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A schematic and structural solution of regenerative-utilization heat use in centralized heat supply systems was previously proposed in order to increase the efficiency of operation of heating networks located in non-passable channels. The impossibility of creating a full-scale experimental setup covering the required range of factors and the area of their change, the complexity of a passive experiment on existing heating mains led to the need to develop a virtual model based on the ANSYS sofware package. A six-factor experiment has been carried out on this virtual model. Regression equations have been obtained to determine the pressure required to provide air purging of the channel, as well as heat exchange with pipelines of direct and return network water located in the channel, and heat exchange with soil around the channel. In addition, a regression relationship has been derived to find the integral heat flux from the listed washed surfaces to the air flow. The transition from dimensionless to natural factors has been made in the paper. The most significant factors are identified with the help of Pareto cards. The obtained dependencies have been verified in Part 1 of the paper. The adequacy of the obtained regression equations has been determined using standard statistical estimation methods based on the calculated values of the Fisher’s, Student’s and other criteria. The response surfaces are presented and analyzed using two dimensional sections for a number of factor values at fixed values of one and a change in two most characteristic, physically significant for a given response function. They have been validated on the basis of the analysis of regression dependencies. The obtained regression equations cover almost the entire range of possible diameters of heating mains, which makes it possible to use them in the development of energy-saving projects.

The mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortions, is reduced to a similar problem for one-dimensional inhomogeneous wave equation. An effective way to solve this problem is based on the use of special functions – polylogarithms, which are complex power series with power coefficients, converging in the unit circle. The exact solution of the problem is expressed in integral form in terms of the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one – in the form of a finite sum in terms of the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All the indicated parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding degrees on an interval of length in the period, which makes it possible to obtain a solution to the problem in elementary functions. In the paper, we study a mixed problem for the telegrapher’s equation which is well-known in applications. This problem of linear substitution of the desired function witha time-exponential coefficient is reduced to a similar problem for the Klein – Gordon equation. The solution of the latter can be found by dividing the variables in the form of a series of trigonometric functions of a line point with time-dependent coefficients. Such a solution is of little use for practical application, since it requires the calculation of a large number of coefficients-integrals and it is difficult to estimate the error of calculations. In the present paper, we propose another way to solve this problem, based on the use of special He-functions, which are complex power series of a certain type that converge in the unit circle. The exact solution of the problem is presented in integral form in terms of second-order He-functions on the unit circle. The approximate solution is expressed in the final form in terms of third-order He-functions. The paper also proposes a simple and effective estimate of the error of the approximate solution of the problem. It is linear in relation to the line splitting step with a time-exponential coefficient. An example of solving the problem for the Klein – Gordon equation in the way that has been developed is given, and the graphs of exact and approximate solutions are constructed.

The paper deals with a mixed problem for the telegraph equation well-known in electrical engineering and electronics, provided that the line is free from distortion. This problem is reduced to the analogous one for the one-dimensional inhomogeneous wave equation. Its solution can be found as the sum of the solution for a mixed homogeneous boundary value problem for the corresponding homogeneous wave equation and for the solution of a non-homogeneous wave equation with homogeneous boundary data and zero initial conditions. Solutions to both problems can be found by separating the variables in the form of a series of trigonometric functions of the line point with time-dependent coefficients. Such solutions are inconvenient for real application because they require calculation of a large number of integrals, and it is difficult to estimate the miscalculation. An alternative method for solving this problem is proposed, based on the use of special functions, viz. polylogarithms, which are complex power-series with power coefficients converging in a unit circle. The exact solution of the problem is expressed in the integral form via the imaginary part of the first-order polylogarithm on the unit circle, and the approximate one is expressed in the form of a finite sum via the real part of the dilogarithm and the imaginary part of the third-order polylogarithm. All these parts of the polylogarithms are periodic functions that have polynomial expressions of the corresponding powers on the segment of the length equal to the period. This makes it possible to effectively find an approximate solution to the problem. Also, a simple and convenient error estimate of the approximate solution of the problem is found. It is linear with respect to the step of splitting the line and the step of splitting the time range in which the problem is considered. The score is uniform along the length of the line at each fixed point of time. A concrete example of solving the problem according to the proposed mode is presented; graphs of exact and approximate solutions are constructed.

