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NORMAL FORMS OF THE DEGENERATE AUTONOMOUS DIFFERENTIAL EQUATIONS WITH THE MAXIMAL JORDAN CHAIN AND SIMPLE APPLICATIONS
стр.5-15
L.R. Kim-Tyan, B.V. Loginov, Yu.B. Rousak
Degenerate differential equations, as part of the differential-algebraic equations, the last few decades cause increasing interest among researchers, both because of the attractiveness of the considered theoretical questions, and by virtue of their applications. Currently, advanced methods developed in this area are used for system modelling and analysis of electrical and electronic circuits, chemical reaction simulations, optimization theory and automatic control, and many other areas. In this paper, the theory of normal forms of differential equations, originated in the works of Poincare and recently developed in the works of Arnold and his school, adapted to the simplest case of a degenerate differential equations. For this purpose we are using technique of Jordan chains, which was widely used in various problems of bifurcation theory. We study the normal forms of degenerate differential equations in the case of the existence of the maximal Jordan chain. Two and three dimensional spaces are studied in detail. Normal forms are the simplest representatives of the degenerate differential equations, which are equivalent to more complex ones. Therefore, normal forms should be considered as a model type of degenerate differential equations.
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МАТЕМАТИЧЕСКОЕ МОДЕЛИРОВАНИЕ ФОРМИРОВАНИЯ РАЗНОСТИ ПОТЕНЦИАЛОВ ПРИ КРИСТАЛЛИЗАЦИИ ОБЛАЧНЫХ КАПЕЛЬ С УЧЕТОМ ФРАКТАЛЬНОСТИ СРЕДЫ
стр.16-24
В работе рассмотрен механизм возникновения разности потенциалов при кристаллизации облачных капель в конвективных облаках с учетом фрактальной структуры среды. Моделирование процесса проводилось на основе дифференциального уравнения в частник производных дробного порядка. Показано, что возникновение разности потенциалов при кристаллизации капель в конвективньк облаках существенно зависит не только от скорости роста льда и дисперсности пузырьков, но и от фрактально-сти среды, где протекает процесс. Приведены результаты моделирования, на основе численного решения построены графики в зависимости от различных значений управляющих параметров.
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К ВОПРОСУ О МАРШРУТИЗАЦИИ ПЕРЕМЕЩЕНИЙ ПРИ ЛИСТОВОЙ РЕЗКЕ ДЕТАЛЕЙ
стр.25-39
А.А. Петунин, А.Г. Ченцов, П.А. Ченцов
Рассматривается решение задачи управления инструментом при листовой резке на машинах с ЧПУ. Предполагается, что исходная постановка осложнена различными ограничениями. Требуется построить решение возникающей задачи маршрутизации, соблюдающее ограничения и минимизирующее аддитивный критерий, включающий стоимости (внешних) перемещений и «внутренних» работ, связанных с резкой деталей по замкнутому контуру. Соблюдение ограничений предполагается обеспечивать за счет специального задания функций стоимости, т.е. (по сути) за счет формирования штрафов за нарушение требуемых условий. Главную роль играет при этом процедура на базе широко понимаемого динамического программирования. Конструируемый на данной основе алгоритм реализован в виде стандартной программы на многоядерной ПЭВМ. Изложение этого алгоритма составляет основную цель настоящей работы.
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ANALYTICAL SOLUTION TO THE PROBLEM OF CONVECTIVE HEAT TRANSFER IN A POROUS RECTANGULAR CHANNEL FOR THERMAL BOUNDARY CONDITIONS OF THE SECOND GENUS
стр.40-53
V.I. Ryazhskikhl, D.A.Konovalovl, A.V.Ryazhskikhl, A.A. Boger, S.V. Dakhin
In the three-dimensional statement, we consider the Brinkman equation together with the equation of heterogeneous heat transfer for an unidirectional flow of the Newtonian fluid under laminar regime through horizontal porous channel having a constant rectangular cross-section with known thermal flows at the boundary and small values of the Darcy numbers. Due to the linearity of the formulated system of model equations, we obtain analytical solution of the system using the Laplace and Fourier integral transformation. The obtained solution allows to estimate the length of the input hydrodynamic section, the coefficient of hydraulic resistance, and the local Nusselt numbers. The results obtained for the hydrodynamic subproblem with a large porosity and thermal subproblem with a stationary temperature field agree with the classical data.
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FORECASTING THE RETURN OF THE LOAN PORTFOLIO ON THE BASIS OF MARKOV MODEL
стр.54-66
We consider the problem of mathematical modelling of flows of loan portfolio payments. We assume that the change in the quality of each loan is described by a simple Markov chain with a finite number of states. In this case, the flow of loan payments is a random process, which depends on the Markov chain. On the basis of the proposed model and known relations of the stochastic systems theory, we describe the expected flows of payments of the entire loan portfolio and construct a method to forecast the expected return (net present value) of the portfolio. We analyze an accuracy of the obtained model and a sensitivity of net present value of the portfolio to a change in the transition probabilities in the Markov chain.
