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Находится множество всех невырожденных двухточечных инвариантов групп движений некоторых феноменологически симметричных двумерных геометрий (плоскости Гельмгольца, псевдогельмгольцевой, дуальногельм-гольцевой и симплициальной плоскостей). Для решения соответствующих функциональных уравнений применяется аналитический метод., In G.G. Mikhaylichenko's classification, along with the well-known geometries, such as the Euclidean plane, Minkowsky plane, two-dimensional sphere, and others, there are two-dimensional Helmholtz type geometries in which the circle does not have the usual pattern, as evidenced by Helmholtz in his work "On the Facts Underlying Geometry," as well as the sim-plicial plane. All these geometries are endowed by group and phenomenological symmetries. The essence of the phenomenological symmetry is in the link between all the mutual distances for a finite number of points. The paper describes a complete system of non-degenerate two-point invariants of groups of motions for some phenomenologically symmetric two-dimensional geometries (Helmholtz plane, pseudo-Helmholtz plane, dual-Helmholtz plane, and simplicial plane) as a solution of corresponding functional equations for a set of two-point invariants of transformation groups. The paper found that every two-point invariant of motion groups of the aforementioned geometries coincides with the metric function of the corresponding plane up to a smooth transformation у( f) f.

Представлены результаты математического моделирования течения в пламени, образующемся при горении дизельного топлива и экспериментальные оценки масштабов турбулентных вихрей в пламени, полученные с применением методов термографии. Путем сравнения результатов численного моделирования и экспериментальных данных показано хорошее согласование основных термодинамических параметров пламени и масштабов турбулентных вихрей в нем., Diffusion combustion generated during combustion both in technological devices and in natural fires is usually implemented during the turbulent flow of combustion products in the flame. Nonstationarity of the process leads to the distortion of the flame shape, which provides surface area extension and combustion rate increase. Turbulence scale and the magnitude of pulsation of the parameters significantly affect the combustion mechanism in turbulent flows. It should be noted that in turbulent conditions the scale of turbulent pulsations and the mixing intensity significantly affect the flame shape, the combustion speed, the thermodynamic parameters of the process, the combustion completeness, and efficiency. The development of the thermography methods gives encouraging results for obtaining reliable temperatures of the flame. Thereby it is possible to visualize the temperature inhomogeneities. Based on the analysis of the flame radiation spectra with the application of high speed infrared cameras, it was found that the temperature in the flame changes repeatedly in time, and there are characteristic frequencies in the range of the temperature changes. These frequencies are caused by the movement of the flame temperature inhomogeneities associated with the structure of the flow. This paper presents results of mathematical modeling of the current in the flame generated during diesel fuel combustion, and experimental estimates of the scale of turbulent eddies in the flame. The results were obtained using the SIMPLEC algorithm and thermography methods. The paper includes the description of the experimental design and data processing. A detailed description of the system of equations used for the mathematical modeling is presented. Comparing the results of numerical simulation and experimental data shows a good correlation of the basic thermodynamic parameters of the flame and the scale of turbulent eddies in it.

Построены новые примеры ассоциированных контактных метрических структур (п, 4, ф, gJ) на 7-мерной единичной сфере S7. Для полученных структур установлено соответствие ассоциированных метрик gJ неинтегри-руемому семейству ассоциированных почти комплексных структур J в 3-мерном комплексном проективном пространстве CP3., In this paper, we construct new examples of associated contact metric structures (n, Ф, gJ) on the 7-dimensional unit sphere S7, other than standard. The construction involved a Hopf bundle n: S7^CP3. This projection maps affinor ф into an almost complex structure J. Therefore, it became necessary to build new examples of associated almost complex structures J in the 3-dimensional complex projective space CP3. Let Ф be a nondegenerate 2-form (a Fubini-Study form). An almost complex structure J is called positively associated with the form Ф if the following conditions are satisfied for any vector fields X, Y: Ф(JX, JY) = Ф^,Y) and Ф^, JX) > 0, if X * 0. Each positively associated almost complex structure J defines a Riemannian metric gJ by the equality g(X,Y) = Ф^, JY); the metric is also called associated. The associated metric has the following properties: g(JX, JY) = g(X, JY), g(JX,Y) = Ф(X,Y). The positively associated almost complex structure can be obtained as follows: J = J 0(1 + R)(1 R) -1, where R is a symmetric endomorphism R: TCP3 ^ TCP3 anticommuting with the standard structure J0, J ~{U 0J° 4 0 -iI In this paper, we have found a series of matrices R satisfying these conditions. Each matrix of this kind defines an associated almost complex structure in the space CP3. One of these matrices, ' 0 н!л 1 R R 0 (1+1 w |)4 f-1 2 3 n n Л www 0 0 r\ 1-2 3 n 0 www 0 0 0 w1w2w where the block Ra =, has been considered in more detail. For this endomorphism, the relevant almost complex structure J and a Hermite metric g have been found in the space CP3. It has been verified that the constructed structure J is not integrable.

При исследовании абелевых групп большое значение имеет свойство гомоморфизмов, отображающих подгруппы данной группы в саму группу, продолжаться до эндоморфизма всей группы. Так, например, (вполне) транзитивные группы без кручения можно определить как группы, в которых все (гомоморфизмы) сохраняющие высоты элементов гомоморфизмы из любой сервантной подгруппы ранга 1 в саму группу продолжаются до (эндоморфизмов) автоморфизмов всей гуппы. Приведены некоторые эквивалентные условия выполнимости свойств для группы быть (вполне) транзитивной, эн-дотранзитивной или слабо транзитивной. Рассмотрены связи между этими понятиями. Известно, что прямое слагаемое вполне транзитивной группы будет вполне транзитивной группой. Существуют транзитивные р-группы, которые имеют нетранзитивное прямое слагаемое. В то же время остаётся открытым вопрос: «Замкнут ли класс транзитивных групп без кручения относительно взятия прямых слагаемых?» Предлагаются некоторые необходимые и достаточные условия, при которых прямое слагаемое произвольной транзитивной группы будет транзитивной группой. Хорошо известен критерий Корнера о (вполне) транзитивности редуцированной р-группы. Ниже данный результат обобщается на произвольные редуцированные абелевы группы., In the study of Abelian groups, the fact that homomorphisms mapping subgroups of a group into this group can be extended to an endomorphism of the whole group is an important property of homomorphisms. For example, (fully) transitive torsion-free groups can be defined as groups in which all (homomorphisms) height-preserving homomorphisms from any pure rank 1 subgroup into this group are extended to (endomorphisms) automorphisms of the group. In this paper, some equivalent feasibility conditions for a group to be (fully) transitive, endo transitive, or weakly transitive are given. Relations between these notions are also shown. It is easy to show that a direct summand of a fully transitive group is a fully transitive group. There exist transitive p-groups which have a nontransitive direct summand. At the same time, the question whether the class of torsion free transitive groups is closed with respect to taking direct summands remains open. In this paper, some necessary and sufficient conditions under which a direct summand of an arbitrary transitive group is a transitive group are proposed. There is a well-known Corner's criterion on (full) transitivity of a reduced p-group. Below, this result is generalized to arbitrary reduced Abelian groups.

Применение формулы Кристоффеля Шварца ограничено трудностями в нахождении прообразов вершин исходного многоугольника. Для определения констант Кристоффеля Шварца использован метод П.П. Куфарева, который сводит проблему нахождения констант к более легкой задаче численного интегрирования обыкновенных дифференциальных уравнений. Описана численная реализация метода П.П. Куфарева и даны примеры гидродинамических расчетов, позволяющие оценить качество представленной математической технологии., Schwarz Christoffel's integral maps simple-connected polygonal domains onto the upper half-plain and is successfully used in various applications related to problems of hydrodynamics, electrodynamics, and elasticity theory. The main difficulty in the practical use of this integral is in the determination of parameters entering into it. This paper investigates P.P. Kufarev's method (1947) for the numerical determination of parameters involved in the Schwarz-Christoffel transformation. In this method, the difficult problem of determining parameters is reduced to an easier one, namely, to numerical integration of a system of ordinary differential equations. These equations describe the motion of preimages of polygon vertices. The paper presents an analysis of Kufarev's equations and describes properties of this transformation. The problem of initial conditions and a way to solve it are also considered. Determining constants in the Schwarz-Christoffel integral by Kufarev's method has a reliable theoretical basis, and the properties of this method are corroborated by calculations.

Проведён анализ решений многомерного уравнения в частных производных первого порядка, содержащего степенные функции от производных. Для исследования данного уравнения применяется метод двухуровневого функционального разделения переменных (РП), являющийся новым вариантом метода функционального РП. В результате получены новые решения рассматриваемого уравнения в неявной форме, содержащие некоторые обобщенные полиномы от независимых переменных., Investigations of nonlinear partial differential equations of the first order with an arbitrary number of independent variables are an important part of up-to-date mathematical physics. For many equations of this class, only solutions of the simplest kind are known, in particular, solutions of the travelling wave type. The present work is devoted to finding solutions of a more complex form for the multi-dimensional equation of the first order with power-law non-linearity in derivatives. To solve this problem, in this paper we propose a new variant of the method of separation of variables the method of two-level functional separation of variables. The characteristic feature of this method is that the desired function depends on a superposition of functions of the first and second levels of one variable, and these functions are determined as the result of solving some ordinary differential equations. Based on the method proposed in the paper, new exact solutions of the considered equation are obtained in an implicit form. The solutions contain some generalized polynomials of independent variables. Conditions of the existence of these solutions are specified. The results of this work can be generalized to other non-linear first order equations and equations of higher orders with many independent variables.

