated risk. The article presents the diagnostics of power generating company risks' and the assessment of the actual aggregated risk as the integral indicator of investment attractiveness of the PGC. The proposed authors' approach to ranking the risk taking into account the level of hazard is based on the calculation of individual limits of risk states variation and risk relative value. The individual risk assessment is based on the Bayes method complemented by a two-step normalization to account for the specificity of PGC. The Merton - Vasicek method and basic principles of the economic capital theory are used in developing the method of the final evaluation of the PGC investment attractiveness. Research veracity is confirmed by the practical implementation. The research results are recommended for use in assessing the current level of the PGC investment attractiveness and development strategy of its increase.
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Computational experiment for a class of mathematical models of magnetohydrodynamics
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The first initial-boundary value problem for the system modelling the motion of the incompressible viscoelastic Kelvin - Voigt fluid in the magnetic field of the Earth is investigated considering that the fluid is under external influence. The problem is studied under the assumption that the fluid is under different external influences depending not only on the coordinates of the point in space but on time too. In the framework of the theory of semi-linear Sobolev type equations the theorem of existence and uniqueness of the solution of the stated problem is proved. The solution itself is a quasi-stationary semi-trajectory. The description of the problem's extended phase space is obtained. The results of the computainal experiment are presented.
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