Статьи журнала - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
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Income modelling of enterprises on the basic of vector prediction
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This paper proposes a new approach to income modelling of enterprises, based on the vector prediction methods. Existing approaches to the income modelling are based on the use of traditional methods of economic dynamics prediction such as the average absolute increase and average growth rate. Quantitative inaccuracy and highly approximate nature of predictions are inherent for traditional methods. The authors propose the economic-mathematical model for the enterprise income planning for a year ahead on a quarterly basis. For revenue prediction two methods of vector prediction are used (the method of orthogonal differences and multiplicative Holt - Winters' method). This model provides prediction for a few steps forward at the same time. Individually, each of these methods doesn't take into account the diversity of the process. Only in conjunction the methods allow to take into account the demolition of trends and seasonal nature of income, thus providing the necessary stability of the prediction. To summarize two predictions their linear combination is calculated, the choice of weighting coefficients being based on the accuracy of private predictions. The accuracy of the private prediction is defined as the average relative error of the forecast.
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Preparation of the target-protein, particularly the protein protonation method can affect considerably the spatial arrangement of the attached hydrogen atoms and the charge state of individual molecular groups in amino acid residues. This means that the calculated protein-ligand binding energies can vary significantly depending on the method of the protein preparation, and it also can lead to the different docked positions of the ligand in the case of docking (positioning of the ligand in the protein active site). This work investigates the effect of the hydrogen atoms arrangement method in the target-protein on the protein-ligand binding energy. All hydrogen atoms of target-protein are fixed or movable. The comparison of the protein-ligand binding energies obtained for the test set of target-proteins prepared using six different programs is performed and it is shown that the protein-ligand binding energy depends significantly on the method of hydrogen atoms incorporation, and differences can reach 100 kcal/mol. It is also shown that taking into account solvent in the frame of one of the two continuum implicit models smooths out these differences, but they are still about 10-20 kcal/mol. Moreover, we carried out the docking of the crystallized (native) ligands from the protein-ligand complexes using the SOL program and showed that the different methods of the hydrogen atoms addition to the protein can give significantly different results both for the positioning of the native ligand and for its protein-ligand binding energy.
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Influence to new formulas gradient for removing impulse noise images
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In conjugate gradient techniques, the conjugate formula is often the primary point of concentration. The conjugate gradient technique is used to solve problems that arise during the process of picture restoration. By using the quadratic model, a brand-new coefficient conjugate will be produced for the operation. The algorithms demonstrate both local and global convergence and descent. The numerical testing revealed that the newly developed method is much superior to the one that came before it. The recently created conjugate gradient strategy has better performance than the FR conjugate gradient technique, which is the industry standard.
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Intellectual mathematical support software and inner architecture of LMS MAI Class.Net
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Distance education prove to be effective in improving the learning and teaching environment. One of the main advantages of distance learning is that web-based courses can be taken anytime and anywhere. The implementation of an e-learning management system (LMS) requires not only good and fast hardware, but also the use of modern software technologies and architectural solutions. This article outlines the main ways of forming the LMS architecture based on a microservice approach, which allows the achievement of high performance and fault tolerance. A distinctive feature of the CLASS.NET system is the presence of a special mathematical software package that allows the optimization of educational processes and tasks (such as students tests generation, students progress analysis, knowledge level assessment, task difficulty analysis, personal learning curve planning). The process of interaction between the LMS system and mathematical software package, as well as the main ways of forming such software as completely independent applications for their further integration into other learning management systems, are thoroughly described. The efficiency of the microservice architecture in terms of scaling, performance and general behavior in case of critical errors in comparison with other systems based on classical architectural approaches is shown. The algorithm of predicting the time a student spends to answer the tasks, which is included in the mathematical software package, is considered.