В марте 2013 г. International Journal of Mathematics and Mathematical Sciences опубликовал статью «Perpendicularity in an Abelian Group». Основная цель статьи введение понятия бинарного отношения ортогональности в произвольной абелевой группе. Нами полностью исследованы прямые суммы групп Z 3 ⊕ Z 3 и Z 5 ⊕ Z 5, построены все их ортогональности. Также нам удалось обобщить полученные результаты для прямых сумм циклических групп Z p ⊕ Z p., More and more geometric concepts penetrate into the space of the study of algebra^ On March 2013, the International Journal of Mathematics and Mathematical Sciences carried an article titled "Perpendicularity in an Abelian Group" written by Haukkanen et al The main objective of that article was to introduce the concept of a binary relation of orthogonality in an arbitrary Abelian group^ The ratio of orthogonality in different algebraic structures has already aroused the interest of mathematicians before^ For example, orthogonality in the rings was studied by Davis in the paper "Rings with orthogonality relation" (Bulletin of the Australian Mathematical Society, vol 4, 1971), Veksler considers orthogonality in lattices and lattice-ordered groups in "Linear spaces with disjoint elements and their conversion into vector lattices" (Leningrad^ Gos^ Ped^ Inst Uchen Zap^, vol 328, pp^ 19-43, 1967) The purpose of the present paper is to get some results about orthogonalities for specific Abelian groups^ We have investigated in detail the orthogonalities of a direct sum of cyclic groups Z p © Z. On the way to answer how many and which orthogonalities this group has, we consider special cases, namely, Z 3 © Z 3, Z 5 © Z 5 and only after that we are able to summarize the results to an arbitrary group Z p © Z · Haukkanen and others introduced the concept of orthogonality in an Abelian group with the help of axioms which are absolutely natural if we give a geometric interpretation to them^ Let G = (G, +) be an additive Abelian Group^ Let 1 be a binary relation in G satisfying: (А1) У a e G : 3b e G : a 1 b, (А2) У a e G\{0}: a / a, (А3) У a, b e G : a 1 b ^ b 1 a, (А4) У a, b,c e G : a 1 b л a 1 c ^ a 1 (b + c), (А5) У a, b e G : a 1 b ^ a 1 -b We call 1 an orthogonality in G^ Definition. We call _L maximal orthogonality in G if it satisfies the axioms (Al) (A5) and after adding thereto any of the other pair, the ratio obtained is not orthogonality in G^ We explore orthogonalities of groups Z 3 © Z 3, Z 5 © Z 5 and consider the general case of a direct sum Z © Z. With the help of our research, we were able to find the number of elementary, maximal, and all possible orthogonalities in the studied groups^ We present our results in the following table^ Group The number of elementary orthogonalities The number of maximal orthogonalities The number of all possible orthogonalities Z 2 © Z 2 3 3 3 z 3 © z 3 6 3 9 z 5 © z 5 15 15 75 Z p © Z p, p > 3 C 2 Cp+i Cp+1 • Cp-1 • Cp-3 • ··· • C4 • C2 (P+ 1)'' p-i 2 C 2 • C 2 • • C 2 Y Cp+1 Cp-1 ··· CP-(2i-1) h (i+1)!

Представлены программно-аппаратные комплексы для анализа гранулометрического состава и проницаемости горных пород, один из которых основан на комплексировании геометрического и гидродинамического методов гранулометрического анализа горных пород, что позволило учитывать коэффициент формы, в том числе и отличной от шарообразной. Другой из рассматриваемых комплексов позволяет проводить контроль проницаемости, основанный на комплексировании данных, получаемых с различных датчиков., There are many methods of control for characteristics of an object in oil and gas industry. Similar characteristics of the object can be controlled by a variety of methods each of which produces an error inherent in the implementation of the principle of an appropriate method, which reduces the reliability of the data. Existing control methods do not provide a complete description of characteristics (such as particle size, shape, and permeability of rocks) that need to be considered in the development and operation of oil and gas fields. In this paper, it is proposed to increase the reliability of the characteristics using the method of combination. Based on the analysis of scientific and methodological researches and combination techniques in different application areas, a classification of combination methods in the aspect of operation techniques is proposed and a cluster characteristic of combination objects with a description of their properties is presented. This classification permits one to choose one or another combination model depending on purposes of the solved problem. This paper presents the hardware and software systems for the analysis of particle size distribution and permeability of rocks. These complexes permit one to control for the particle size with allowance for the shape factor and determine the permeability in low-permeability rocks by enhancing the range specified by the liquid flow (0.0001-30 ml/min). The designed device for monitoring the particle size distribution is based on a combination of geometric and hydrodynamic methods, which makes it possible to measure the coefficient of the particle shape, including other than spherical. As a result, the device allows measuring the particle size in the range of 1-60 microns, repeatability, and reliability of results within the error of 5% (by Student''s t-test), the accuracy of the created complex increased by 2% with respect to the particle size distribution regardless of the shape. The proposed device allows one to find the median of the obtained distribution and specific surface area of particles in accordance with their shapes, differential particle size distribution, cumulative particle size distribution function, weight percentage of particle fractions, particle number percentage, and formfactor for each measurement range. The author of the article also proposes a device allowing one to determine the permeability in low-permeability rocks based on a combination of data from different sensors (pressure, differential pressure, and temperature) in a single system for collecting and analyzing information. This allows one to automatically maintain the pressure (for any sensors, depending on the method) in the experiments.

Исследованы свойства хороших колец обобщенных матриц. Показано, что любая обобщенная матрица есть сумма диагональной и обратимой обобщенных матриц. Получено одно условие k-хорошести произвольного кольца обобщенных матриц., This paper concerns properties of k-good formal matrix rings Kn of order n with rings R1, R2,..., R n on the main diagonal and R rR,-bimodules My on other places. In the ring theory, various matrix rings play an important role. Above all I mean formal matrix rings. Formal matrix rings generalize a notion of matrix ring of order n over a given ring. Every ring with nontrivial idem-potents is isomorphic to some formal matrix ring. The endomorphism ring of a decomposable module also is a formal matrix ring. The studies of such rings are quite useful for solving some problems on endomorphism rings of Abelian groups. In this paper I show that every matrix form Kn is the sum of diagonal matrix and invertible matrix. Also I give one condition when Kn is the k-good ring.

Исследуются возникающие в задачах теплопроводности двумерные граничные интегральные уравнения, операторные ядра которых выражаются через пространственно-временную С 0 -полугруппу. При условии 5Q е C k + 2 доказана устойчивая разрешимость интегральных уравнений в пространствах k раз непрерывно дифференцируемых на границе Sfl векторных функций со значениями в пространствах типа Соболева, определяемых степенями генератора C 0 -полугруппы., In this paper, we study two-dimensional vector boundary Fredholm integral equations of the second kind with an operator kernel expressed in terms of a spatial-temporal C 0 -semigroup. Such two-dimensional integral equations allow one to obtain solutions of vector boundary value problems of the first, second, and third kind for linear differential-operator equations Д 2и = Bu in a planar bounded simply connected domain Q+ or its exterior Q = R 2 \ Q+. The operator coefficient B is a generator of the C 0 -semigroup in space L 2(I Y x I T) (I Y = [0,Y], I T = [0,T]). In turn, these boundary value problems are possible formulations of initial boundary value problems of heat conduction on the time interval I T in a homogeneous cylinder О+ х I Y or х I Y with inhomogeneous boundary conditions of the first, second, and third kind on the lateral surface of the cylinder, zero boundary conditions of the first, second, or third kind (depending on the operator B) on the cylinder bases and zero initial conditions. The main result of this paper is as follows: under condition дО е C k+ 2, the spaces C k(дО, H n B (I Y х I T)) are invariant with respect to direct and inverse operators of the integral equations, and such operators are bounded in these spaces. Here, C k(дО, H n B (I Y х I T)) is the space of vector functions, k times continuously differentiable on the border DO. with values in the Sobolev type space H n B (I Y х I T) defined by powers n + 1 of the operator B.

Для проведения расчетов в обоснование надежности работы ТВЭЛов необходимо знать как параметры микроструктуры распределяются по объему таблетки. Было предложено использовать штатные методы контроля микроструктуры с последующим пересчетом характеристик в 3D. В данной работе определены требования к исходным данным, описан алгоритм пересчета в 3D, оценены возможные ошибки и пути их минимизации. В статье приведены результаты исследования и пересчета пористости топливных таблеток, изготовленных из различных порошков., In order to verify reliability of fuel assembly exploitation, it''s necessary to know the volumetric distribution of microstructure parameters in 3D. In this paper, we propose to use standard methods of microstructure analysis with further recalculation of characteristics in 3D. This paper represents initial data requirements, algorithm of 3D-conversion, possible errors, and ways of their minimization. Results of the investigation and conversion of porosity of fuel pellets produced from different powders are presented.