In this paper we introduce a new class of foliations on Rie-mannian 3-manifolds, called B-foliations, generalizing the class of foliations of non-negative curvature. The leaves of B-foliations have bounded total absolute curvature in the induced Riemannian metric. We describe several topological and geometric properties of B-foliations and the structure of closed oriented 3-dimensional manifolds admitting B-foliations with non-positive curvature of leaves.

The study of non-stationary rarefied gas flows is, currently, attracting a great deal of attention. Such an interest arises from creating the pulsed jets used for deposition of thin films and special coatings on the solid surfaces. However, the problems of non-stationary rarefied gas flows are still understudied because of their large computational complexity. The paper considers the computational aspects of investigating non-stationary movement of gas reflected from a wall and flowing through a suddenly formed gap. The study objective is to analyse the possible numerical kinetic approaches to solve such problems and identify the difficulties in their solving. When modeling the gas flows in strong rarefaction one should consider the Boltzmann kinetic equation, but its numerical implementation is rather time-consuming. In order to use more simple approaches based, for example, on approximation kinetic equations (Ellipsoidal-Statistical model, Shakhov model), it is important to estimate the difference between the solutions of the model equations and of the Boltzmann equation. For this purpose, two auxiliary problems are considered, namely reflection of the gas flow from the wall and outflow of the free jet into the rarefied background gas.A numerical solution of these problems shows a weak dependence of the solution on the type of the collision operator in the rarefied region, but at the same time a strong dependence of a behavior of the macro-parameters on the velocity grid step. The detailed velocity grid is necessary to avoid a non-monotonous behavior of the macro-parameters caused by so-called ray effect. To reduce computational costs of the detailed velocity grid solution, a hybrid method based on the synthesis of model equations and the Boltzmann equation is proposed. Such an approach can be promising since it reduces the domain in which the Boltzmann collision integral should be used.The article presents the results obtained using two different software packages, namely a Unified Flow Solver (UFS) [13] and a Nesvetay 3D software complex [14-15]. Note that the UFS uses the discrete ordinate method for velocity space on a uniform grid and a hierarchical adaptive mesh refinement in physical space. The possibility to calculate both the Boltzmann equation and the model equations is realized. The Nesvetay 3D software complex was created to solve the Shakhov model equation (S-model) for calculations based on non-structured non-uniform grids, both in velocity space and in physical one.

The purpose of this paper is to compare the shortcomings of the widely used inequality coefficients that appear when working with real (ie, knowingly incomplete) data and searching for alternative quantitative methods for describing inequalities that lack these shortcomings.Research methods:– consideration of an extensive range of as full as possible real data on the population distribution by income, expenditure, property (ie data on the economic structure of society);– revealing the specific shortcomings of these data on the economic structure of society, finding out which information is missing or presented disproportionately;– comparison of the values of the most widely used indices of inequality calculated on real data on the economic structure, with a view to establishing the suitability of these indicators for problems of inequality estimation;– development of an index of inequality that adequately describes the real economic structure of society.Research data:– official data of Rosstat and the Federal Tax Service on incomes of Russian citizens;– specialized sites of announcements about the prices for real estate and cars;– Credit Suisse Research Institute data on the distribution of Russian citizens by property level;– Forbes data on income and wealth of the richest people in Russia.It is shown that the income data are essentially incomplete and fragmentary – the width of the income range (i.e., the income of therichest member of society) is known, but the filling of rich cohorts is not known, since the incomes of the richest members of society are hidden.We proposed the next (criteria as) requirements for an inequality index:– possibility of calculating the index of inequality for arbitrary quantization;– invariance of the value of the inequality index for different quantization of the same data;– sensitivity of the index to the width of the income range.It is noted that only the exponential function describes societies with high social inequality enough well (the intensity of the exponential distribution is more than 10).For the presented population distributions, the next indices of inequality are calculated:– decile coefficient of funds;– Gini coefficient;– Pareto index;– indicators of total entropy (zero, first or Tayle index, and second orders);– the ratio of maximum income (property value) to the modal;– intensity of exponential distribution.It is shown, that:– the value of the Pareto index does not have a unique relationship with the inequality;– the coefficients of the funds (decile, quintile, etc.) are not computable for arbitrary quantization, and therefore are unsuitable for comparing data from various sources and have different quantization;– The Gini index requires complete data on the rich;– from all considered criteria of inequality the first three indicators of the total entropy, as well as the ratio of maximum income (property) to the modal strongly depend on data quantization.Therefore they are unsuitable for comparison data from various sources with different quantization. It is concluded that the intensity of the exponential distribution does not possess the listed disadvantages and can be recommended as an index of inequality.