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The paper is devoted to dosimetric modelling of the human red bone marrow (RBM) internal exposure due to beta-emitting 90Sr incorporated in spongiosa bone. The dose factor calculation (absorbed dose rate due to unit specific activity of 90Sr) is based on the modelling of radiation transport in segments of the skeleton bones with active hematopoiesis. Segmentation considerably simplifies the modelling, but can lead to an underestimation due to electron emission from the neighboring parts of the bone adjacent to the studied segment. The objective of the study is to determine this cross-fire effect on the absorbed dose in RBM. For this purpose, we analyze the results of the numerical experiment on modelling of dose absorption within the bone segments of various shape and size that were parts of the computational phantoms of skeletons of people of different sex and age. We analyze dose factor dependencies on the area of the spongiosa bone surface and the ratio of weights of bone and RBM. It is found that if the area of the spongiosa surface (SS) > 6 cm2, then the effect of neighboring bone parts exposure is negligible. For a smaller SS the extension of the linear dimensions of the spongiosa bone by 2 mean electron path lengths results in dose factor increase proportional to the ratio of the extended spongiosa bone surface area to the original one to the power of 0,28. For human computational phantoms, these values are in the range 1,03-1,21 and are used as adjustment coefficients for the dose factors. Relative standard uncertainty of the adjustment coefficient is 5%.
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This work considers hypersonic aircraft open-loop control problem in a presence of terminal and phase constraints. By the discretization process this problem is transformed into a nonlinear programming problem which is solved numerically by the interval explosion search algorithm. This algorithm belongs to metaheuristic algorithms of interval global optimization. Desired control is constructed in a class of interval piecewise-constant and piecewise-linear functions. Also this work demonstrates the comparison of results obtained by the proposed method and by Galerkin projection technique. This comparison confirms the efficiency of the interval based control algorithm.
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Introducing a power of the operator in direct spectral problems
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The resolvent method, proposed by Sadovnichiy and Dubrovsky in the 1990s, is successfully applied in the direct spectral problem to calculate the asymptotics of eigenvalues of the perturbed operator, find formulas for the regularized trace, and recover perturbation. But the application of this method faces difficulties when the resolvent of the unperturbed operator is non-nuclear. Therefore, a number of physical problems could only be considered on the interval. This article describes a justification of the transition to the power of an operator in order to expand the area of possible applications of the resolvent method. Considering the problem of calculating the regularized trace of the Laplace operator on a parallelepiped of arbitrary dimension, we show that for every fixed dimension it is possible to choose the required power of the operator and to calculate the regularized traces. These studies are relevant due to the need to study important applied problems, particularly in hydrodynamics, electronics, elasticity theory, quantum mechanics, and other fields.
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Invariant Manifolds of the Hoff Model in "Noise"
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The work is devoted to the study the stochastic analogue of the Hoff equation, which is a model of the deviation of an I-beam from the equilibrium position. The stability of the model is shown for some values of the parameters of this model. In the study, the model is considered as a stochastic semilinear Sobolev type equation. The obtained results are transferred to the Hoff equation, considered in specially constructed “noise” spaces. It is proved that, in the vicinity of the zero point, there exist finite-dimensional unstable and infinite-dimensional stable invariant manifolds of the Hoff equation with positive values of parameters characterizing the properties of the beam material and the load on the beam
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Invariant description of control in a Gaussian one-armed bandit problem
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We consider the one-armed bandit problem in application to batch data processing if there are two alternative processing methods with different efficiencies and the efficiency of the second method is a priori unknown. During the processing, it is necessary to determine the most effective method and ensure its preferential use. Processing is performed in batches, so the distributions of incomes are Gaussian. We consider the case of a priori unknown mathematical expectation and the variance of income corresponding to the second action. This case describes a situation when the batches themselves and their number have moderate or small volumes. We obtain recursive equations for computing the Bayesian risk and regret, which we then present in an invariant form with a control horizon equal to one. This makes it possible to obtain the estimates of Bayesian and minimax risk that are valid for all control horizons multiples to the number of processed batches.