Приведены результаты экспериментальных исследований по определению силы лобового сопротивления и коэффициента лобового сопротивления двухлопастного ветродвигателя на основе эффекта Магнуса с гладкими вращающимися цилиндрами в диапазоне скоростей воздушного потока 4-13 м/с (Re = 26800-105000) при постоянном числе вращения цилиндра вокруг собственной оси. Результаты показали, что увеличение числа Рей-нольдса приводит к уменьшению коэффициента лобового сопротивления, так как коэффициент лобового сопротивления прямо пропорционален силе лобового сопротивления, однако обратно пропорционален квадрату скорости., In this paper, we consider aerodynamic characteristics of two rotating cylinders transversely streamlined by an air flow. The paper presents the results of experimental studies on determining the force of drag and drag coefficient of a two-bladed wind turbine based on the Magnus effect with smooth rotating cylinders in the range of air flow velocities from 4 to 13 m/s (Re = 26800105000) at a constant number of the cylinder rotation around its own axis. From the experimental data, the optimum value for the development of wind turbines operating based on the Magnus effect is determined. The results showed that an increase in the Reynolds number leads to a decrease in the drag coefficient since the drag coefficient is proportional to the force of drag but inversely proportional to the square of the speed value. It is shown that the coefficient of drag and the coefficient of lift depend on the Reynolds number and the number of revolutions of the cylinder. Conditions under which the Magnus effect helps to maximize the lifting force and, consequently, to increase the efficiency of the wind turbine are experimentally determined.

Рассматривается численное моделирование закрученного ламинарного течения вязкого газа в вихревой камере, которая используется для разделения тонкодисперсных порошков на мелкую и крупную фракции по заданному размеру частиц. Для повышения эффективности процесса фракционного разделения частиц предлагается дополнительно закручивать стенки вихревой камеры, что способствует выравниванию поля окружной скорости и тем самым стабилизирует баланс центробежных и аэродинамических сил, действующих на сепарируемые частицы. При такой постановке задачи существенное влияние оказывает выбор геометрических параметров вихревой камеры. Достоверность результатов и выводов, представленных в работе, основывается на тестовых расчетах, а также на сопоставлении численных решений, полученных в переменных вихрь функция тока окружная составляющая скорости и в переменных скорость давление., In this paper, numerical modeling of the swirling laminar viscous flow in the vortex chamber used for the separation of fine powders into small and large fractions by a given particle size is considered. To enhance the efficiency of particle separation, it is proposed to additionally tighten the walls of the vortex chamber to equalize the circumferential velocity field and thereby to stabilize the balance of centrifugal and aerodynamic forces acting on the separated particles. In this formulation of the problem, the choice of geometrical parameters of the vortex chamber is of great importance. Reliability of the results and conclusions presented in the paper is based on test calculations, as well as on a comparison of the numerical solutions obtained in the vortex stream function circumferential velocity variables and in the speed-pressure variables.

Рассматривается топологическое пространство S A, которое является модификацией прямой Зоргенфрея S и определяется следующим образом: если точка x е A с S, то базой окрестностей точки x является семейство полуинтервалов {[a,b): a,b е R,a A гомеоморфно пространству S. Кроме того, получено, что пространство S A гомеоморфно пространству S для любого замкнутого подмножества A с К.. Подобные вопросы рассматривались в работе V.A. Chatyrko, Y. Hattori, где топология «стрелки» на множестве A заменялась на евклидову топологию., In this paper, we consider a topological space S A which is a modification of the Sorgenfrey line S and is defined as follows: if a point x e A с S, then the base of neighborhoods of the point x is a family of intervals {[a, b): a, b e R, a A is homeomorphic to the space S. In addition, it was found that the space S A is homeomorphic to the space S for any closed subset A с К.. Similar problems were considered by V.A. Chatyrko and Y. Hattori in [4], where the "arrow" topology on the set A was replaced by the Euclidean topology. In this paper, we consider two special cases: A is a closed subset of the line in the Euclidean topology and the closure of the set A in the Euclidean topology of the line is countable. The following results were obtained: Let a set A be closed in R. Then the space S A is homeomorphic to the space S. Let a countable set A с К. be such that its closure A is countable relatively to R. Then S A is homeomorphic to S. Let A be a countable closed subset in S. Then S A is homeomorphic to S.

Показывается, что координаты частиц жидкости и давление можно выразить через одну произвольную функцию у так, что условие несжимаемости будет выполнено при любом ее выборе. Введение такой функции в качестве искомой величины существенно облегчает получение аналитических и численных решений уравнений гидродинамики, записанных в переменных Ла-гранжа., The paper shows that coordinates of liquid particles and pressure can be expressed in terms of one arbitrary function у so that the incompressibility condition is satisfied for any choice of this function. Introducing this function features as the unknown one significantly simplifies obtaining analytical and numerical solutions of the hydrodynamic equations written in Lagrangian variables. An incompressible flow can be written in Eulerian or Lagrangian variables. Both forms of these equations have been known for a long time but scientists usually prefer to use the Euler variables. This is explained by the unusualness of Lagrange equations. They include nonlinear terms in a form that is inconvenient for numerical and analytical calculations. Until now, hydrodynamicists did not try to exclude the incompressibility condition from the Lagrange equations with the aim of reducing the number of unknown variables. Therefore, in this paper we show that the incompressi-bility condition can be satisfied automatically if the particle coordinates x(a, b, t) andy(a, b, t) are expressed in terms of the same arbitrary function of coordinates and time. In Lagrangian variables, such a function plays the same role as the function of the current in Euler variables. In this paper, as an example of the exact solution, a solution of the problem of standing waves in a liquid layer is presented. The problem is solved using Lagrange variables. To do this, it is necessary to select an area of the Lagrangian variables in the form of an infinite strip the lower edge of which corresponds to a solid wall. Similarly, the problem is solved for a traveling wave.

Продолжается развитие геометрического метода для изучения свойств отображений с s-усредненной характеристикой, основанного на специальном характеристическом законе искажения модулей семейств кривых., In this paper we continue to develop the geometric method of studying properties of space mappings with an s-averaged characteristic. The method is based on the characteristic distortion law for modules of families of curves. In recent decades, the theory of mappings with bounded distortion in the n-dimensional Euclidean space is one of the most meaningful and intensively developed branches of the function theory. These mappings were introduced and systematically studied in works by Yu.G. Reshet-nyak, published since 1966. A part of Yu. G. Reshetnyak''s results is contained in [1]. The most powerful tools used in the study of space mapping properties are methods that study invariance properties of conformal capacity or the module of families of curves. The equivalence of analytical and geometrical (expressed in terms of the conformal capacity of condensers) definitions of mappings with bounded distortion was introduced by O. Martio, S. Rickman and J. Vaisala [2]. In [3], the equivalence was proved for definitions of homeomorphic mappings with distortion bounded on the average. The equivalence for mappings nonhomeomorphic with distortion bounded on the average was considered in [4]. In the presented paper, we give a geometric definition of mappings with an s-averaged characteristic using the concept of a spherical p-module of a family of curves. This definition can also be interpreted as a generalization of the method of modules for mappings with an s-averaged characteristic. We also study these mappings properties and prove the equivalence of the geometric and analytic definitions using the distortion theorems.

Представлена математическая модель, описывающая нестационарный процесс горения сверхбогатой метановоздушной смеси в реакторе фильтрационного горения газов для получения синтез-газа с учетом эффектов дисперсии и межфазного теплообмена. Численно исследован состав продукта, температура в зоне горения, скорость распространения волны фильтрационного горения в широком диапазоне параметров процесса в диапазоне давлений от 1 до 8 атм. Проведено сравнение результатов моделирования с экспериментальными данными., The paper presents a mathematical model describing the non-static process of ultrarich methane air mixture combustion in a filtration gas combustion reactor with allowance for elects of gas dispersion and interphase heat exchange. To consider the fluid-solid interaction, we performed a decomposition of the calculation zone into two domains. The dependence of gas and solid physical properties on temperature was included in the consideration. The GRI 3.0 kinetics mechanism of methane combustion is used with allowance for the dependence of kinetics parameters on pressure. Intensification of diffusion processes through gas phase dispersion and radiation heat transfer are taken into account. The composition of reaction products, combustion temperature, and thermal wave velocity are numerically investigated for a wide range of process parameters. The influence of pressure on the combustion process is studied in a range from 1 to 8 atm. Numerical results are compared with the experimental data. The mathematical model presented in this work can be used for optimizing porous media reactors. In the paper, we show the possibility of porous media methane conversion under high pressure conditions. High performance optimum parameters are obtained.

На основе разработанных ранее режимов ориентации космических аппаратов (КА) «Ямал-200» с использованием модели инерциальных измерителей построен режим, в котором угловые параметры получаются в результате обработки сигналов бортового ретрансляционного комплекса. Предложена схема формирования угловых параметров, структура и схема программно-аппаратного комплекса оценки углов ориентации. Приведены данные по реализации работы этой системы на реальном изделии., On the basis of previously developed attitude control modes of Yamal-200 spacecrafts using inertial meters model, a mode was built that derives angular parameters from onboard repeater complex (ORC) signal processing. This paper proposes angular parameters generating circuit, structure and diagram of software & hardware package for the estimate of attitude control angles. This paper provides data on the implementation of this system in an actual product.