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АЛГОРИТМЫ ПОСТРОЕНИЯ ОПТИМАЛЬНЫХ УПАКОВОК В ЭЛЛИПСЫ
стр.67-79
В.Н. Ушаков, П.Д. Лебедев, Н.Г. Лавров
В задачах теории управления часто требуется проводить аппроксимацию множеств наборами из конгруэнтных элементов. Одним из вариантов такой аппроксимации служит упаковка в фигуры на плоскости набора кругов равного радиуса. В статье рассмотрены два варианта задачи о построении оптимальной упаковки в эллипсы различной формы: в первом фиксировано число элементов и требуется максимизировать их радиус, во втором фиксирован радиус кругов и требуется максимизировать их число. В первом варианте применяются итерационные методы, имитирующие отталкивание центров кругов друг от друга и от границы множества. В них используются конструкции чебышевского центра, ортогональных проекций и отталкивания точек. Во втором - рассматриваются упаковки с гексагональной решеткой, которые близки к оптимальным. Реализован программный комплекс построения упаковок для эллипсов с различным соотношением осей.
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GRAVITATIONAL MODEL OF POPULATION DYNAMICS
стр.80-93
M. Y. Khavinson, M.P. Kulakov
We consider the author’s mathematical model of population dynamics of territories, taking into account the migration between the territories. The model is a system of ordinary differential equations with constant coefficients. The main idea of the presented modelling is to take into account the migration interactions of territories in the form of nonlinear terms (they are the pair products of phase variables of the territories population). On the one hand, the approach allows to consider the model as a variant of expanding the gravitational approach in migration studies. On the other hand, this approach allows to apply the approaches of mathematical biology, which are successfully used in econophysics and sociodynamics. In order to verify the model, we use statistical data on population and migration between federal districts of the Russian Federation. The results of the modelling show the significance of the "repulsion" of migrants arriving in the Central and NorthWestern federal districts, mainly in the nearby regions (Southern, North Caucasian and Volga federal districts). Model evaluations of the migration balance are obtained. The evaluations exceed statistical ones by dozens of times and, to all appearances, describe the "latent" migration of the population, covering both long-term and short-term movements. An analysis of the change in the stationary values of the population for a linear change in the parameters is carried out. It is shown that there are such values of the parameters of migration attractiveness of the Russian Federation federal districts, under which the population increases both in the whole in the Russian Federation and in individual districts. It is established that such changes can occur due to significant differences in the opportunities, which are "provided" by different federal districts for migrants (e.g., living and working conditions, upbringing and education of children, etc.), and intra-Russian migration.
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INFLUENCE OF THE METHOD OF HYDROGEN ATOMS INCORPORATION INTO THE TARGET PROTEIN ON THE PROTEIN-LIGAND BINDING ENERGY
стр.94-107
D.C. Kutov, E.V. Katkova, A.V. Sulimov, O.A. Kondakova, V.B. Sulimov
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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МОДЕЛИРОВАНИЕ И ОЦЕНИВАНИЕ ТРАЕКТОРИИ ДВИЖУЩЕГОСЯ ОБЪЕКТА
стр.108-119
И.В. Семушин, А.В. Цыганов, Ю.В. Цыганова, А.В. Голубков
Построена линейная дискретная модель кругового движения влево/вправо с заданным радиусом. Сформулирован и реализован новый алгоритм математического моделирования траектории движущегося объекта, состоящей из отрезков прямолинейного и кругового движения в условиях неполных зашумленных измерений. Показано, как применять алгоритмы оптимальной дискретной фильтрации для оценивания такой траектории. Разработан программный комплекс «Моделирование и оценивание траектории подвижного объекта vl.O» для моделирования и линейного оценивания траектории движущегося объекта на основе дискретных моделей равномерного прямолинейного и кругового движения. Результаты работы могут найти применение при решении практических задач судовождения, робототехники, обработки сигналов со сканирующих дальномеров и др.
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CIRCULAR SHIFT OF LOOP BODY — PROGRAMME TRANSFORMATION, PROMOTING PARALLELISM
стр.120-132
The article deals with the programme transformation executing the circular shift of loop body statements. It can be used for vectorizing or parallelizing. This becomes possible due to the fact that when the order of loop body statements is changed, some of the bottom-up arcs become top-down arcs. Besides, sometimes loop carried dependence arcs are substituted by loop independent ones. It should be pointed out that in executing the circular shift the number of loop iterations is reduced by one. The transformation can be used both independently and in conjunction with other transformations promoting parallelism. These could be "forward substitution", "scalar expansion", "privatization", "array expansion", etc. The transformation under consideration in this article can be used both in hand parallelization and added to a paralleling (optimizing) compiler. Moreover, the application of the transformation results in the equivalent code only for the loops where loop unrolling is the equivalent transformation. Thus, they can contain nested loops, if statements and other programming language statements.
