Обзорные статьи. Рубрика в журнале - Вестник Южно-Уральского государственного университета. Серия: Математическое моделирование и программирование
A model of "nonadditive" routing problem where the costs depend on the set of pending tasks
Статья обзорная
We consider a generalization of the bottleneck (minimax) routing problem. The problem is to successively visit a number of megalopolises, complicated by precedence of constraints imposed on the order of megalopolises visited and the fact that the cost functions (of movement between megalopolises and of interior tasks) may explicitly depend on the list of tasks that are not completed at the present time. The process of movement is considered to be a sequence of steps, which include the exterior movement to the respective megalopolis and the following completion of (essentially interior) jobs connected with the megalopolis. The quality of the whole process is represented by the maximum cost of steps it consists of; the problem is to minimize the mentioned criterion (which yields a minimax problem, usually referred to as a "bottleneck problem"). Optimal solutions, in the form of a route-track pair (a track, or trajectory, conforms to a specific instance of a tour over the megalopolises, which are numbered in accordance with the route; the latter is defined by the transposition of indices), are constructed through a "nonstandard" variant of the dynamic programming method, which allows to avoid the process of constructing of all the values of the Bellman function whenever precedence constraints are present.
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The linear model of plane-parallel thermal convection in a viscoelastic incompressible Kelvin-Voigt material amounts to a hybrid of the Oskolkov equations and the heat equations in the Oberbeck-Boussinesq approximation on a two-dimensional region with Bénard's conditions. We study the solvability of this model with the so-called multipoint initial-final conditions. We use these conditions to reconstruct the parameters of the processes in question from the results of multiple observations at various points and times. This enables us, for instance, to predict emergency situations, including the violation of continuity of thermal convection processes as a result of breaching technology, and so forth. For thermal convection models, the solvability of Cauchy problems and initial-final value problems has been studied previously. In addition, the stability of solutions to the Cauchy problem has been discussed. We study a multipoint initial-final value problem for this model for the first time. In addition, in this article we prove a generalized decomposition theorem in the case of a relatively sectorial operator. The main result is a theorem on the unique solvability of the multipoint initial-final value problem for the linear model of plane-parallel thermal convection in a viscoelastic incompressible fluid.
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Analysis of biharmonic and harmonic models by the methods of iterative extensions
Статья научная
The article describes the results of recent years on the analysis of biharmonic and harmonic models by the methods of iterative extensions. In mechanics, hydrodynamics and heat engineering, various stationary physical systems are modeled using boundary value problems for inhomogeneous Sophie Germain and Poisson equations. Deflection of plates, flows during fluid flows are described using the biharmonic model, i.e. boundary value problem for the inhomogeneous Sophie Germain equation. Deflection of membranes, stationary temperature distributions near the plates are described using the harmonic model, i.e. boundary value problem for the inhomogeneous Poisson equation. With the help of the developed methods of iterative extensions, efficient algorithms for solving the problems under consideration are obtained.
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Evolution of the Viola - Jones Object Detection Method: a Survey
Статья научная
The Viola and Jones algorithm is one of the most well-known methods of object detection in digital images. Over the past 20 years since the first publication, the method has been extensively studied, and many modifications of the original algorithm and its individual parts have been proposed by researchers and engineers. Some ideas popularized by Paul Viola and Michael Jones became the basis for many other algorithms of object localization in images. This paper presents a description of Viola and Jones algorithm, the history of its development and modifications in the context of various problems of object localization in images, as well as a description of the current state of affairs: the method’s place in the era of convolutional neural networks extensive application
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Invariant manifolds of semilinear Sobolev type equations
Статья научная
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 spectral problems and mathematical models of continuum mechanics
Статья научная
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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Mathematical modelling economy
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The article presents an overview of the main methods of economic modelling used in scientific research over the past twenty years. This overview does not claim to cover all areas, methods and models used in scientific research in the field of economics, since it is impossible to do within a single article. We consider mathematical modelling of only two branches of economic theory: macroeconomics and microeconomics. At the same time, we present no literature review of methods and models of research in the section of microeconomics, which take place in the tools of scientific research, but were described in the section of macroeconomics. We believe that this review is useful to scientists engaged in the indirect study of economic phenomena and processes.
