Математическое моделирование. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
Статья научная
In this note we consider the population the model of which, derived on the basis of ethnographical accounts, includes a projection matrix with both positive and negative entries. Interpreting the eventually negative trajectories as representing the collapse of the population, we use some classical tools from convex analysis to determine a cone containing the initial conditions that give rise to the persistence of both the population and its social structure.
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Positive solutions to Sobolev type equations with relatively p-sectorial operators
Статья научная
The article describes sufficient conditions for the existence of positive solutions to both the Cauchy problem and the Showalter-Sidorov problem for an abstract linear Sobolev type equation. A distinctive feature of such equations is the phenomenon of non-existence and non-uniqueness of solutions. The research is based on the theory of positive semigroups of operators and the theory of degenerate holomorphic semigroups of operators. The merger of these theories leads to a new theory of degenerate positive holomorphic semigroups of operators. In spaces of sequences, which are analogues of Sobolev function spaces, the constructed abstract theory is used to study a mathematical model. The results can be used to study economic and engineering problems.
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Probabilistic solutions to the problem of rational consumer choice with random income
Статья научная
Probabilistic solutions are used when the amount of decision-makers is large. Each of them chooses the optimal solution independently of the others by solving his optimization problem. In this case, the optimal solution constructed by a randomly selected person (e.g. a consumer of goods) can be considered as a random vector. In particular, probabilistic solutions arise naturally in the rational consumer choice problem if income is assumed to be random. The problem of the utility function maximization at a time when the income of a randomly selected consumer is described as a random variable is considered as the stochastic optimization problem. The properties and distribution of the probabilistic solution of the consumer choice problem for various types of the utility function and income distribution are studied.
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Procedure for constructing soft models of complex systems by time series
Статья научная
The problem of creating models of complex systems for assessing their state is considered. The analysis of approaches to construction of diagnostic models is given and their features are marked. For a complex system with a hierarchical structure, a procedure for constructing the models to assess its state using a scalar time series is proposed. In this case, each hierarchical level is described by a lumped-parameter differential equation. The procedure is based on the concept of soft modelling. The efficiency of the proposed procedure is demonstrated by the example of constructing a model for assessing the state of a complex heart rhythm regulation system.
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Quantitative estimates on Jacobians for hybrid inverse problems
Статья научная
We consider σ-harmonic mappings, that is mappings U whose components ui solve a divergence structure elliptic equation div(σui)=0, for i=1,...,n. We investigate whether, with suitably prescribed Dirichlet data, the Jacobian determinant can be bounded away from zero. Results of this sort are required in the treatment of the so-called hybrid inverse problems, and also in the field of homogenization studying bounds for the effective properties of composite materials.
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Статья научная
Sectorial operators that act in complex Banach spaces and map real subspaces into themselves should be called real sectorial operators. These operators have already been used implicitly in the study of various diffusion equations. Meanwhile, in the Lojasiewicz - Simon theory which provides longtime convergence of solutions to stationary solutions, the real valued Lyapunov functions play an important role. In order to make general methods for studying longtime convergence problems on the basis of the Lojasiewicz - Simon theory, it may therefore be meaningful to give an explicit definition for these real sectorial operators and to show their basic properties that are inherited from those of complex sectorial operators.
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Recent results on the Cahn - Hilliard equation with dynamic boundary conditions
Статья научная
The pure or viscous Cahn - Hilliard equation with possibly singular potentials and dynamic boundary conditions is considered and the well-posedness of the related initial value problem is discussed. Then, a boundary control problem for the viscous Cahn - Hilliard system is studied and first order necessary conditions for optimality are shown. Moreover, the same boundary control problem is addressed for the pure Cahn - Hilliard system, by investigating it and passing to the limit in the analogous results for the viscous Cahn - Hilliard system as the viscosity coefficient tends to zero.
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Regularity results and solution semigroups for retarded functional differential equations
Статья научная
We show that the solutions of the retarded functional differential equations in a Banach space, whose existence and uniqueness are established in paper of A. Favini and H. Tanabe, have some further regularity properties if the initial data and the inhomogeneous term satisfy some smootheness assumptions. Some results on the solution semigroups analogous to the one of G. Di Blasio, K. Kunisch and E. Sinestrari and to the one of E. Sinestrari are also obtained.