Two Tychonoff spaces X and Y are said to be l-equivalent (u-equivalent) if Cp(X) and Cp(Y) are linearly (uniformly) homeomorphic. N.V.Velichko proved that the Lindelцf property is preserved by the relation of l-equivalence. A. Bouziad strengthened this result and proved that the Lindelцf number is preserved by the relation of l-equivalence. In this paper the concept of the support different variants of which can be founded in the papers of S.P. Gul'ko and O.G. Okunev is introduced. Using this concept we introduce an equivalence relation on the class of topological spaces. Two Tychonoff spaces X and Y are said to be fu-equivalent if there exists an uniform homeomorphism h: Cp(Y) → Cp(X) such that supphx and supph-1x are finite sets for all x∈X and y∈Y. This is an intermediate relation between relations of uand l-equivalence. In this paper it has been proved that the Lindelцf number is preserved by the relation of fu-equivalence., Two Tychonoff spaces X and Y are said to be l-equivalent (u-equivalent) if CP(X) and Cp(Y) are linearly (uniformly) homeomorphic. N.V. Velichko proved that the Lindelцf property is preserved by the relation of l-equivalence. A. Bouziad strengthened this result and proved that the Lindelцf number is preserved by the relation of l-equivalence. In this paper the concept of the support different variants of which can be founded in the papers of S.P. Gul'ko and O.G. Okunev is introduced. Using this concept we introduce an equivalence relation on the class of topological spaces. Two Tychonoff spaces X and Y are said to be fu-equivalent if there exists an uniform homeomorphism h: Cp(Y) Cp(X) such that supph x and supph-1x are finite sets for all x∈X and y∈Y. This is an intermediate relation between relations of uand l-equivalence. In this paper it has been proved that the Lindelцf number is preserved by the relation of fu-equivalence.

Определены отношения ширины к глубине канала шнека экструдера, при котором является обоснованным предположение об отсутствии влияния боковых стенок на профиль скорости циркуляционного течения в серединном сечении канала, который считается прямоугольным. Приводится методика получения решения для профиля вектора скорости псевдопластичной жидкости без учета влияния боковых стенок. Достоверность результатов подтверждается сравнением с данными, представленными в литературе. Для численного решения с учетом влияния боковых стенок используется непрямой метод граничных элементов. Определено отношение ширины к глубине канала шнека, при котором допустимо пренебрегать влиянием боковых стенок на течение в середине канала., This paper is devoted to definition of a width-to-depth ratio of a single-screw extruder channel when sidewalls do not affect the velocity profiles of circulatory high-viscosity non-Newtonian fluid flow at the mid of the channel. The channel has a rectangular cross-section. Power-law model is used to describe fluid behavior. The indirect boundary element method is used for numerical solution taking into account of sidewalls. Comparing of obtained velocity profiles with the known results showed a good agreement. Research is performed in the range of power-law index from 0.4 to 1.0. The method for obtaining the velocity component profiles for the case of shear-thinning fluid flow without considering influence of sidewalls is presented. The width-to-depth ratio of a single-screw extruder channel when it is acceptable to neglect influence of side-walls on flow at the mid of the channel is defined.

Теоретически рассмотрено влияние толщины золотого покрытия на диаграмму изгиба вольфрамовой микропроволоки диаметром 15 мкм. Показано, что покрытие толщиной примерно до 2 мкм практически не сказывается на диаграмме изгиба вольфрамовой микропроволоки, а следовательно, практически не должно сказываться на физико-механических свойствах отражающей поверхности антенны, связанной из такой микропроволоки., Knitted materials made of a microwire are widely used in space systems as reflecting surfaces of transformable parabolic reflector antennas. To increase the reflecting properties and improve the radio-technical characteristics of the antenna, the microwire is usually covered with a gold layer of about 0.2 microns thick. However, according to the theoretical calculations and experimental investigations, when a large antenna hits streams of stationary plasma engines of the satellites, the dispersion (erosion) of the gold covering occurs. The erosion magnitude can reach several microns. Since the covering thickness which defines the reflection factor of the radar-reflecting surface is about one-tenth of a micron, the erosivity of the specified streams can significantly reduce the efficiency of the antenna. In this regard, an increase in the coating thickness to at least 1 micron or even more is required. Since large antennas are knitted from a tungsten microwire with 15 microns in diameter, it is necessary to estimate the effect of the gold covering of 1 micron thick or more on the bending moment of such microwires, shape recoverability after a bend, and permanent deformation after a plastic bending. In this paper, the effect of the gold coating thickness on the bending diagram of a tungsten microwire of 15 microns in diameter is theoretically considered. It has been revealed that the coating of about 2 microns thick has essentially no effect on the bending diagram of the tungsten microwire and, therefore, does not significantly affect the physical-mechanical properties of the reflecting surface of antennas knitted from such a microwire.

Известна полная классификация двумерных феноменологически симметричных геометрий. Она содержит как хорошо известные геометрии (евклидова, псевдоевклидова, симплектическая, сферическая и т.д.), так и неизвестные (собственно гельмгольцева, псевдогельмгольцева, дуальногельм-гольцева и симплициальная). Простой анализ доказывает однородность метрической функции псевдогельмгольцевой и дуальногельмгольцевой геометрий. Поэтому данные геометрии принадлежат классу финслеровых пространств. В данной работе применяются методы финслеровой геометрии для исследования псевдогельмгольцевой и дуальногельмгольцевой двумерных геометрий: проверяются финслеровы аксиомы, находится финслеров метрический тензор, финслеровы основной и дополнительный тензоры, вычисляются финслеров скаляр и специальный тензор кривизны., There exists the complete classification of two-dimensional phenomenologically symmetric geometries, i.e., geometries for which the six mutual distances between the four arbitrary points are functionally connected. In these geometries, the distance is understood in a generalized sense as the value of a function called the metric function. Axioms of a metric are not obligatorily satisfied. For all these geometries, groups of motion are three-dimensional. The classification of such two-dimensional geometries includes both well-known geometries (Euclidean, pseudo-Euclidean, symplectic, spherical, etc.), and unknown ones (the properly Helmholtz, pseudo-Helmholtz, dual Helmholtz, and simplicial geometries). In this paper, we use methods of Finsler geometry to study the pseudo-Helmholtz and dual Helmholtz two-dimensional phenomenologically symmetric geometries. In particular, in the first section, we introduce the definition of pseudo-Helmholtz and dual Helmholtz planes, and then prove that they are positive definite Finsler spaces (homogeneity and positivity of the metric function, as well as the positive definiteness of the Finsler metric tensor are verified), though, in contrast to the actual Helmholtz geometry, with some restrictions on the domain. In the second section, the psevdo-Helmholtz two-dimensional manifold is defined and it is proved that it is a positive definite Finsler space for |P| > 1 in a certain domain. Then, the metric tensor gy, basic Finsler tensor Cy, and additional tensor Ayk are calculated. With these tensors, the Finsler scalar J is obtained and it is proved that the special Finsler curvature tensor Siyki for the two-dimensional pseudo-Helmholtz manifold is zero. In the third section, the dual Helmholtz two-dimensional manifold is defined and it is proved that it is a positive definite Finsler space in the domain of definition. Then, as in the second section, the metric tensor, basic Finsler tensor Cy, and additional A iJk tensor are calculated. Then, it is proved that J = 2 and the special Finsler curvature tensor SiykI = 0.

Класс линий на параболоиде, введенный ранее автором по названием SG-линий, применен при раскрое сетеполотна для офсетного рефлектора. Рассмотрены два случая взаимного расположения осей родительского параболоида и вырезающего круглого цилиндра (оси либо параллельны, либо нет)., The term 'tailoring' means not only 'cutting' of a metallic mesh but also the method of its attachment to the supporting structures, which generally means a decrease in root-mean-square deviation (RMSD) of the reflector's real surface from the ideal surface of the parent paraboloid. The approach of tailoring offered in this paper is based on application of SG-lines. SG-lines showed up as a solution of a (non-formal) problem to find a class of lines lying on a paraboloid of revolution and satisfying the following conditions. Locally, they should be close to geodesic lines in some reasonable sense. They can be referred to the natural parameter without significant computational problems. They should be uniquely determined by its endpoints defined on the paraboloid. These lines are used for metallic mesh tailoring for an offset reflector. An algorithm the input values of which are the design parameters of the reflector and metallic mesh has been composed. The algorithm returns a set of petals with dimensions required for tailoring. The case in which the axis of the parent paraboloid is parallel to that of the cutting cylinder is considered, as well as the case of nonparallel axes. The computational experiment carried out for specific conditions shows an error in area of 0.3653%, and the error in perimeter is of 0.0745%.

Показано, что каждая вполне инертная подгруппа вполне разложимой группы G конечного ранга соизмерима с некоторой вполне инвариантной подгруппой тогда и только тогда, когда типы прямых слагаемых ранга 1 группы G либо равны, либо несравнимы, причем все прямые слагаемые ранга 1 группы G не делятся ни на одно простое число p., A subgroup H of an Abelian group G is said to be fully inert in G if the subgroup HfфH has a finite index in фH for any endomorphism ф of the group G. Subgroups H and K of the group G are said to be commensurable if the subgroup KfH has a finite index in H and in K. Some properties of fully inert and commensurable groups in the context of direct decompositions of the group and operations on subgroups are proved. For example, if a subgroup H is commensurable with a subgroup K, then H is commensurable with HfK and with H + K; if a subgroup H is commensurable with a subgroup K, then the subgroup fH is commensurable with fK for any homomorphism f. The main result of the paper is that every fully inert subgroup of a completely decomposable finite rank torsion-free group G is commensurable with a fully invariant subgroup if and only if types of rank 1 direct summands of the group G are either equal or incomparable, and all rank 1 direct summands of the group G are not divisible by any prime number p.

