Publications

We consider the regularized 3D Navier-Stokes-Cahn-Hilliard equations describing isothermal flows of viscous compressible two-component fluids with interphase effects. We construct for them a new energy dissipative finite-difference discretization in space, i.e., with the non-increasing total energy in time. This property is preserved in the absence of a regularization. In addition, the discretization is well-balanced for equilibrium flows and the potential body force. The sought total density, mixture velocity and concentration of one of the components are defined at nodes of one and the same grid. The results of computer simulation of several 2D test problems are presented. They demonstrate advantages of the constructed discretization including the absence of the so-called parasitic currents.

It is an important feature of a monotone measure that it is not additive in general. In the paper, we propose the mathematical tool, based on canonical sequences of monotone measures, for analyzing additivity of monotone measures on subalgebras and give a way of generating such monotone measures. It turns out that the generating rule can be considered as an effect of a linear operator defined on the set of monotone measures. We also investigate in what cases the sequence of such operators behaves commutatively and preserve continuity properties from the generating monotone measure.

Over the past years, there is a deep interest in the analysis of different communities and complex networks. Identification of the most important elements in such networks is one of the main areas of research. However, the heterogeneity of real networks makes the problem both important and problematic. The application of SRIC and LRIC indices can be used to solve the problem since they take into account the individual properties of nodes, the possibility of their group influence, and topological structure of the whole network. However, the computational complexity of such indices needs further consideration. Our main focus is on the performance of SRIC and LRIC indices. We propose several modes on how to decrease the computational complexity of these indices. The runtime comparison of the sequential and parallel computation of the proposed models is also given.

Usually DEA methods are used for the assessment of the regions disaster vulnerability. However, most of these methods work with precise values of all the characteristics of the regions. At the same time, in real life, quite often most of the data consists of expert estimates or approximate values. In this regard, we propose to use modified DEA methods, which will take into account inaccuracy of the data. We apply these methods to the evaluation of wildfire preventive measures in regions of the Russian Federation.

We consider an application of long-range interaction centrality (LRIC) to the problem of the influence assessment in the global retail food network. Firstly, we reconstruct an initial graph into the graph of directed intensities based on individual node’s characteristics and possibility of the group influence. Secondly, we apply different models of the indirect influence estimation based on simple paths and random walks. This approach can help us to estimate node-to-node influence in networks. Finally, we aggregate node-to-node influence into the influence index. The model is applied to the food trade network based on the World International Trade Solution database. The results obtained for the global trade by different product commodities are compared with classical centrality measures.

Using the SIPRI Arms Transfers Database covering all trade in military equipment over the period 1950–2018, we examine the relationship between countries from a novel empirical perspective. We consider the arms transfers network as a multiplex network where each layer corresponds to a particular armament category. First, we analyze how different layers overlap and elucidate main ties between countries. Second, we consider different patterns of trade in order to identify countries specializing on particular armament categories and analyze how they change their export structure in dynamic. We also examine how countries influence each other at different layers of multiplex network. Finally, we analyze the influence of countries in the whole network.

We obtain criteria for the L2-dissipativity of finite-difference schemes based on regularizations of 1D barotropic and full gas dynamics systems of equations that are linearized at a constant solution. Bibliography: 8 titles.

We study an explicit in time and symmetric in space finite-difference scheme with a kinetic regularization for the 2D and 3D gas dynamics system of equations linearized at a constant solution (with any velocity). We derive both necessary and sufficient conditions for $L^2$-dissipativity of the Cauchy problem for the scheme by the spectral method. The Courant number is uniformly bounded with respect to the Mach number in them.

Generalized Pauli’s theorem, proved by D. S. Shirokov for two sets of anticommuting elements of a real or complexified Clifford algebra of dimension 2n, is extended to the case, where both sets of elements depend smoothly on points of Euclidean space of dimension r. We prove that in the case of even n there exists a smooth function such that two sets of Clifford algebra elements are connected by a similarity transformation. All cases of connection between two sets are considered in the case of odd n. Using the equation for the spin connection of general form, it is shown that the problem of the local Pauli’s theorem is equivalent to the problem of existence of a solution of some special system of partial differential equations. The special cases n = 2, r ≥ 1 and n ≥ 2, r = 1 with simpler solution of the problem are considered in detail.

We consider binary gas mixture flows with viscous compressible components in the absence of chemical reactions. We aggregate the previously derived regularized equations for inhomogeneous mixtures and thus derive new simpler regularized equations for homogeneous ones (i.e., with the common velocity and temperature). The entropy balance equation with the non-negative entropy production is stated for the new equations. They are constructed for numerical simulations of flows.

In this paper, we present all constant solutions of the Yang-Mills equations with SU(2) gauge symmetry for an arbitrary constant non-Abelian current in Euclidean space Rn of arbitrary finite dimension n. Using the invariance of the Yang-Mills equations under the orthogonal transformations of coordinates and gauge invariance, we choose a specific system of coordinates and a specific gauge fixing for each constant current and obtain all constant solutions of the Yang-Mills equations in this system of coordinates with this gauge fixing, and then in the original system of coordinates with the original gauge fixing. We use the singular value decomposition method and the method of two-sheeted covering of orthogonal group by spin group to do this. We prove that the number (0, 1, or 2) of constant solutions of the Yang-Mills equations in terms of the strength of the Yang-Mills field depends on the singular values of the matrix of current. The explicit form of all solutions and the invariant F^2 can always be written using singular values of this matrix. The relevance of the study is explained by the fact that the Yang-Mills equations describe electroweak interactions in the case of the Lie group SU(2). Nonconstant solutions of the Yang-Mills equations can be considered in the form of series of perturbation theory. The results of this paper are new and can be used to solve some problems in particle physics, in particular, to describe physical vacuum and to fully understand a quantum gauge theory.