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Relay races along a pair of selectable routes
Статья научная
Case of two teams competition, which should overcome the distance divided onto stages, is considered. In the case under consideration, every stage has its own number of routes, which the participants of the team may select to overcome. It is shown, that competition bears the character of the relay race, and two-parallel semi-Markov process is the natural approach to modelling of the situation. From all possible routes two were selected. The conception of switching space, which display all possible switching trajectories is proposed. The formula for calculation of switching trajectories number is acquired. It is shown, that ordinary semi-Markov process with the use of the recursive procedure may be obtained from the complex two-parallel semi-Markov process, which describes the wandering through selected routes. The formulae for realization of the recursion are proposed. Conception of distributed forfeit is proposed. It is shown, that forfeit depends on difference of stages, teams overcome at current time, and routes, on which participants solved to overcome stage. The formula for estimation of total forfeit, which one team pays to other team is obtained. It is shown, that the sum of forfeit may be used as the optimization criterion in the game strategy optimization task.
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Research of the Optimal Control Problem for One Mathematical Model of the Sobolev Type
Статья научная
The article is devoted to the study of optimal control for one mathematical model of the Sobolev type, which is based on the model equation, which describes various processes (for example, deformation processes, processes occurring in semiconductors, wave processes, etc.) depending on the parameters and can belong either to the class of degenerate (for > 0) equations or to the class of nondegenerate (for < 0) equations. The article is the first attempt to study the control problem for mathematical semilinear models of the Sobolev type in the absence of the property of non-negative definiteness of the operator at the time derivative, i.e. the construction of a singular optimality system in accordance with the singular situation caused by the instability of the model. Conditions for the existence of a control-state pair are presented, and conditions for the existence of an optimal control are found.
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Resource allocation in cloud computing via optimal control to queuing systems
Статья научная
We consider resource allocation problem in the cloud computing. We use queuing model to model the process of entering into the cloud and to schedule and to serve incoming jobs. In this paper, the main problem is to allocate resources in the queuing systems as a general optimization problem for controlled Markov process with finite state space. For this purpose, we study a model of cloud computing where the arrival jobs follow a stochastic process. We reduce this problem to a routing problem. In the case of minimizing, cost is given as a mixture of an average queue length and number of lost jobs. We use dynamic programming approach. Finally, we obtain the explicit form of the optimal control by the Bellman equation.
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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
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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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 parabolic-hyperbolic type differential equations with spectral analysis method
Статья научная
The study investigates a parabolic-hyperbolic type differential equation with nonlocal boundary and initial conditions. The problem is approached using the spectral analysis method, allowing the solution to be expressed as a series expansion in terms of eigenfunctions of the associated spectral problem. The existence, uniqueness, and stability of the solution are rigorously established through analytical techniques, ensuring the well-posedness of the problem. Furthermore, the study carefully examines the issue of small denominators that arise in the series representation and derives sufficient conditions to guarantee their separation from zero. These results contribute to the broader mathematical theory of mixed-type differential equations, providing valuable insights into their structural properties. The findings have practical applications in various fields of physics and engineering, particularly in modeling wave propagation, heat conduction, and related dynamic processes. The theorems obtained ensure that under appropriate assumptions on the given data, the problem admits a unique and stable solution, reinforcing its theoretical and practical significance.
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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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Stability of solutions to the stochastic Oskolkov equation and stabilization
Статья научная
This paper studies the stability of solutions to the stochastic Oskolkov equation describing a plane-parallel flow of a viscoelastic fluid. This is the equation we consider in the form of a stochastic semilinear Sobolev type equation. First, we consider the solvability of the stochastic Oskolkov equation by the stochastic phase space method. Secondly, we consider the stability of solutions to this equation. The necessary conditions for the existence of stable and unstable invariant manifolds of the stochastic Oskolkov equation are proved. When solving the stabilization problem, this equation is considered as a reduced stochastic system of equations. The stabilization problem is solved on the basis of the feedback principle; graphs of the solution before stabilization and after stabilization are shown.
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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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Stationary electrochemical machining simulation applying to precision technologies
Статья научная
The modification of statement of electrochemical formation problem is offered for mathematical modelling of the precision technologies. As an example the process of cutting with a plate electrode-tool is considered. For the description of the technologies with high localization of the processes a stepwise function of current efficiency is used. It realizes for simulation of the anode dissolution process in passivating electrolytes under short impulse current. This function determines the movement rate of the anode boundary in the areas of an active electrochemical dissolution and also it defines the boundaries of the areas where dissolution is absent. The stationary and limiting-stationary machining problems are formulated and solved on the base of the offered model. The limiting model describes the maximum localization process. The stationary problem is characterized by the presence of anode surface part, on which the current density is equal to a critical value. Investigations in the whole range of ratio of the maximal and critical values of electrical field strength on the anode surface are carried out.
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