Proceedings of the National Academy of Sciences of Belarus. Series of Physical-Mathematical Sciences / Известия Национальной академии наук Беларуси. Серия физико-математических наук
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ДИНАМИКА ФАЗОВЫХ ПЕРЕХОДОВ 1-ГО РОДА В ФИНСЛЕРОВОМ КОНФИГУРАЦИОННОМ ПРОСТРАНСТВЕ ЛЕНГМЮРОВСКОГО МОНОСЛОЯ
Get PDFIn the article, the Euler – Lagrange equations, which describe first-order phase transition dynamics in a con-figuration Finsler space of a Langmuir monolayer, have been obtained. An approximate method for analysis of the equations has been developed. The method is based on a combination of analytical and numerical calculations using the zero-order approximation with a fixed relaxation time and the more exact approximation with a model distribution of relaxation times. Heterogeneous dynamics of the system has been demonstrated. Such dynamics corresponds to the monolayer metastable state with different relaxation times of phase nuclei. The relaxation time distribution has a maximum and a maximum height depends on a monolayer compression rate. The increase of the maximum height at enhancement of a compression rate is accompanied by an explicit plateau of the isotherm that displays the characteristic behavior of the monolayer isotherm in the region of phase transition. The dynamics of a two-dimensional phase transition has been numerically studied at the compression rate as being sufficiently low, and a comparative analysis of the system behavior at two approximations (the approximation of fixed relaxation time and the approximation of model distribution of relaxation times) has been made. It has been found that the presence of phase nuclei with different relaxation times causes an effective centrifugal force, the magnitude of which depends on the gradient of electrocapillary forces
МОНОТОННЫЕ РАЗНОСТНЫЕ СХЕМЫ ДЛЯ МОДЕЛИ ШНЭКЕНБЕРГ
Get PDFIn this article the canonical form of the vector-difference schemes is constructed. The definition of the monotonicity of difference schemes is given. This definition is related to the positivity property of the difference solution. Based on this definition, the monotone difference schemes for the Schnakenberg model with the Dirichlet and Neumann boundaryconditions are constructed. This model is a semi-nonlinear reaction-diffusion system, and it plays an important role in mathematical modeling in the fields of physical chemistry and biology. In constructing a monotone difference scheme for this model with the Neumann boundary condition, the idea of half-integral nodes at the boundary points under the secondkind boundary conditions is used. The results of numerical experiments have confirmed the effectiveness of the suggested methods. the numerical solution without nonphysical oscillation is obtained.В настоящей работе построена каноническая форма векторно-разностных схем. Дано определение монотонности таких разностных схем, связанное со свойством положительности разностного решения. На основе этого определения построены монотонные разностные схемы для модели Шнэкенберг с граничными условиями Дирихле и Неймана. Эта модель представляет собой полунелинейную реакционно-диффузную систему и играет важную роль при математическом моделировании в областях физической химии и биологии. При построении монотонной разностной схемы для указанной модели с граничным условием Неймана основная идея состоит в том, чтобы использовать полуцелые узлы в граничных точках задания краевых условий второго рода. Представлены результаты вычислительных экспериментов, подтверждающих эффективность предложенных методов. численное решение получено без нефизических осцилляций
О ПСЕВДОЛИПШИЦЕВОСТИ МНОЖЕСТВА РЕШЕНИЙ ПАРАМЕТРИЧЕСКИХ ЗАДАЧ ОПТИМИЗАЦИИ
Get PDFThe study of the properties of solution mappings in parametrical optimization problems represents an urgent problem. Particularly, considerable efforts are directed to finding the conditions of different types of generalized Lipschitzian continuity of solution mappings, namely their calmness and pseudo-Lipschitzian continuity (also referred to as the Aubin property) [1]. A new interesting approach to investigating the calmness of solution mappings has recently been proposed by Canovas et al. [2] for parametrical linear programming problems and applied to a much wider range of problems by Klatte and Kummer [3]. In this approach, the calmness of solution mappings is related to the calmness of an associated system representing a constraint on the level set of the objective function on the domain of the problem. In our note, we propose to expand the use of the approach [3] for investigating the pseudo-Lipschitzian continuity of solution mappings. Several sufficient conditions for the pseudo-Lipschitzian continuity of solution mappings, as well as the generalization of the Hoffman lemma are presented.Исследование свойств множества решений параметрических задач оптимизации представляет собой достаточно актуальную проблему. Значительные усилия направлены, в частности, на поиск условий различных типов обобщенной липшицевости множества решений, в частности условий их устойчивости (calmness) и псевдолипшицевости (Aubin property) [1]. Новый интересный подход к исследованию устойчивости множества решений предложен в работе М. Кановас и др. [2] в случае параметрической задачи линейного программирования и распространен Д. Клатте и Б. Куммером [3] на существенно более широкий круг задач. В данном подходе устойчивость множества решений связывается с устойчивостью некоторой ассоциированной системы, представляющей ограничение множества уровня целевой функции на множестве допустимых точек задачи. В настоящей статье предлагается расширить применение подхода [3] на исследование псевдолипшицевости множества решений; представлены некоторые достаточные условия псевдолипшицевости множества решений, а также обобщение леммы Хоффмана.