A numerically stable direct multiplicative method for solving systems of linear equations that takes into account the sparseness of matrices presented in a packed form is considered. The advantage of the method is the calculation of the Cholesky factors for a positive definite matrix of the system of equations and its solution within the framework of one procedure. And also in the possibility of minimizing the filling of the main rows of multipliers without losing the accuracy of the results, and no changes are made to the position of the next processed row of the matrix, which allows using static data storage formats. The solution of the system of linear equations by a direct multiplicative algorithm is, like the solution with LU-decomposition, just another scheme for implementing the Gaussian elimination method. The calculation of the Cholesky factors for a positive definite matrix of the system and its solution underlies the construction of a new mathematical formulation of the unconditional problem of quadratic programming and a new form of specifying necessary and sufficient conditions for optimality that are quite simple and are used in this paper to construct a new mathematical formulation for the problem of quadratic programming on a polyhedral set of constraints, which is the problem of finding the minimum distance between the origin ordinate and polyhedral boundary by means of a set of constraints and linear algebra dimensional geometry. To determine the distance, it is proposed to apply the known exact method based on solving systems of linear equations whose dimension is not higher than the number of variables of the objective function. The distances are determined by the construction of perpendiculars to the faces of a polyhedron of different dimensions. To reduce the number of faces examined, the proposed method involves a special order of sorting the faces. Only the faces containing the vertex closest to the point of the unconditional extremum and visible from this point are subject to investigation. In the case of the presence of several nearest equidistant vertices, we investigate a face containing all these vertices and faces of smaller dimension that have at least two common nearest vertices with the first face.

In 1994, in [13], N. Papaghiuc introduced the notion of semi-slant submanifold in a Hermitian manifold which is a generalization of CR- and slant-submanifolds. In particular, he considered this submanifold in Kaehlerian manifolds, [13]. Then, in 2007, V. A. Khan and M. A. Khan considered this submanifold in a nearly Kaehler manifold and obtained interesting results, [11]. Recently, we considered semi-slant submanifolds in a locally conformal Kaehler manifold and gave a necessary and sufficient conditions for two distributions (holomorphic and slant) to be integrable. Moreover, we considered these submanifolds in a locally conformal Kaehler space form, [4]. In this paper, we define 2-kind warped product semi-slant submanifolds in a locally conformal Kaehler manifold and consider some properties of these submanifolds.

To synthesize Maxwell optics systems, the mathematical apparatus of tensor and vector analysis is generally employed. This mathematical apparatus implies executing a great number of simple stereotyped operations, which are adequately supported by computer algebra systems. In this paper, we distinguish between two stages of working with a mathematical model: model development and model usage. Each of these stages implies its own computer algebra system. As a model problem, we consider the problem of geometrization of Maxwell's equations. Two computer algebra systems---Cadabra and FORM---are selected for use at different stages of investigation., in Russian; in English

International audience; http://mech.math.msu.su/~fpm/ps/k14/k143/k14303.pdfIn this paper, a computerized model for morphological analysis of languages with word-formation based on affixation processes is proposed. The main idea consists in defining structural patterns of words and corresponding lists of suffixes. First, a detaileddescription of a stemming algorithm, its modification, and the technique of determining grammatical characteristics of word-forms are given. The next part of this work focuses on the application of the proposed algorithms for the French language. Finally, some results of execution of these algorithms are provided.; http://mech.math.msu.su/~fpm/ps/k14/k143/k14303.pdfВ работе предлагается метод структурных схем в качестве модели морфологического анализа словоформ естественного языка с развитым аффиксальным словообразованием и словоизменением. Дано описание алгоритма выделения псевдоосновы, его модификация, а также алгоритм восстановления грамматических характеристик словоформ. Описано применение предложенного метода для анализа словоформ французского языка. Представлены результаты работы предложенных алгоритмов.