Доказывается, что для начального регулярного несчетного ординала т и произвольных начальных ординалов а, р, а, In this paper, we prove that for the elementary regular ordinal and arbitrary ordinals а, p, а

Описывается биологическая модель для воспроизведения динамики биомасс планктона в период развития весеннего речного термобара в глубоком озере. Численное моделирование гидробиологических процессов осуществляется с помощью модели Франкса и др. «нутриент фитопланктон зоопланктон». Проведена верификация разработанной модели на основе имеющихся в литературе результатов расчётов других авторов. На примере канадского озера Камлупс исследовано влияние притока отдельных биологических составляющих модели планктона за счёт стока реки Томпсон на их пространственно-временное распределение в водоёме., In this paper, a mathematical model for simulating the hydrodynamic and hydrobiological processes in a temperate water body during the evolution of the spring riverine thermal bar is described. A thermal bar is a narrow zone in a lake where the water, which has a maximum density, sinks from the surface to the bottom. Numerical simulation of the dynamics of plankton ecosystems in case of Kamloops Lake (British Columbia, Canada) is accomplished by using the nutrient phytoplankton zooplankton model of Franks et al. (1986). The hydrodynamic model, which includes the Coriolis force due to Earth's rotation, is written in the Boussinesq approximation with the continuity, momentum, energy, and salinity equations. Closure of the simultaneous equation system is performed with a two-parameter Wilcox k-ю turbulence model and algebraic relations for the coefficients of turbulent diffusion. The convection-diffusion equations are solved by a finite volume method to satisfy the integral conservation laws. The numerical algorithm for the flow and temperature fields' indication is based on a Crank Nicolson difference scheme. In the equations, the convective terms are approximated with the QUICK second-order upstream scheme. The systems of grid equations are solved by the under-relaxation method at each time step. The data from numerical experiments have shown qualitative agreement with results obtained by Holland et al. (2003). Simulations with the variable values of the concentrations of the biological components, coming from the Thompson River, have demonstrated that the high riverine nutrient concentrations do not play a significant role in dynamics of the phytoplankton and zooplankton biomasses; increasing of the phytoplankton in the river leads to a reduction of the nutrient at the location of the thermal bar, and the monotone growth of the riverine zooplankton incoming has a negative impact on the phytoplankton population.

Приведена математическая модель задачи о нелинейных колебаниях вязко-упругого трубопровода с протекающей через неё жидкостью. С помощью метода Бубнова Галеркина математическая модель задачи сводится к решению системы обыкновенных интегродифференциальных уравнений, решаемая численным методом исключения слабосингулярных особенностей в интегральных и интегродифференциальных уравнениях. Установлено, что для выявления влияния вязкоупругих свойств материала конструкций на колебания трубопровода, необходимо использовать слабосингулярные ядра наследственности типа Абеля., In this paper, mathematical models of nonlinear dynamic problems with fluid and gas flows through pipelines have been developed based on the Boltzmann Volterra integral models with weakly singular hereditary kernels. Using the Bubnov-Galerkin method for the boundary conditions, the resulting nonlinear integrodifferential equations with partial derivatives are reduced to solving systems of nonlinear ordinary integrodifferential equations with both constant and variable coefficients as functions of time. It is proposed to investigate oscillating processes occurring in a pipeline by a numerical algorithm for solving the nonlinear integrodifferential equations with weakly singular hereditary kernels, which is convenient for a computer implementation. On the basis of the developed computational algorithm, a complex of computer application programs allowing one to explore a completely new class of mathematic simulation problems, such as an oscillatory process of viscoelastic thin-walled pipelines with a large diameter, in terms of the shell theory is designed. The influence of a singularity in the hereditary kernels on oscillations of the construction with viscoelastic properties has been numerically investigated. When simulating the nonlinear problems, a number of new dynamic effects were explored. It was found that the determination of the effect of viscoelastic properties of the construction material on vibrations of the pipeline with a flowing liquid requires applying weakly singular hereditary kernels with an Abel type singularity.

Рассматривается задача оптимального управления для параболического уравнения с интегральным граничным условием и с управлениями в коэффициентах. Исследованы вопросы корректности постановки задачи, доказана дифференцируемость по Фреше функционала цели, найдено выражение для его градиента и установлено необходимое условие оптимальности., In this paper, an optimal control problem for a parabolic equation with an integral boundary condition and controls in coefficients is considered. Let it be required to minimize the functional i J(u) = f I м(X,T; и) У(x) |2 dx 0 on the solutions м = м( x, t) = м( x, t; o) of the boundary value problem ut (k (x, t )mx )x + q( x, t )m = f (x, t), (x, t) е QT ={(x,t):0 < x < 1, 0 < t < T} м (x,0) = ф(x), 0 < x < I, i mx(0,t) = 0, k(l,t)ux(I,t) = f H(x)mx(x,t)dx + g(t), 0 < t < T, 0 corresponding to all allowable controls о = o(x,t) = (k(x,t), q(x,t)) from the set V = {u( x, t) = (k (x, t), q( x, t)) е H = W21(QT) x L2(QT) :0 < v < k (x, t )< ц, |kx (x, t)0 are given numbers and y(x),ф(x)е №(0,1), H(x)е W2(0,1), f (x, t )е L2(Qt ), and g (t )е№2'(0,Г) are known functions. The work deals with problems of correctness in formulating the considered optimal control problem in the weak topology of the space H = W2(QT) x L2(QT). Examples showing that this problem is incorrect in the general case in the strong topology of the space H are presented. The objective functional is proved to be continuously Frechet differentiable and a formula for its gradient is found. A necessary condition of optimality is established in the form of a variational inequality.

Исследуется установившееся течение неньютоновской жидкости в плоском канале с внезапным расширением. Реологическое поведение среды описывается моделью Балкли Гершеля. Для нахождения стационарных полей скорости и давления используется численный алгоритм, в основе которого лежат метод установления и процедура SIMPLE. С целью ускорения расчета применяется технология параллельных вычислений MPI. Приводятся распределения основных характеристик потока с выделением зоны двумерного течения в области скачка сечения. Показано влияние основных параметров задачи на картину распределения квазитвердых ядер в потоке жидкости., In this paper, the steady-state flow of a non-Newtonian fluid in a planar channel with sudden expansion is investigated. The rheological behavior of this media is described by the Herschel Bulkley model. To determine the static velocity and pressure fields, a numerical algorithm based on the relaxation method and SIMPLE procedure are used. The MPI technique of parallel programming is used to accelerate the computation. Regularization of the rheological model is used to provide algorithm stability and limit viscosity increase at low deformation rates. The mathematical problem statement involves non-dimensional parameters: the Reynolds number, Bingham number (non-dimensional viscoplasticity parameter), and power-law index. We report results of numerical simulation in a range of 1 < Re < 40 for the Reynolds number, 0 < Se < 2 for the Bingham number, and 0.4 < k < 2 for the power-law index (shear thinning and shear thickening fluids). Main characteristic distribution of the fluid flow with a two-dimensional localization in the expansion zone is presented. The impact of main parameters of the problem on the dead zone distribution in the fluid flow is shown.

Разработан метод решения обратной задачи Кристоффеля Шварца конформное отображение наперед заданной многоугольной области на каноническую область единичный круг. В основе метода лежит применение обобщенной функции Жуковского для решения модельных задач отображения на верхнюю полуплоскость луночки, в общем случае заданной сектором произвольной области с гладкой границей в расширенной плоскости комплексного переменного. Работоспособность решения проверена на конкретных примерах., The method for solving the inverse Schwarz Christoffel problem conformal mapping of a given polygonal area on the canonical domain, the unit circle is developed in the paper. The method is based on the use of a sequence of conformal mappings related to the mapping of the polygon onto the upper half with discarded segments by a Cayley linear fractional transformation followed by sequential addition of the discarded segments to the upper half-plane. Model problems consider conformal mappings of circular, elliptical, and hyperbolic lunes to the upper half-plane. The solution of the presented model problems is based on the generalized Zhukovsky function; the obtained results expand the applicability of the Zhukovsky function as compared to existing methods of its application in the implementation of conformal mappings. The working ability of the solution was tested by specific examples in solving the problem of conformal mapping of quadrilateral and heptagonal domains to the unit circle. Recommendations for the algorithmic implementation of the method are presented.

Проведено теоретическое исследование процесса вытеснения метана из га-зогидратного пласта путем закачки углекислого газа в пласт, с последующим замещением метана из состава гидрата двуокисью углерода. Рассмотрен случай, когда интенсивность образования гидрата диоксида углерода лимитируется диффузией углекислого газа через образовавшийся гидратный слой между потоком газовой смеси и гидратом метана. Исследована динамика основных параметров процесса и расходов закачиваемого и выходящего углекислого газа и добываемого метана., In this paper, a numerical simulation of the process of methane displacement from a gas hydrate reservoir during the injection of carbon dioxide followed by methane substitution by carbon dioxide in the composition of the hydrate was performed. The process of gas replacement was assumed to occur in a stable field of methane hydrates and carbon dioxide with CO2 in a gaseous state. The intensity of carbon dioxide hydrate formation was determined by diffusion of carbon dioxide across the hydrate layer formed between a stream of the gas mixture and the methane hydrate in the porous channels. The general pressure of the system, the partial pressures of the carbon dioxide and methane, the saturations of hydrates of carbon dioxide and methane, and the temperature as functions of time were obtained. In the initial stage of the process, free methane was forced out, and subsequently the methane received after the gas replacement process was extracted. This resulted in a complete transition of methane hydrate into carbon dioxide hydrate and a complete extraction of methane from the porous reservoir.

