Математическое моделирование. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
On the mean-value property for polyharmonic functions
Статья научная
The mean-value property for normal derivatives of polyharmonic function on the unit sphere is obtained. The value of integral over the unit sphere of normal derivative of mth order of polyharmonic function is expressed through the values of the Laplacian's powers of this function at the origin. In particular, it is established that the integral over the unit sphere of normal derivative of degree not less then 2k-1 of k-harmonic function is equal to zero. The values of polyharmonic function and its Laplacian's powers at the center of the unit ball are found. These values are expressed through the integral over the unit sphere of a linear combination of the normal derivatives up to k-1 degree for the k-harmonic function. Some illustrative examples are given.
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On the one-dimensional harmonic oscillator with a singular perturbation
Статья научная
In this paper we investigate the one-dimensional harmonic oscillator with a left-right boundary condition at zero. This object can be considered as the classical selfadjoint harmonic oscillator with a singular perturbation concentrated at one point. The perturbation involves the delta-function and/or its derivative. We describe all possible selfadjoint realizations of this scheme in terms of the above mentioned boundary conditions. We show that for certain conditions on the perturbation (or, equivalently, on the boundary conditions) exactly one non-positive eigenvalue can arise and we derive an analytic expression for the corresponding eigenfunction. These eigenvalues run through the whole negative semi-line as the perturbation becomes stronger. For certain cases an explicit relation between suitable boundary conditions, the non-positive eigenvalue and the corresponding eigenfunction is given.
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On the regularizability conditions of integral equations
Статья научная
Solving of integral equations of the first kind is an ill-posed problem. It is known that all problems can be divided into three disjoint classes: correct problems, ill-posed regularizable problems and ill-posed not regularizable problems. Problems of the first class are so good that no regularization method for them is needed. Problems of the third class are so bad that no one regularization method is applicable to them. A natural application field of the regularization method is the problems from the second class. But how to know that a particular integral equation belongs to the second class rather than to the third class? For this purpose a large number of sufficient regularizability conditions were constructed. In this article one infinite series of sufficient conditions for regularizability of integral equations constructed with the help of duality theory of Banach spaces is investigated. This method of constructing of sufficient conditions proved to be effective in solving of ill-posed problems. It is proved that these conditions are not pairwise equivalent even if we are restricted by the equations with the smooth symmetric kernels.
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On the stability of two-dimensional flows close to the shear
Статья научная
We consider the stability problem for two-dimensional spatially periodic flows of general form, close to the shear, assuming that the ratio of the periods tends to zero, and the average of the velocity component corresponding to the "long" period is non-zero. The first terms of the long-wavelength asymptotics are found. The coefficients of the asymptotic expansions are explicitly expressed in terms of some Wronskians and integral operators of Volterra type, as in the case of shear basic flow. The structure of eigenvalues and eigenfunctions for the first terms of asymptotics is identified, a comparison with the case of shear flow is made. We study subclasses of the considered class of flows in which the general properties of the qualitative behavior of eigenvalues and eigenfunctions are found. Plots of neutral curves are constructed. The most dangerous disturbances are numerically found. Fluid particle trajectories in the self-oscillatory regime in the linear approximation are given.
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On time selection for track possession assignment at the railway station
Статья научная
We consider the problem on track possession assignment at the railway station. The problem is to determine the time during which the train traffic is interrupted for repair works at several railway tracks. To this end, we use a traffic schedule of the station in order to solve the problem on searching for the time interval having the following two properties. First, during this time interval, all track sections that are necessary to be repaired are vacant simultaneously. Second, this time interval has the maximum length. In addition, we solve two problems to determine the time interval having length that is not less than the length of the specified time interval in the following two cases. First, the desired time interval has the minimum number of occupied track sections that are necessary to be repaired. Second, the desired time interval has the minimum number of delayed (transferred) passenger/freight trains going through the tracks that are necessary to be repaired. All problems are solved by methods of mixed integer linear programming.