We consider consecutive aggregation procedures for individual preferences 𝔠 ∈ ℭ_r (A) on a set of alternatives A, |A| ≥ 3: on each step, the participants are subject to intermediate collective decisions on some subsets B of the set A and transform their a priori preferences according to an adaptation function 𝒜. The sequence of intermediate decisions is determined by a lot J, i.e., an increasing (with respect to inclusion) sequence of subsets B of the set of alternatives. An explicit classification is given for the clones of local aggregation functions, each clone consisting of all aggregation functions that dynamically preserve a symmetric set 𝔇 ⊆ ℭ_r (A) with respect to a symmetric set of lots 𝒥. On the basis of this classification, it is shown that a clone ℱ of local aggregation functions that preserves the set ℜ_2(A) of rational preferences with respect to a symmetric set 𝒥 contains nondictatorial aggregation functions if and only if 𝒥 is a set of maximal lots, in which case the clone ℱ is generated by the majority function. On the basis of each local aggregation function f, lot J, and an adaptation function 𝒜, one constructs a nonlocal (in general) aggregation function f_J,A that imitates a consecutive aggregation procesure. If f dynamically preserves a set 𝔇 ⊆ ℭ_r (A) with respect to a set of lots 𝒥, then the aggregation function f_J,A preserves the set 𝔇 for each lot J ∈ 𝒥. If 𝔇 = ℜ_2(A), then the adaptation function can be chosen in such a way that in any profile c ∈ (ℜ_2(A))n, the Condorcet winner (if it exists) would coincide with the maximal element with respect to the preferences f_J,A (c) for each maximal lot J and f that dynamically preserves the set of rational preferences with respect to the set of maximal lots.

This article treats the use of modern information technologies in the classroom educational process. It emphasizes the need to combine symbolic and visual mathematics, describes the problems associated with this issue, provides a review of the existing systems and a list of the requirements a modern mathematical visualization system must meet. The article is conceived as a manifesto for the use of mathematical visualization in education. The article then goes on to describe the developments of the authors' research group. The functionality of visualmath.ru website is described. This resource contains an ample collection of visual and text modules for teachers to create presentations based largely on visual materials. The most important part of the article is the description of fast and powerful JavaScript visulatization libraries developed specifically for the project: Skeleton and Grafar. The former is designed to display two-dimensional graphs, while the latter visualizes three-dimensional objects with transparency and illumination effects. Both libraries are capable of processing large element sets in near-real time. In conclusion, selected examples of visualizations created using the libraries use are provided, including the ones used in courses on mathematical analysis and analytical geometry.

Power of nodes has been studied in many works, in particular, using centrality concepts. However, in some applications, a large flow between two nodes implies that these nodes become too interdependent on each other. For instance, in trade networks, the possible shortage of flow between two countries may lead to the deficit of goods in the importing country but, on the other hand, it may also affect the financial stability of the exporting country. This feature is not captured by existing centrality measures. Thus, we propose an approach that takes into account interdependence of nodes. First, we evaluate how nodes influence and depend on each other via the same flow based on their individual attributes and a possibility of their group influence. Second, we present several models that transform information about direct influence to a single vector with respect to the network structure. Finally, we compare our models with centrality measures on artificial and real networks.

We analyze export/import food trade network that contains several layers. Each layer accounts for a particular commodity that countries trade with. The network has directed weighted edges. We look at statistical and topological similarity of layers in order to detect dependencies between different products trade. The measures include the estimation of out-degree correlation as well as the analysis of communities. We apply a normalization technique to the initial graphs that takes into account individual attributes of nodes and the possibility of groups formation. The most important elements of the networks are considered in order to compare different layers. Additionally, we analyze the network in time and detect the most similar periods of trade. The analysis of trade in dynamics gives the opportunity to track changes in export/import patterns. The results may have a significant contribution to the further analysis of food security of countries and the development of trade processes.

The paper examines the choice problem when the total number of observations and criteria is too large. There are many different procedures, which are used for decision-making process under multiple criteria; however, most of them cannot be applied to large datasets due to their computational complexity while others provide sufficient accuracy. To solve the problem, we consider the idea of superposition, which consists in the sequential application of choice functions where the result of the previous function is the input for the next function. Among the main benefits of the superposition are its manageable computational complexity and high performance. We analyze normative properties of the superposition that characterize how stable and sensible the final choice is. We also consider the application of superposition to tornado prediction and search problems. As a result, we show that superposition of choice functions provides higher efficiency values compared to traditional solutions.

The article presents the main approaches to the study of the stochastic process of population dynamics with continuous time and space and with fixed individuals, a countable system of integro-differential equations corresponding to the dynamics of spatial moments of this process is derived, and a method for finding an approximate solution using the method of moments is described.