Для апробации предложенной ранее методики профилирования кольцевых сверхзвуковых воздухозаборников изоэнтропического сжатия разработана конструкция открывающегося бокового секторного воздухозаборника. Проведены испытания воздухозаборника в аэродинамической трубе, в которых впервые осуществлен процесс открывания и закрывания бокового воздухозаборника в сверхзвуковом набегающем потоке. По фотографиям течения видно, что в исследованном диапазоне чисел Маха (М = 2.5-4) воздухозаборник после открывания находится в запущенном состоянии., In this paper, the practical implementation of the previously proposed design method of the annular supersonic air inlet with isentropic compression is considered. In the beginning, the contour of the annular supersonic nozzle is calculated. Here, the ideal gas is isentropically accelerated from M = 1 to M = 3. When the flow direction turns around 180 degrees, the walls of the nozzle and each flow line can be considered as compression surfaces for the supersonic air inlet. In this study, two flow lines with relative mass flow rate equal to 1.0 and 0.6 were chosen as inlet compression surfaces. Then, the theoretical contour was transformed into a real inlet profile in accordance with the given restrictions. The profile was transformed into four sectors with possibilities of both folding and unfolding inside and outside the central cylindrical body of 120 mm in diameter. The movement of the sectors is implemented by pneumatic cylinder at air pressure up to 10 MPa. The resulting air inlet model was tested in the wind tunnel at Mach number varying from M = 2.5 to M = 4. Over the course of experiments, the process of both inlet folding and unfolding in the supersonic air flow was demonstrated. The time of folding and unfolding was 0.25 and 0.05 s, respectively. According to the shadow visualization results in each inlet sector in the working position, the flow pattern complies with theoretical predictions.

Оценивается состояние низкоэнергетических молекул, находящихся вблизи кристаллической структуры открытой нанотрубки. Систематические расчеты проведены для трубок малых размеров, для которых установлено, что адсорбция молекул газа поверхностными кристаллами это не захват их частью поверхности нанообъекта, а вовлечение молекул в сложное орбитальное движение около частицы., In this paper, the state of low-energy molecules in the vicinity of an open nanotube's crystal structure is assessed. A considerable attention is paid to the correlation of the discrete and the continual approaches in the description of the interaction energy from carbon graphene structures. The small size tubes are conveniently modeled using the discrete approach. However, there exist big tubes containing hundreds of thousands of nodes in their crystal lattice, as well as systems of nanotubes including about a million carbon atoms. In this case, it is reasonable to use the continuum approach. In order to get that done, the energy of large tubes is obtained in the work by means of integration. However, the systematic calculations are carried out for tubes of smaller dimension for which it is found that adsorption of gas molecules by surface crystals is not a case of capturing them by a part of the nanoobject's surface but that of involving the molecules in a complex orbital motion around the particle.

Представлено дальнейшее развитие геометрического метода модулей семейств кривых для изучения свойств негомеоморфных пространственных отображений отображений с s-усредненной характеристикой. Обобщается теорема, известная для случая n = 2 как Iversen Tsuji's Theorem и доказываются характеристические свойства для сферического модуля семейств кривых, асимптотических для некоторого особого граничного множества., In this paper, we continue to develop the geometric method of modules of curve families for studying analytical and geometrical properties of nonhomeomorphic mappings with s-averaged characteristic. We consider the question of the erasure of special sets under mappings with s-averaged characteristic. In this work, in contrast to previous results which require that the mapping is homeomorphic or the capacity of singular points is zero, nonhomeomorphic mappings with s-averaged characteristic are considered and a weaker condition is taken as constraints. We generalize the theorem which is known in the case n = 2 as Iversen-Tsuji's theorem for the case n > 3. There are well-known examples demonstrating the existence of essential singularities for which Hausdorffs measure AB * 0 at some p * 0 for mappings with an s-averaged characteristic. The work presents some examples which illustrate distinctive properties of the considered class of mappings. A theorem about the module distortion for families of curves under mappings with allowance for multiplicity and, as a consequence, the characteristic property of the spherical module of families of curves asymptotic to a special boundary set is proved. The mappings are extended to continuous ones if the dimension of the set of singular points I dim I < n-2 and s > 1. The results are applicable to many classes of mappings of subclasses W„1(U).

Изучаются неэлементарные функции типа интегралов Френеля, определенные в виде степенных рядов с использованием возрастающих и центральных факториальных степеней. Установлены некоторые свойства этих функций, приведены их графики. Выведены обыкновенные дифференциальные уравнения, решениями которых являются новые функции., Replacing in the well-known series cosx = Y----, sinx = Y---falling n=0 (2n)! n=0 (2n +1)! factorial powers (m! = mm ) by rising and central factorial powers (mm and m[m] respectively), да (_1)nx2n да (_1)nx2n+1 да (_1)nx2n we obtain real functions Cos x = Y --, Sin x = Y ----=, Cosc x = Y --.-„-, n=0 (2n)2n n=0(2n + 1)2n+1 П=0(2п)[2п] да (_1)nx2n+1 and Sinc x = Y^---tz-rr. П=0(2п + 1)[2n+1] In this paper, we consider the non-elementary Fresnel-type integral functions Cj(x) = |Cos12dt, S1(x) = |Sin12dt, C2(x) = JCosc12dt, S2(x) = JSinc12dt. We prove the 0 0 0 0 following formulas: Q(x) = 4 (cos^C + srn^S f|))-x, = 4 (srn^C g)-co^S f|)j, C2(x) = x Xl^ fl,5;l,5,9;-xl], S2(x) = xL^ f 3,1;5,7,7;_xL 2 20 2 3 ^ 4 3 3 4 27 / 2 3 2 3 ^4 6 6 4 27 where C(p) and S(p) are Fresnel integrals and 2F3(a1,a2;b1,b2,b3;z) is a generalized hypergeometric function. We also show that functions C1(x), S1(x) are solutions of the ordinary linear second-order differential equations 4xy" 4y'+x3y =x4 4 and 4xy'4y'+x3y = 4x2, respectively, and the functions C2( x), S2( x) are solutions of the ordinary linear fourth-order differential equations 27x3yIV -135x2y"+(16x5 + 339)y" -384y'=-384 and 27x3yIV 81x2y"'+ (16x5 +177x)y' + + (32x4 -192)y' = 0, respectively.

Построена математическая модель нагнетания холодного газа в пласт, в исходном состоянии насыщенный снегом и газом, сопровождаемого гидрато-образованием. Получено условие, при котором существует минимальный нагрев системы «газ+снег+гидрат», обеспечивающий полный переход снега в гидратное состояние. Установлено, что в зависимости от начального состояния пласта и параметров, определяющих нагнетание газа, возможны режимы полного образования гидрата в объемной области и на фронтальной поверхности, а также частичного образования гидрата. Выявлено, что с ростом начальной снегонасыщенности пласта, процесс образования гидрата происходит интенсивнее, в связи с чем протяженность прогретой зоны уменьшается., In this paper, the problem of cold gas injection into the reservoir initially saturated with snow and gas with simultaneous formation of hydrate is considered. The mathematical model assumes gas injection in the equilibrium state. The self-similar solutions describing the distribution of the basic parameters (pressure fields, temperature, hydrate saturation, snow and gas saturation) in the reservoir are constructed. It is found that, depending on the initial state of the system «snow + gas» and the intensity of gas injection, different zones in the filtering area, such as «gas + hydrate», «gas + hydrate + snow», and «gas + snow», can be generated. The existence condition of maximum heating of «gas + snow + hydrate» system which provides a complete transition of snow in the hydration state is obtained. It is shown that initial state of the reservoir and parameters of gas injection define the conditions of full or partial hydrate formation in a volume region and at the front surface. It is revealed that the hydrate formation is more intensive and the length of the heated area decreases with increasing initial snow saturation in the reservoir.

Доказывается негомеоморфность двух топологических пространств, а именно, прямой Зоргенфрея S и ее модификации Sq, где Q множество рациональных чисел на прямой. При доказательстве используется монотонность гомеоморфизма ф: S ^ S на некотором интервале (a, b) с S. Этот факт установил E. K. Van Douwen. Вопросы о гомеоморфизме прямой Зоргенфрея и ее модификаций рассматривались в работе V.A. Chatyrko, Y. Hattory, где топология «стрелки» на некотором множестве A заменена на евклидову топологию, а также в работе Е.С. Сухачевой, Т.Е. Хмылевой, где доказывается гомеоморфность пространств S и SA, если A это подмножество счетного замкнутого множества на прямой К. и пространство SA определяется аналогично пространству SQ., In this paper, it is proved that two topological spaces, namely, the Sorgenfrey line S and its modifications Sq, where Q is the set of rational numbers on the real line, are nonhomeomorphic. Topology of the space Sq is defined as follows: if х е Q с S, then the base of neighborhoods of the point х is the family of semiintervals {[х,х + е): е > 0},and if х е S \ Q, then the base of the neighborhood is a family of semiintervals {(х -е, х]: е > 0}. The proof of this fact uses monotonicity of the homeomorphism ф: S ^ S on some interval (a, b) с S (E.K. Van Douwen, 1979).