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Invariant manifolds of semilinear Sobolev type equations
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The article is devoted to a review of the author's results in studying the stability of semilinear Sobolev type equations with a relatively bounded operator. We consider the initial-boundary value problems for the Hoff equation, for the Oskolkov equation of nonlinear fluid filtration, for the Oskolkov equation of plane-parallel fluid flow, for the Benjamin-Bon-Mahoney equation. Under an appropriate choice of function spaces, these problems can be considered as special cases of the Cauchy problem for a semilinear Sobolev type equation. When studying stability, we use phase space methods based on the theory of degenerate (semi)groups of operators and apply a generalization of the classical Hadamard-Perron theorem. We show the existence of stable and unstable invariant manifolds modeled by stable and unstable invariant spaces of the linear part of the Sobolev type equations in the case when the phase space is simple and the relative spectrum and the imaginary axis do not have common points.
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Inverse problem for Sobolev type mathematical models
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The work is devoted to the study of an inverse problem for the linear Sobolev type equation of higher order with an unknown coefficient depending on time. Since the equation might be degenerate the phase space method is used. It consists in construction of projectors splitting initial spaces into a direct sum of subspaces. Actions of operators also split. Therefore, the initial problem is reduced to two problems: regular and singular. The regular one is reduced to the first order nondegenerate problem which is solved via approximations. The needed smoothness of the solution is obtained. Then it is substituted into the singular problem which is solved using the methods of relatively polynomially bounded operator pencils theory. The main result of the work contains sufficient conditions for the existence and uniqueness of the solution to the inverse problem for a complete Sobolev type model of the second order. This technique can be used to investigate inverse problems of the considered type for Boussinesq-Love mathematical model.
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Inverse problem for a linearized quasi-stationary phase field model with degeneracy
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The inverse problem for a linearized quasi-stationary phase field model is considered. The inverse problem is reduced to a linear inverse problem for the first order differential equation in a Banach space with a degenerate operator at the derivative and an overdetermination condition on the degeneracy subspace. The unknown parameter in the problem dependens on the source time function. The theorem of existence and uniqueness of classical solutions is proved by methods of degenerate operator semigroup theory at some additional conditions on the operator. General results are applied to the original inverse problem.
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We consider inverse problems of evolution type for mathematical models of quasistationary electromagnetic waves. It is assumed in the model that the wave length is small as compared with space inhomogeneities. In this case the electric and magnetic potential satisfy elliptic equations of second order in the space variables comprising integral summands of convolution type in time. After differentiation with respect to time the equation is reduced to a composite type equation with an integral summand. The boundary conditions are supplemented with the overdetermination conditions which are a collection of functionals of a solution (integrals of a solution with weight, the values of a solution at separate points, etc.). The unknowns are a solution to the equation and unknown coefficients in the integral operator. Global (in time) existence and uniqueness theorems of this problem and stability estimates are established.
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Inverse problems for some Sobolev-type mathematical models
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The present article is devoted to the study of mathematical models based the Sobolev-type equations and systems arising in dynamics of a stratified fluid, elasticity theory, hydrodynamics, electrodynamics, etc. Along with a solution we determine an unknown right-hand side and coefficients in a Sobolev-type equations of the forth order. The overdetermination conditions are the values of a solution in a collection of points of a spatial domain. The problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. The existence and uniqueness theorems of solutions for the linear and nonlinear cases are proven. In the linear case the result is global in time and it is local in the nonlinear case. The main spaces in question are the Sobolev spaces.
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Inverse spectral problems and mathematical models of continuum mechanics
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The article contains results in the field of spectral problems for mathematical models with discrete semi-bounded operator. The theory is based on linear formulas for calculating the eigenvalues of a discrete operator. The main idea is to reduce spectral problem to the Fredholm integral equation of the first kind. A computationally efficient numerical method for solving inverse spectral problems is developed. The method is based on the Galerkin method for discrete semi-bounded operators. This method allows to reconstruct the coefficient functions of boundary value problems with a high accuracy. The results obtained in the article are applicable to the study of problems for differential operators of any order. The results of a numerical solution of the inverse spectral problem for a fourth-order perturbed differential operator are presented. We study some mathematical models of continuum mechanics based on spectral problems for a discrete semi-bounded operator.