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On a class of Sobolev-type equations
Статья научная
The article surveys the works of T.G. Sukacheva and her students studying the models of incompressible viscoelastic Kelvin-Voigt fluids in the framework of the theory of semilinear Sobolev-type equations. We focus on the unstable case because of greater generality. The idea is illustrated by an example: the non-stationary thermoconvection problem for the order 0 Oskolkov model. Firstly, we study the abstract Cauchy problem for a semilinear nonautonomous Sobolev-type equation. Then, we treat the corresponding initial-boundary value problem as its concrete realization. We prove the existence and uniqueness of a solution to the stated problem. The solution itself is a quasi-stationary semi-trajectory. We describe the extended phase space of the problem. Other problems of hydrodynamics can also be investigated in this way: for instance, the linearized Oskolkov model, Taylor''s problem, as well as some models describing the motion of an incompressible viscoelastic Kelvin-Voigt fluid in the magnetic field of the Earth.
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On evolutionary inverse problems for mathematical models of heat and mass transfer
Статья научная
This article is a survey. The results on well-posedness of inverse problems for mathematical models of heat and mass transfer are presented. The unknowns are the coefficients of a system or the right-hand side (the source function). The overdetermination conditions are values of a solution of some manifolds or integrals of a solution with weight over the spatial domain. Two classes of mathematical models are considered. The former includes the Navier-Stokes system, the parabolic equations for the temperature of a fluid, and the parabolic system for concentrations of admixtures. The right-hand side of the system for concentrations is unknown and characterizes the volumetric density of sources of admixtures in a fluid. The unknown functions depend on time and some part of spacial variables and occur in the right-hand side of the parabolic system for concentrations. The latter class is just a parabolic system of equations, where the unknowns occur in the right-hand side and the system as coefficients. The well-posedness questions for these problems are examined, in particular, existence and uniqueness theorems as well as stability estimates for solutions are exposed.
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Статья научная
This article is a survey devoted to inverse problems of recovering sources and coefficients (parameters of a medium) in mathematical models of heat and mass transfer. The main attention is paid to well-posedness questions of the inverse problems with pointwise overdetermination conditions. The questions of this type arise in the heat and mass transfer theory, in environmental and ecology problems, when describing diffusion and filtration processes, etc. As examples, we note the problems of determining the heat conductivity tensor or sources of pollution in a water basin or atmosphere. We describe three types of problems. The first of them is the problem of recovering point or distributed sources. We present conditions for existence and uniqueness of solutions to the problem, show non-uniqueness examples, and, in model situations, give estimates on the number of measurements that allow completely identify intensities of sources and their locations. The second problem is the problem of recovering the parameters of media, in particular, the heat conductivity. The third problem is the problem of recovering the boundary regimes, i. e. the flux through a surface or the heat transfer coefficient.
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This paper addresses some classes of linear and quasi-linear partial differential algebraic equations (PDAEs), i.e. systems of partial differential equations with singular matrices multiplying the higher derivatives of the desired vector-function. Such systems do not belong to the class of the Cauchy - Kovalevskaya equations, and therefore do not not comply with known existence theorems. The current research focuses on the first order evolutionary systems with one variable and investigates PDAEs depending on the parameter. The concept of index for PDAEs is introduced and various statements of initial boundary problems are considered. The results obtained are used to simulate and analyze the heat and mass exchange processes in power plants.
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Optimal control in linear Sobolev type mathematical models
Статья научная
The article presents a review of the work of the Chelyabinsk mathematical school on Sobolev type equations in studying the optimal control problems for linear Sobolev type models with initial Cauchy (Showalter-Sidorov) conditions or initial-final conditions. To identify the nonemptiness of the set of feasible solutions to the control problem we use the phase space method, which has already proved itself in solving Sobolev type equations. The method reduces the singular equation to a regular one defined on some subspace of the original space and applies the theory of degenerate (semi)groups of operators to the case of relatively bounded, sectorial and radial operators. Here mathematical models are reduced to initial (initial-final) problems for an abstract Sobolev type equation. Abstract results are applied to the study of control problems for the Barenblatt-Zheltov-Kochina mathematical model, which describes fluid filtration in a fractured-porous medium, the Hoff model on a graph simulating the dynamics of I-beam bulging in a construction, and the Boussinesq-Löve model describing longitudinal vibrations in a thin elastic rod, taking into account inertia and under external load, or the propagation of waves in shallow water.