О СТАБИЛИЗАЦИИ КОЛИЧЕСТВА ОРБИТ КЭМЕРОНОВСКИХ МАТРИЦ БОЛЬШОГО РАНГА
Get PDFA quadratic (0,1)-matrix of degree n with just n units among its elements will be called a Cameron matrix. The orbits of the natural action of the group Sn× Sn (the square of the symmetric group of degree n) on the set of Cameron matrices of degree n (an independent action on the rows and the columns of matrices) are considered. It is proved that for fixed d < n, the number of such orbits for matrices of rank n – d is constant for n ≥ 3d and grows with the growth of n if n < 3d. For each orbit, its representative in a quasi-Jordan form is indicated. Назовем квадратную (0,1)-матрицу порядка n, среди элементов которой ровно n единиц, кэмероновской матрицей. Рассматриваются орбиты естественного действия группы Sn× Sn (квадрат симметрической группы степени n) на множестве кэмероновских матриц порядка n (независимое действие на строках и столбцах матриц). Установлено, что для фиксированного d < n число таких орбит для матриц ранга n – d постоянно при n ≥ 3d и растет с ростом n при n < 3d. Для каждой орбиты указан ее представитель в квазижордановой форме.
РЕШЕНИЕ СМЕШАННЫХ ЗАДАЧ МЕТОДОМ ХАРАКТЕРИСТИК ДЛЯ ВОЛНОВОГО УРАВНЕНИЯ С ИНТЕГРАЛЬНЫМ УСЛОВИЕМ
Get PDFThe mixed problem for the wave equation with one integral condition and one Dirichlet’s condition on the right boundary of the domain is considered in the one-dimensional case. It is proved that the fulfillment of the matching conditions is necessary and sufficient for existence and uniqueness of the classical solution of the given mixed problem under certain smoothness conditions for the given functions. The method of characteristics is used for analysis of the problem. This method is reduced to partitioning the original domain by characteristics line in sub-domains where the solution of the given problem is constructed with the help of initial, boundary and integral conditions. However, in some sub-domains the solution of the problem is reduced to Volterra’s second-type equation. For this equation, the theorems of correct solvability are fulfilled. Matching conditions are obtained by equating the values of the solution and its derivatives up to the second-order, including on characteristics. The obtained results allow building either the analytical solution of the given problem if Volterra’s equation solution can be constructed in explicit form, or the approximate solution with the help of numerical methods. However, in building the approximate solution, the additional conjugation conditions for solution and its derivatives should be introduced on characteristics.
ШЫРЫНЯ ЗАБАРОНЕНАЙ ЗОНЫ І АЖЭ-РЭКАМБІНАЦЫЯ Ў СВЯТЛОДЫЁДАХ НА АСНОВЕ GaInAsSb ПРЫ ТЭМПЕРАТУРАХ 10–300 К
Get PDFThe parameters for a temperature dependence of the band gap in the temperature range of 10–300 K and for a temperature dependence of spin-orbit splitting energy were obtained using the experimental emission spectra for LEDs based on Ga1–xInxAsySb1–y/AlGaAsSb heterostructures. For temperatures of 10–80 K, the rise of the emission intensity is limited by the Auger recombination process, for which the recombination energy of an electron-hole pair is transferred to a hole with its transition to the spin-orbital band. With an increase in a temperature of more than 100 K, there is a rise of the coefficient of the Auger recombination process, for which the energy released by the recombination of an electron-hole pair excites another electron in the conduction band. The sum of these processes results in quenching the LED emission with increasing temperature over 150 K.