При обработке поверхностей деталей передаточного механизма используются различные типы фрез (сферические, тороидальные и др.). Стоимость специального профильного инструмента несколько выше, чем таких фрез. Но увеличение стоимости инструмента окупается значительным сокращением машинного времени обработки деталей. Построена математическая модель поверхности профильной фрезы (как поверхности вращения) для обработки деталей цилиндрической передачи с эксцентриково-циклоидальным зацеплением. Создана компьютерная программа для определения радиусов круговых сечений фрезы для заданного набора смещений по оси вращения., Various types of cutters (spherical, toroidal, etc.) are used for processing surfaces of transmission gear parts. The cost of a special forming tool is somewhat higher than that of such cutters. However, the increase in the cost pays a significant reduction in time necessary for processing parts. The paper presents mathematical simulation of a profile cutter (as a surface of revolution) for processing parts of a cylindrical transmission gear with an eccentrically cycloidal gearing (EC-gearing). In part 1 this problem is solved for the input part. The surface of the cutter is constructed as a family of circles with increasing radii in planes perpendicular to the axis of the cutter, the centres lying on this axis. We have obtained an equation for the family of curves, which are cross sections of the tooth surface of the gear by these planes. It is these curves that must be touched by circles forming the surface of the profile cutter. The requirement of the circles touching the curves of the family leads to a system of equations which allows finding the radii of the circles depending on the height of the cross section rise. The solution to this system is found analytically, which eventually leads to one equation for one unknown. The root of this transcendental equation is found numerically. A similar scheme is used in part 2 to find the equation of the profile cutter's surface for the output part. A computer program has been made aiming to specify the radii of the cutter's circular cross-sections for a given set of displacements along the axis of rotation. The work provided substantial assistance in manufacturing EC-engagement parts for gears of various types.

Разработана одномерная математическая модель пьезодвигателя с учетом механической и акустической передачи энергии. Модель позволяет рассчитать частотные характеристики механоакустической колебательной системы, по которым выбираются материалы элементов и конструкция двигателя, позволяющие согласовать источник энергии с нагрузкой изделия., At present, micro electromechanical systems appear to be progressively applicable in engineering. The problem of reducing weight-and-dimensional characteristics of the spacecraft (SC) system is of great importance in the space field. One of the solutions is replacement of electromechanical motors in different SC units by piezo motors. It is necessary to take into account the condition of consistency of PZT stack actuators (the sources of mechanical energy) with the load when calculating and designing the piezo motors of high energy transfer coefficient. In this case, the most relevant is mathematical modeling of a product design with correct consideration of the specific drive load in an actual industrial field. Mathematical modeling by the finite element method in combination with 3D numerical simulation is most accurate and appropriate. However, this method is laborious and time-consuming. In practice, designers and constructors prefer to use simple linear one-dimensional mathematical models, which provide an accurate information about materials and product design. In linear one-dimensional models of oscillatory systems for the range of frequencies applied in piezo motors, it is necessary to consider the acoustic and mechanical principles of energy conversion and transfer from the source to the load. In this paper, a one-dimensional mathematical model of the piezo motor with consideration of mechanical and acoustic energy transfer is developed. This model allows calculating the frequency-response characteristics of mechanical acoustic oscillating system according to which the materials of elements and design of the drive are chosen.

Исследования направлены на решение задачи динамического взаимодействия реального газа с движущимся в нем телом в условиях значительного изменения кинематических параметров. Рассмотрена задача расчета нелинейных аэродинамических характеристик острого конуса. Наиболее полной среди численных моделей является модель, описываемая уравнениями На-вье-Стокса, позволяющая учесть эффекты сжимаемости, вязкости и теплопроводности газа с применением нескольких моделей турбулентности и диффузии. Приведены сравнения полученных расчетов с результатами баллистических экспериментов, Creation of reliable methods for calculating the aircraft aerodynamic characteristics with allowance for their essentially non-linear and sometimes ambiguous dependence on the motion parameters is of great interest, both in the aspect of developing non-stationary supersonic aerodynamics and for solving some practical problems of control for the motion of bodies under study. The investigations are intended to solve the problem of dynamic interaction between a real gas and a body moving within it in conditions of a significant change in kinematic parameters. In this paper, the problem of calculating nonlinear aerodynamic characteristics of a sharp cone is considered. The most complete numerical model is that described by Navier-Stokes equations, which allows taking into account the effects of gas compressibility, viscosity, and thermal conductivity with application of several models of turbulence and diffusion. The obtained results of calculations are compared with results of ballistic experiments.

Проведено исследование движения частицы в форме вытянутого эллипсоида вращения в закрученном потоке. Установлено, что в зависимости от ориентации частицы в пространстве возможно как ее восходящее, так и нисходящее движение. С использованием модели дрейфа определены условия восходящего и нисходящего движения. Предложены зависимости, позволяющие определить компоненты вектора скорости частицы., In this paper, the investigations of the prolate ellipsoidal particles movement in a swirling flow were carried out. The motion of small particles relative to the carrier fluid flow is insignificant. As a result, the smallest particles move along a helical path at an approximately cylindrical surface. Large particles affected by centrifugal force move to the walls of the outer cylinder along a conical surface. Increasing of the angular velocity of the cylinders leads to an increase in the centrifugal force which intensifies the radial motion of the particles to the outer cylinder. It is found that, depending on the particle orientation in the space, both upward and downward motions are possible. The movement is caused by particular drag force influence on the ellipsoidal particle. The drag force is characterized by the horizontal and vertical components. The horizontal component for heavy particles with density value greater than that of the carrier medium is directed towards the axis of symmetry. The direction of the vertical component of the drag force depends on the orientation of the particles in the space. Utilizing the particle slip velocity model, the regimes of upward and downward movement are defined. Dependencies for the determination of particle velocity components are proposed. It is found that the gravitational settling velocity of the particle decreases with an increase in the value of the Froude number. In the case of a strong swirling flow, the motion of the particle becomes upward.

Уравнение Шредингера описывает квантово-механические процессы, происходящие при движении частиц через потенциальный барьер. В такой задаче, нужно находить плотность вероятности частиц и проследить ее эволюцию во времени. Результаты такой теории находят применения в ряде вопросов теоретической физики, например в нанотехнологиях, где нужно вычислять накопления материальных частиц в потенциальных ямках. Зависящее от времени уравнение Шредингера имеет прямую аналогию с уравнением теплопроводности. Поэтому в качестве численного алгоритма применяется метод матричной экспоненты, применяемый ранее для решения уравнения теплопроводности., The Schrodinger equation describes quantum mechanics processes occurring when particles pass through a potential barrier. In this problem, it is necessary to find the probability density of particles and to track its evolution in time. In this paper, it is shown that time-dependent Schrodinger's equation has a direct analogy to the heat conductivity equation, differing from it in the imaginary time. As a numerical method of the decision, it is offered to apply the method of matrix exponential function in which a finite difference analogue of the one-dimensional Laplacian is considered as a matrix operating on a vector. This way of the solution allows one to consider potential barriers of any form in the Schrodinger equation. Time is included now into the decision as a parameter, and it allows one to get rid of the necessity of time quantization and to do it only on a spatial variable. In this aspect, this way favorably differs from traditional ways of solving evolutionary equations which use quantization both on time and on a spatial variable. Results of numerical experiments show that the greatest amplitudes of probability are localized in the field of minima of potential barriers.

Изучается зависимость решения неоднородной краевой задачи для нелинейной системы дифференциальных уравнений составного типа, моделирующей обтекание компактного препятствия потоком смеси вязких сжимаемых жидкостей. Полученные результаты могут служить основой для исследования свойств функционалов от этих решений (например, функционала сопротивления), в частности для вычисления производной по области от функционала и последующего построения численного алгоритма поиска оптимальной формы тела, обтекаемого потоком смеси., In this paper, it is studied how the solution of the boundary value problem for the motion equations of a mixture of compressible viscous fluids depends on the shape of the domain. Such a problem arises in connection with the problem of searching for the optimum shape of the obstacle which is flown around by a stream of the mixture. The solution is reduced to studying the dependence of solutions of a nonlinear system of compound-type partial differential equations on the matrix setting the deformation of the domain. Properties of coefficients of the linear system obtained for a difference of two possible solutions (corresponding to two different matrices) allow one to construct only its very weak solutions. Therefore, there appears the necessity to consider the conjugate problem and to construct its solutions (weak and strong ones). The basic results of the work are estimations allowing one to assert that the mapping associating the solution of the abovementioned boundary value problem to the matrix is a Lipschitz mapping. In particular, this implies the uniqueness of the solution of the inhomogeneous boundary value problem for the initial system of equations. On the basis of the obtained results, differentiability of the functional reflecting the drag force of the streamlined obstacle can be established, as well as an explicit formula representing the derivative of the functional.