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In this paper, we study the problem of boundary control and final observation for one degenerate mathematical model of motion speed potentials distribution of filtered liquid free surface with the Showalter-Sidorov initial condition. The mathematical model is based on the degenerate Boussinesq equation with an inhomogeneous Dirichlet condition. This model belongs to the class of semilinear Sobolev-type models in which the nonlinear operator is p-coercive and s-monotone. In the paper, the problem of boundary control and final observation for a semilinear Sobolev-type model is considered and conditions for the existence of a control-state pair of the problem are found. In applied studies of a research problem, it is allowed to find such a potentials distributionof filtered liquid free surface, at which the system transitions from the initial condition to a given final state within a certain period of time T.
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Investigation of the transient responses of a beam on an elastic polymeric foundation
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The negative impact of vibrations on various devices and mechanisms can be significant, so it is important to take this factor into account when designing, operating and maintaining various equipment and engineering systems. Various methods and technologies can be used to protect against the negative effects of vibrations. Special damping materials are often used. This research paper is devoted to the analysis of the effectiveness of vibration reduction taking into account the physical parameters of elastic polymeric materials. To conduct the study, a mathematical model describing motion of the beam resting on an elastic polymeric foundation is constructed. The model is based on a system of nonlinear differential equations. An algorithm was developed and applied for the numerical solution of this system of equations. Numerical experiments were carried out for the study of the system reaction to different cases of accelerations. As a result, the deflection structure for materials with different physical characteristics were obtained. These results can serve as a starting point for a deeper study of materials and creation of more complex structures.
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The study of the phase space morphology of the mathematical model deformation of an I-beam, which lies on smooth Banach manifolds with singularities (k-Whitney assembly) depending on the parameters of the problem, is devoted to the paper. The mathematical model is studied in the case when the operator at time derivative is degenerate. The study of the question of non-uniqueness of the solution of the Showalter-Sidorov problem for the Hoff model in the two-dimensional domain is carried out on the basis of the phase space method, which was developed by G.A. Sviridyuk. The conditions of non-uniqueness of the solution in the case when the dimension of the operator kernel at time derivative is equal to 1 or 2 are found. Two approaches for revealing the number of solutions of the Showalter-Sidorov problem in the case when the dimension of the operator kernel at time derivative is equal to 2 are presented. Examples illustrating the non-uniqueness of the solution of the problem on a rectangle are given.
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Analysis of complex hydraulic networks, electric circuits, electronic schemes, chemical processes etc. often results in a system of interconnected differential and algebraic equations. If the process under study has after-effect, then the system includes integral equations. This paper addresses simulation of hydraulic networks by means of the theory for singular systems of integral differential equations. We present theoretical tools that help investigate qualitative properties of such systems and search for effective methods of solution. A mathematical model for the straight through boiler circuit has been developed and a numerical method for its solution has been constructed. Experimental results showed that the theory for singular systems of integral differential equations performs well when applied to simulation of the hydraulic networks.
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In the design of iterative learning control (ILC) algorithm for stochastic nonlinear networked systems, the underlying assumption is differentiability of the system dynamics. In many cases, in reality, stochastic nonlinear networked systems have non-differentiable dynamics, but their dynamics functions after discretization by using conventional methods have global Lipschits’ continuous (GLC) condition. In this paper, we apply an ILC algorithm for stochastic nonlinear networked systems that have the GLC condition. We demonstrate that to design the ILC algorithm, differentiability of the system dynamics is not necessary, and the GLC condition is sufficient for designing the ILC algorithm for stochastic nonlinear networked systems with non-differentiable dynamics. We investigate the analysis of convergence and the tracking performance of the proposed update law for stochastic nonlinear networked systems with GLC condition. We show that there exists no limited condition for the stochastic data dropout probabilities in the convergence investigation of the input error. Then, the results are reviewed and confirmed with a numerical example.
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