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Optimal measurement of dynamically distorted signals
Статья научная
There has been suggested new approach to measure a signal distorted as by inertial measurement transducer, as by its resonances.
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Optimal solutions for inclusions of geometric Brownian motion type with mean derivatives
Статья научная
The idea of mean derivatives of stochastic processes was suggested by E. Nelson in 60-th years of XX century. Unlike ordinary derivatives, the mean derivatives are well-posed for a very broad class of stochastic processes and equations with mean derivatives naturally arise in many mathematical models of physics (in particular, E. Nelson introduced the mean derivatives for the needs of Stochastic Mechanics, a version of quantum mechanics). Inclusions with mean derivatives is a natural generalization of those equations in the case of feedback control or in motion in complicated media. The paper is devoted to a brief introduction into the theory of equations and inclusions with mean derivatives and to investigation of a special type of such inclusions called inclusions of geometric Brownian motion type. The existence of optimal solutions maximizing a certain cost criterion, is proved.
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Optimization of training modules choice during multipurpose training of specialists
Статья научная
The article is devoted to the problems of mathematical modelling of the processes of organizing multipurpose learning. Under the multipurpose training is understood such an organization of the educational process, in which in one group specialists are trained in several related areas of activity, trajectory of training for which at certain intervals intersect. In order to reduce the total training time, as well as the cost or resources required in the training process, it is expedient to carry out the dynamic grouping of students by subgroups in order to master certain competences. The development of the mathematical apparatus used to optimize the multipurpose learning process has not been completely studied at present. To reduce the dimension of the overall task of optimizing the process of organizing multipurpose training, its step-by-step solution is proposed. The article describes the approach to calculating estimates of the possibility of training specialists in the areas of training and selection of training modules available in the educational organization. The paper considers the options for optimizing the selection of modules for training specialists on the following criteria: minimizing the total duration of training, the cost of training and resources used for training. The algorithm with the help of which it is possible to form an optimal group of students is proposed.
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Статья научная
This paper is concerned with the oscillatory properties of second order non-linear dynamic equation with integro forcing term on an arbitrary time scales. We reduce our original dynamic equation into an alternate equation by introducing a function of forward jump operator. To study oscillations we establish some crucial Lemmas and employ generalized Riccati transformation technique which transforms our second order dynamic equation into the first order dynamic equation on an arbitrary time scales. These results also guarantee that the solution of our equation oscillates. Furthermore, we establish the Kamenev-type oscillation criteria of our system. At the end, we consider a second order dynamic equation on time scales with deviating argument and compare it with our result which gives the sufficient conditions of oscillation of it.
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Parameter identification and control in heat transfer processes
Статья научная
The article is devoted to the study of some mathematical models describing heat transfer processes. We examine an inverse problem of recovering a control parameter providing a prescribed temperature distribution at a given point of the spatial domain. The parameter is a lower order coefficient depending on time in a parabolic equation. This nonlinear problem is reduced to an operator equation whose solvability is established with the help of a priori estimates and the fixed point theorem. Existence and uniqueness theorems of solutions to this problem are stated and proved. Stability estimates are exposed. The main result is the global (in time) existence of solutions under some natural conditions of the data. The proofs rely on the maximum principle. The main functional spaces used are the Sobolev spaces.
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Performance bounds and suboptimal policies for multi-class queue
Статья научная
In this paper, we consider a general class of a queuing system with multiple job types and flexible service facility. We use a stochastic control policy to determine the performance loss in multi-class M/M/1 queue. The considered system is originally a Markov decision processes (MDP). The author showed how to compute performance bounds for the stochastic control policy of MDP with an average cost criteria. In practice, many authors used heuristic control policies due to some hardness in computing and running mathematically optimal policies. The authors found bounds on performance in order to an optimal policy where the goal of this job is to compute the difference of optimality and a specific policy. In other words, this study shows that, the optimal bounds of the average queue length for any non-idling policies can be found by a factor of service rates.
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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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