Представлены результаты математического моделирования течения ньютоновской жидкости в канале с конструктивным элементом, моделирующим наличие затвора. Математическая постановка задачи формулируется в переменных вихрь функция тока. Численное решение рассматриваемой задачи осуществляется конечно-разностным методом на основе схемы переменных направлений. Представлены результаты параметрических исследований кинематики течений как для полностью открытого, так и частично закрытого затвора., In this paper, results of mathematical modeling of a steady flow of a Newtonian fluid in a flat channel are presented. At some distance from the inlet and outlet of the channel, a valve having a specified diameter and capable to change the width of the flow cross section depending on the position of the locking element is situated. On the walls, the adhesion conditions are performed; at the input, the profile corresponding to the steady flow of a predetermined constant flow is assigned, and the Neumann boundary conditions are used at the output. The mathematical statement of a problem is formulated using a dimensionless stream function and vorticity variables. The numerical solution of the problem is implemented with the application of an original finite-difference method based on the scheme of alternating directions. The physical flow area with a curved boundary is converted into a rectangular domain by introducing new coordinates, and the stationary solution of the problem is obtained using the relaxation method. The calculations demonstrate that in the vicinity of the inlet and outlet boundaries, the one-dimensional flow areas corresponding to a steady fluid flow in an infinite channel occur, and in the vicinity of the shutter, a two-dimensional flow with recirculations is observed. The effect of the Reynolds number on the distribution of the kinematic characteristics is investigated. The results of the parametric studies of the flow pattern in relation to the opening position and diameter of the locking element are presented.

Построена математическая модель инерционного датчика для измерения плотности газожидкостной среды. На основе теоремы о моменте количества движения для относительного движения около подвижной оси, проходящей через центр масс системы, получено определяющее уравнение для крутильных колебаний рамки расходомера. Найдены аналитическое и численное решения полученного уравнения, на базе которых оценивается влияние плотности смеси на величину амплитуды крутильных колебаний рамки., The Emerson transport company (USA), which is engaged in delivery of oil to consumers, has developed an inertial sensor for medium density measurements. The sensor operating principle is based on Coriolis forces the action of which causes the deflection of a vibrating tube with a moving liquid-gas mixture in it. The real sensor (Emerson) consists of two metal tubes with a gas-liquid mixture flow. In the central part of the system, there are two electromagnets which create axial vibrations of the frames. In this paper, a single pipe and a rigid stand with electromagnets between them are used. The rounded frame is replaced by piecewise linear sections and fixed onto the main channel by rubber elements. On the basis of the angular momentum theory for relative motion around a mobile axis passing through the mass center of the system, the governing equation for torsional vibrations of the meter frame has been obtained. The analytical and numerical solutions of this equation are obtained and used as a basis for estimating the effect of the mixture density on the amplitude of frame torsional vibrations.

В работе представлены результаты математического моделирования воздействия струй продуктов сгорания двигательной установки (ДУ) на поверхность Марса при посадке десантного модуля (ДМ) «ЭкзоМарс». Проведены численные исследования разрушения различных типов грунтов (сыпучего, комковатого, кусковатого и песка) под действием истекающих сверхзвуковых струй продуктов сгорания ДУ в процессе посадки десантного модуля на поверхность Марса. Рассмотрены два режима работы ДУ посадочной платформы «ЭкзоМарс»: режим максимальной и минимальной тяги. По результатам параметрических исследований выявлено расстояние, с которого начинается эрозия поверхности Марса, более разрушительную силу сверхзвуковые струи ДУ проявляют на высоте с 1 м и ниже. Показано, что в момент касания ДМ поверхности Марса при работе ДУ на максимальном режиме возможна эрозия всех типов грунтов, за исключением сыпучего грунта, а на минимальном эрозия не наблюдается., The paper presents results of mathematical modeling of the effect caused by supersonic propulsion system jets on the surface of Mars during the descent of the ExoMars module. The action of propulsion system supersonic jets on the Mars surface was investigated numerically using the OpenFOAM software. According to results of the force action of supersonic jets on the Mars surface, the possible erosion of the Mars soil is estimated using Coulomb's law of shearing resistance. The soil model involves empirical constants which were obtained from previous Viking, Pathfinder, and MER space programs. The soil erosion during the landing of the spacecraft on the surface of Mars under the influence of the propulsion system at the maximum and minimum mode for loose, lumpy soil, and sand was investigated. During the numerical research, the distance from which soil erosion begins was revealed. The destructive force of propulsion system supersonic jets is greatest at one meter height and below. At the time of the spacecraft contacts with the surface of Mars at the maximum mode of the propulsion system, erosion of all soil types is possible except for the loose soil; for the minimum mode of the propulsion system, erosion is not observed.

Свертки мер и функций, преобразование Фурье мер на локально компактных абелевых n-арных группах были введены в работе [1]. Развитие гармонического анализа на n-арных алгебраических объектах, наделенных топологией, тесно связано с существованием на подобных объектах ненулевой инвариантной меры. Инвариантные меры на топологических n-арных полугруппах рассматривались в [2] и [3]. В теореме 2 данной работы установлены необходимые и достаточные условия существования левоинвариантной меры на топологических n-арных подполугруппах бинарных групп. Ее можно рассматривать, как распространение результатов работы [4] на случай топологических n-арных полугрупп. Теорема 1 устанавливает результат представляющий интерес для топологической алгебры., Convolutions of measures and functions, as well as the Fourier transform of measures on locally compact Abelian n-ary groups were introduced in [1]. Development of harmonic analysis on n-ary algebraic objects endowed with a topology is closely related to the existence of a non-zero invariant measure on such objects. Invariant measures on topological n-ary semigroups were considered in [2] and [3]. In Theorem 2 of this paper, we establish necessary and sufficient conditions for the existence of a left-invariant measure on topological n-ary subsemigroups of binary groups. It can be treated as an extension of the results of [4] to the case of n-ary topological semigroups. The result established in Theorem 1 establishes is interesting for topological algebra.

Предложен подход к математическому моделированию пакетного пьезоак-тюатора с учетом массы и механических потерь движения пьезопластин на основе аналоговых электрических схем замещения. Данный подход допустим для решения одномерных задач и может быть полезен для предварительных расчетов колебательных систем микролинейных пакетных пьезо-преобразователей., It is now an urgent task to minimize the weight and size of all the elements of space technology; this applies to all electric drives functioning in the spacecraft equipment. A significant alternative to electromechanical devices are microlinear piezoactuators. Replacing all the drives in the spacecraft by microlinear piezoactuators is not only a technical objective but also a significant import substitution in an important industry sector of the country. The analysis of such a transducer can be performed in two ways. The first method is to calculate the power of elastic waves using the equations of wave propagation in each medium with appropriate boundary conditions. The second method is based on the representation of the oscillating system in the form of electrical circuits, and application of Kirchhoff s laws. In this paper, an approach to the mathematical modeling of a batch piezoactuator with allowance for the mass and mechanical losses of piezoplate motion based on analog electrical equivalent circuit are proposed. This approach is valid for the solution of one-dimensional problems and can be useful for preliminary calculations of oscillating systems of microlinear batch piezoelectric transducers. According to the study it was determined that: frequency characteristics of the analog electrical equivalent circuit with allowance for masses and losses significantly differ from those of the analog electrical equivalent circuit of the oscillating system without regard to masses and losses; for a more accurate analysis of the vibration system, it is necessary to use a complete analog electrical equivalent circuit; frequency characteristics of forces show the presence of resonances inside of the batch piezoelectric transducer. All piezoplates resonate at different frequencies and the amplitude of forces on the individual piezoelectric elements can exceed the resultant force of the entire batch piezoelectric transducer.

Для параболической антенны рефлекторного типа актуальна задача прикрепления отражающего сетеполотна таким образом, чтобы как можно менее страдали радиофизические характеристики устройства. Ясно, что при различных избранных схемах раскроя указанная задача решается (приближенно) разными способами. Схема раскроя порождает задачу об оптимуме раскроя, который может пониматься в различных смыслах. Известен подход, основанный на уменьшении, насколько возможно, СКО (среднего квадратичного отклонения от идеальной поверхности). Положим в основу оптимизации другой критерий степень однородности натяжения сетеполотна в различных областях рефлектора (например, степень растяжения сетеполотна вблизи технологического центра должна составлять некоторую долю от степени его натяжения на периферии). Обычно же предполагается, что приемлемый результат достигается, если во всех точках рефлектора максимальное натяжение находится в некоторых заранее указанных пределах и минимальное тоже в пределах (своих). Представлено решение задачи об оптимальном раскрое, причем оптимальность предполагает условия: экстремальные относительные искажения длины должны быть максимально близки к единице, при этом чем ближе к центру параболоида, тем «строже» должно быть соблюдено указанное условие., For a parabolic reflector-type antenna, an attachment of a reflecting metallic mesh sheet in a manner ensuring the least possible impact on radiophysical properties of the device is a crucial problem. It is clear that under different selected schemes of tailoring, the specified problem is solved (approximately) in different ways. The scheme of tailoring gives rise to a problem of optimum tailoring. Optimum tailoring can be understood in different ways. There is a well-known approach based on reducing the RMSD as much as possible, (root-mean-square deviation from an ideal surface). We however take different criteria as a basis for optimization the degree of homogeneity of tension for the metallic mesh in different areas of the reflector (for example, the degree of metallic mesh stretching near the technological center should be a certain fraction of its degree of tension on the periphery). Usually, it is assumed that an acceptable result is achieved if there is a maximum tension at all the points of the reflector within some pre-specified limits, and minimal also within the desirable limits (own ones). In this paper, we solve a problem of optimal tailoring, and optimality suggests the following conditions: extreme distortion of the relative length should be as close to unity, while the closer to the center of the paraboloid, the "stricter" specified condition must be satisfied.