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Oskolkov models and Sobolev-type equations
Статья научная
This article is a review of the works carried out by the author together with her students and devoted to the study of various Oskolkov models. Their distinctive feature is the use of the semigroup approach, which is the basis of the phase space method used widely in the theory of Sobolev-type equations. Various models of an incompressible viscoelastic fluid described by the Oskolkov equations are presented. The degenerate problem of magnetohydrodynamics, the problem of thermal convection, and the Taylor problem are considered as examples. The solvability of the corresponding initial-boundary value problems is investigated within the framework of the theory of Sobolev-type equations based on the theory for -sectorial operators and degenerate semigroups of operators. An existence theorem is proved for a unique solution, which is a quasi-stationary semitrajectory, and a description of the extended phase space is obtained. The foundations of the theory of solvability of Sobolev-type equations were laid by Professor G.A. Sviridyuk. Then this theory, together with various applications, was successfully developed by his followers.
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Semilinear Sobolev type mathematical models
Статья научная
The article contains a review of the results obtained in the scientific school of Georgy Sviridyuk in the field of semilinear Sobolev type mathematical models. The paper presents results on solvability of the Cauchy and Showalter-Sidorov problems for semilinear Sobolev type equations of the first, the second and higher orders, as well as examples of non-classical models of mathematical physics, such as the generalized Oskolkov model of nonlinear filtering, propagation of ion-acoustic waves in plasma, propagation waves in shallow water, which are studied by reduction to one of the above abstract problems. Methods for studying the semilinear Sobolev type equations are based on the theory of relatively -bounded operators for equations of the first order and the theory of relatively polynomially bounded operator pencils for equations of the second and higher orders in the variable . The paper uses the phase space method, which consists in reducing a singular equation to a regular one defined on some subspace of the original space, to prove existence and uniqueness theorems, and the Galerkin method to construct an approximate solution.
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Sobolev type equations in spaces of differential forms on Riemannian manifolds without boundary
Статья научная
The article contains a review of the results obtained by the author both independently and in collaboration with other members of the Chelyabinsk scientific school founded by G.A. Sviridyuk and devoted to Sobolev-type equations in specific spaces, namely the spaces of differential forms defined on some Riemannian manifold without boundary. Sobolev type equations are nonclassical equations of mathematical physics and are characterized by an irreversible operator at the highest derivative. In our spaces, we need to use special generalizations of operators to the space of differential forms, in particular, the Laplace operator is replaced by its generalization, the Laplace-Beltrami operator. We consider specific interpretations of equations with the relatively bounded operators: linear Barenblatt-Zheltov-Kochina, linear and semilinear Hoff, linear Oskolkov ones. For these equations, we investigate the solvability of the Cauchy, Showalter-Sidorov and initial-final value problems in different cases. Depending on the choice of the type of equation (linear or semi-linear), we use the corresponding modification of the phase space method. In the spaces of differential forms, in order to use this method based on domain splitting and the actions of the corresponding operators, the basis is the Hodge-Kodaira theorem on the splitting of the domain of the Laplace-Beltrami operator.
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System analysis of classification of prime knots and links in thickened surfaces of genus 1 and 2
Статья научная
In this paper, we present a system analysis of approaches to classification of prime knots and links in thickened surfaces of genus 1 and 2 obtained by the author in collaboration with S.V. Matveev and V.V. Tarkaev in 2012 - 2020. The algorithm of the classification forms structure of the present paper. The results of classification are considered within system analysis of the main ideas of key steps of the algorithm. First, we construct prime projections. To this end, we define a prime link projection, enumerate graphs of special type which embedding in the surface can be a prime projection, enumerate projections in the surface, and show that all obtained projections are prime and not equivalent using some invariants of projections. Second, we construct prime links. To this end, we define a prime link, construct a preliminary set of diagrams, use invariants of links to form equivalence classes of the obtained diagrams and show that the resulting diagrams are not equivalent, and prove primality of the obtained links. At that, at each step, the used methods and the introduced objects are characterized from viewpoints of two cases (genus 1 and 2), and we distinguish properties that are common for both cases or characteristics of only one of two cases. Note consolidated tables, which systematize the classified projections with respect to their properties: generative graph, genus, number of components and crossings, existence and absence of bigon that simplifies the further work with the proposed classification of projections and links.