О НЕСТАЦИОНАРНОМ РАСПРЕДЕЛЕНИИ ВЕРОЯТНОСТЕЙ СОСТОЯНИЙ МАРКОВСКОЙ СЕТИ С БЕСКОНЕЧНОЛИНЕЙНЫМИ СИСТЕМАМИ ОБСЛУЖИВАНИЯ В УСЛОВИЯХ ВЫСОКОЙ Н
Get PDFThe object of research is the Markov queuing network with infinite-server queues. The disciplines of the customer’s service in queuing systems (QS) are FIFO (first come – first served), service rates of customers are distributed exponentially with their own rates for each QS in each line of QS. The purpose of the research is to obtain sufficient conditions for representability of non-stationary state probabilities of such a network operating within the heavy-traffic regime in the multiplicative form. In the introduction, the field of applications of Markov networks with infinite-server queues has been described; the relevance of this work has also been indicated; a brief overview of the previous results on this subject has been given. In the main part, the network has been shown; the system of Kolmogorov’s difference-differential equations for the state probabilities of the network conditions has been derived. The main result of this article is as follows, i.e. the multiplicative form of the non-stationary state probabilities of the above-mentioned Markov network operating within the heavy-traffic regime is formulated and proved as a theorem. The obtained results can be used for modeling the behavior of information and computer systems and networks, transportation systems, insurance companies, banking networks and other facilities, the stochastic models which are the queuing networks.
ИССЛЕДОВАНИЕ ТОНКИХ ПЛЕНОК Cu2ZnSnSe4 МЕТОДОМ АТОМНО-СИЛОВОЙ МИКРОСКОПИИ
Get PDFIn comparison to the traditional use of glass substrates, the thin films onto metal substrates offer improved device cooling, economical large-scale roll-to-roll processing, and applicability in lightweight, as well as flexible products. However, unlike glass, metal foils tend to exhibit rough surfaces. This article studies the substrate-type (Mo/glass and Мо-foil) effect on the topographic characteristics of the Cu2ZnSnSe4 films by atomic force microscopy (AFM). Cu2ZnSnSe4 thin films were prepared by the electrodeposition of stack copper/tin/copper/zinc (Cu/Sn/Cu/Zn) precursors, followed by selenization. AFM wasused to study the topographic characteristics of thin films, including grain size, surface roughness, and maximum height of the profile. It is shown that the films obtained on Mo/glass and Mo-foil substrates have similar roughness and in the both cases the grain structure is formed. The Cu2ZnSnSe4 thin films show relatively high surface roughness and maximum roughness profile height compared to Cu-Zn-Sn precursors. The increase in the surface roughness of the films was caused by the growth of grains during annealing and selenization processes.Методом атомно-силовой микроскопии исследовано влияние типа подложки на структуру и шероховатость поверхности пленок Cu2ZnSnSe4, полученных методом селенизации металлических прекурсоров Cu-Zn-Sn на подложках из стекла с подслоем молибдена и молибденовой фольги (Мо/стекло, Мо-фольга). Обнаружено, что пленки Cu2ZnSnSe4 на подложках Мо/стекло и Мо-фольга имеют близкие значения шероховатости и зернистую структуру. Пленки Cu2ZnSnSe4 имеют более высокие значения шероховатости и максимальной высоты неровности профиля, чем металлические прекурсоры Cu-Zn-Sn. Увеличение шероховатости при формировании пленок Cu2ZnSnSe4 из прекурсоров происходит за счет роста зерен в процессе отжига и селенизации
СУЩЕСТВОВАНИЕ ИЗМЕРИМЫХ СОГЛАСОВАННЫХ СЕЛЕКТОРОВ МНОГОЗНАЧНЫХ ОТОБРАЖЕНИЙ
Get PDFThe present article is devoted to considering measurable set-valued random functions that are adapted to a fixed filtration of σ-algebras and the values of which are closed subsets of some complete separable metric space. For such functions, a criterion of measurability and adaptation is proved, which is analogous to Castain’s well-known criterion of measurability of set-valued functions. A theorem on existence of measurable and adapted selectors of set-valued random functions, which approximate some measurable adapted random function, is obtained. This theorem is improved in the case of set-valued functions with compact values. The generalization of Filippov’s theorem about the inverse function to the set-valued measurable random functions is proved. The obtained results can be useful both for proving the existence and for considering the properties of the solutions of stochastic differential inclusions.В настоящей статье рассматриваются измеримые многозначные случайные отображения, согласованные с заданным потоком σ-алгебр, значениями которых являются замкнутые подмножества некоторого полного сепарабельного метрического пространства. Для них установлен критерий измеримости и согласованности, аналогичный известному критерию Кастэна измеримости многозначных отображений. Доказана теорема о существовании у случайных многозначных отображений измеримых и согласованных селекторов, с заданной точностью аппроксимирующих некоторую однозначную измеримую и согласованную случайную функцию. Данная теорема усилена в случае, когда рассматриваемое многозначное отображение принимает компактные значения. Доказана теорема, обобщающая на многозначные измеримые случайные отображения теорему Филиппова об обратной функции. Полученные результаты могут быть использованы при доказательстве существования и исследовании свойств решений стохастических дифференциальных включений.