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MODEL OF POROUS MATERIAL SINTERING
стр.133-141
0.V. Shults, V.A. Simonenko, S.P. Drovosekov, P.V. Kapustina, I.A. Belobrova
The mathematical model of inert porous materials sintering was proposed. As an initial data, the model uses the initial distribution of pore surface elements and time-temperature conditions of sintering. Sintering is modelled by the system of kinetic equations describing shrinkage of a material depending on its initial properties and a time-temperature condition of the process. Experiments on aluminum oxide powder sintering have been carried out in several time-temperature conditions in which linear shrinkage was traced in time. The initial powder particles size distribution was previously measured. A value of main parameters used in the model was determined according to the sample linear shrinkage data measured at the 10°C/min heating rate. Time dependences of linear shrinkage were calculated for the heating rates of 5 and 15°C/min. Comparison of the calculated dependences with experimental data for heating rates 5 and 15 °C/min has shown the good qualitative and quantitative concordance.
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SEQUENTIAL APPLICATION OF THE HIERARCHY ANALYSIS METHOD AND ASSOCIATIVE TRAINING OF A NEURAL NETWORK IN EXAMINATION PROBLEMS
стр.142-147
O.S. Avsentiev, T.V. Meshcheryakova, V.V. Navoev
We propose development of examination methodology based on a sequential application of the MAI method (i.e., the hierarchy analysis method) and associative training of neural networks. The proposed method is an alternative to the usual methods to solve a direct examination problem.
We present a methodological approach to the examination problem. The approach allows to save information about all objects and consider their indicators in total. Therefore, there is the soft maximum principle (softmax), based on the model of expert evaluations mixing. This approach allows different interpretations of the examination results, which save quality unchanged overall picture of the examination object indicators ratio, and to get more reliable examination results, especially in cases where the objects characteristics are very different.
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COMPUTER SIMULATION OF THE VOLGA RIVER HYDROLOGICAL REGIME: PROBLEM OF WATER-RETAINING DAM OPTIMAL LOCATION
стр.148-155
E.O. Agafonnikova, A. Yu. Klikunova, A.V. Khoperskov
We investigate of a special dam optimal location at the Volga river in the area of the Akhtuba left sleeve beginning (7 km to the south of the Volga Hydroelectric Power Station dam). We claim that a new water-retaining dam can resolve the key problem of the Volga-Akhtuba floodplain related to insufficient water amount during spring floodings due to the overregulation of the Lower Volga. Using a numerical integration of Saint-Vanant equations we study the water dynamics across the northern part of the Volga-Akhtuba floodplain taking into account its actual topography. As the result we found an amount of water VA passing to the Akhtuba during spring period for a given water flow through the Volga Hydroelectric Power Station (so-called hydrograph which characterises the water flow per unit of time). By varying the location of the water-retaining dam xd, yd we obtained various values of VA(xd, yd) as well as various flow spatial structure on the territory during the flood period. Gradient descent method provides the dam coordinated with the maximum value of VA. Such approach to the dam location choice let us find the best solution, that the value VA increases by a factor of 2. Our analysis demonstrates a good potential of the numerical simulations in the field of hydraulic works.
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SPECTRAL PROBLEMS ON COMPACT GRAPHS
стр.156-162
S.I. Кadchenko, S.N. Kakushkin, G.A. Zakirova
The method of finding the eigenvalues and eigenfunctions of abstract discrete semibounded operators on compact graphs is developed. Linear formulas allowing to calculate the eigenvalues of these operators are obtained. The eigenvalues can be calculates starting from any of their numbers, regardless of whether the eigenvalues with previous numbers are known. Formulas allow us to solve the problem of computing all the necessary points of the spectrum of discrete semibounded operators defined on geometric graphs. The method for finding the eigenfunctions is based on the Galerkin method. The problem of choosing the basis functions underlying the construction of the solution of spectral problems generated by discrete semibounded operators is considered. An algorithm to construct the basis functions is developed. A computational experiment to find the eigenvalues and eigenfunctions of the Sturm - Liouville operator defined on a two-ribbed compact graph with standard gluing conditions is performed. The results of the computational experiment showed the high efficiency of the developed methods.
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ОБУЧАЮЩИЙ ВИДЕОКУРС КАК ЭЛЕМЕНТ ЭЛИТНОГО МАТЕМАТИЧЕСКОГО ОБРАЗОВАНИЯ
стр.163-165
А.В. Кунгурцева, Е.И. Назарова, Н.Н. Овчинникова, М.А. Сагадеева
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