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Космические проекты: информационно-математический аспект и супервычисления (история и перспективы)
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Речь идет об информационно-математическом обеспечении проблем космонавтики: исторический экскурс и современные перспективы. Отмечена важнейшая роль Главного Теоретика космонавтики академика М.В. Келдыша, который фактически определил основные направления становления и развития космических исследований и дистанционного аэрокосмического зондирования Земли как планеты, называемых в настоящее время во всем мире кратко REMOTE SENSING.
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Математические модели и оптимальное управление процессами фильтрации и деформации
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В статье представлен обзор работ автора по изучению задачи оптимального управления для полулинейных моделей соболевского типа с s-монотонным и p-коэрцитивным операторами. Приводятся теоремы существования и единственности слабого обобщенного решения задачи Коши или задачи Шоуолтера - Сидорова для одного класса вырожденных неклассических моделей математической физики. Представленная теория базируется на методе фазового пространства и методе Галеркина - Петрова. Разработанная схема численного метода позволяет находить приближенные решения задачи Коши и задачи Шоуолтера - Сидорова для рассматриваемых моделей. Строится абстрактная схема изучения задачи оптимального управления данного класса моделей. На основе абстрактных результатов доказывается существование оптимального управления процессами фильтрации и деформации. Приводятся необходимые условия оптимального управления.
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Статья обзорная
Разработаны математические модели экструзионных процессов и программный комплекс, помогающий управленческому производственному персоналу решать задачи ресурсоэнергосберегающего управления экструдерами, применяемыми в гибких многоассортиментных производствах упаковочных полимерных материалов. Для комплексной оценки производительности, энергопотребления процесса и показателей качества материала в экструдерах, отличающихся аппаратной гибкостью и сложностью структуры потоков, предложен комбинированный метод моделирования, базирующийся на использовании статических и динамических моделей. Программный комплекс настраивается на метод производства, тип продукции, требования к производительности, энергопотреблению. Он включает банк данных характеристик экструзионных процессов, модуль выбора аппаратурного оформления процесса, конструктор виртуальных моделей шнеков экструдеров, подсистему структурно-параметрического синтеза математической модели и расчета выходных параметров процесса, подсистему визуализации результатов, в частности, в виде трендов выходных параметров и 3D графиков их зависимостей от управляющих воздействий. Тестирование комплекса по данным производств поливинилхлоридных, полиэтиленовых пленок на заводах в России, Германии подтвердило его работоспособность и возможность использования в качестве эффективного инструмента поддержки принятия решений при определении конфигураций и режимов работы экструдеров, обеспечивающих заданные потребительские характеристики продукции.
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Математические модели соболевского типа высокого порядка
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Статья содержит обзор результатов автора в области математических моделей на основе уравнений соболевского типа высокого порядка. Теория построена на основе известных фактов по разрешимости начальных (начально-конечных) задач для уравнений соболевского типа первого порядка. Идея базируется на обобщении теории вырожденных (полу)групп операторов на случай уравнений соболевсого типа высокого порядка: расщеплении пространств, действий всех операторов, построении пропагаторов и фазового пространства однородного уравнения, а также множества допустимых начальных значений для неоднородного уравнения. Мы используем уже хорошо зарекомендовавший себя при решении уравнений соболевского типа метод фазового пространства, заключающийся в редукции сингулярного уравнения к регулярному, определенному на некотором подпространстве исходного пространства. В работе проводится редукция математических моделей к начальным (начально-конечным) задачам для абстрактного уравнения соболевского типа высокого порядка. Полученные результаты могут найти дальнейшее применение при исследовании задач оптимального управления, нелинейных математических моделей, а также для построения теории уравнений соболевского типа высокого порядка в квазибанаховых пространствах.
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