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A numerical simulation of fluid flow in a blood pump was performed using the FlowVision software package. This test problem, provided by the Center for Devices and Radiological Health of the US. Food and Drug Administration, involved considering fluid flow according to several design modes. At the same time for each case of calculation a certain value of liquid flow rate and rotor speed was set. Necessary data for calculations in the form of exact geometry, flow conditions and fluid characteristics were provided to all research participants, who used different software packages for modeling. Numerical simulations were performed in FlowVision for six calculation modes with the Newtonian fluid and standard $k-\varepsilon$ turbulence model, in addition, the fifth mode with the $k-\omega$ SST turbulence model and with the Caro rheological fluid model were performed. In the first stage of the numerical simulation, the convergence over the mesh was investigated, on the basis of which a final mesh with a number of cells of the order of 6 million was chosen. Due to the large number of cells, in order to accelerate the study, part of the calculations was performed on the Lomonosov-2 cluster. As a result of numerical simulation, we obtained and analyzed values of pressure difference between inlet and outlet of the pump, velocity between rotor blades and in the area of diffuser, and also, we carried out visualization of velocity distribution in certain cross-sections. For all design modes there was compared the pressure difference received numerically with the experimental data, and for the fifth calculation mode there was also compared with the experiment by speed distribution between rotor blades and in the area of diffuser. Data analysis has shown good correlation of calculation results in FlowVision with experimental results and numerical simulation in other software packages. The results obtained in FlowVision for solving the US FDA test suggest that FlowVision software package can be used for solving a wide range of hemodynamic problems.

The grid-characteristic method is successfully used for solving hyperbolic systems of partial differential equations (for example, transport / acoustic / elastic equations). It allows to construct correctly algorithms on contact boundaries and boundaries of the integration domain, to a certain extent to take into account the physics of the problem (propagation of discontinuities along characteristic curves), and has the property of monotonicity, which is important for considered problems. In the cases of two-dimensional and three-dimensional problems the method makes use of a coordinate splitting technique, which enables us to solve the original equations by solving several one-dimensional ones consecutively. It is common to use up to 3-rd order one-dimensional schemes with simple splitting techniques which do not allow for the convergence order to be higher than two (with respect to time). Significant achievements in the operator splitting theory were done, the existence of higher-order schemes was proved. Its peculiarity is the need to perform a step in the opposite direction in time, which gives rise to difficulties, for example, for parabolic problems.

In this work coordinate splitting of the 3-rd and 4-th order were used for the two-dimensional hyperbolic problem of the linear elasticity. This made it possible to increase the final convergence order of the computational algorithm. The paper empirically estimates the convergence in L1 and L∞ norms using analytical solutions of the system with the sufficient degree of smoothness. To obtain objective results, we considered the cases of longitudinal and transverse plane waves propagating both along the diagonal of the computational cell and not along it. Numerical experiments demonstrated the improved accuracy and convergence order of constructed schemes. These improvements are achieved with the cost of three- or fourfold increase of the computational time (for the 3-rd and 4-th order respectively) and no additional memory requirements. The proposed improvement of the computational algorithm preserves the simplicity of its parallel implementation based on the spatial decomposition of the computational grid.

The nonlinear restrictive (with restrictions of the inequalities type) dynamic mathematical model of the committed competition vacant market of many goods in conditions of the goods deliveries time-lag and of the linear dependency of the demand vector from the prices vector is offered. The problem of finding of prices and deliveries of goods into the market which are optimal (from seller’s profit standpoint) is formulated. It is shown the seller’s total profit maximum is expressing by the continuous piecewise smooth function of vector of volumes of deliveries with breakup of the derivative on borders of zones of the goods deficit, of the overstocking and of the dynamic balance of demand and offer of each of goods. With use of the predicate functions technique the computing algorithm of optimization of the goods deliveries into the market is built.

The probabilistic topic model of a text document collection finds two matrices: a matrix of conditional probabilities of topics in documents and a matrix of conditional probabilities of words in topics. Each document is represented by a multiset of words also called the “bag of words”, thus assuming that the order of words is not important for revealing the latent topics of the document. Under this assumption, the problem is reduced to a low-rank non-negative matrix factorization governed by likelihood maximization. In general, this problem is ill-posed having an infinite set of solutions. In order to regularize the solution, a weighted sum of optimization criteria is added to the log-likelihood. When modeling large text collections, storing the first matrix seems to be impractical, since its size is proportional to the number of documents in the collection. At the same time, the topical vector representation (embedding) of documents is necessary for solving many text analysis tasks, such as information retrieval, clustering, classification, and summarization of texts. In practice, the topical embedding is calculated for a document “on-the-fly”, which may require dozens of iterations over all the words of the document. In this paper, we propose a way to calculate a topical embedding quickly, by one pass over document words. For this, an additional constraint is introduced into the model in the form of an equation, which calculates the first matrix from the second one in linear time. Although formally this constraint is not an optimization criterion, in fact it plays the role of a regularizer and can be used in combination with other regularizers within the additive regularization framework ARTM. Experiments on three text collections have shown that the proposed method improves the model in terms of sparseness, difference, logLift and coherence measures of topic quality. The open source libraries BigARTM and TopicNet were used for the experiments.

Most biomechanical tasks of interest to clinicians can be solved only using personalized mathematical models. Such models allow to formalize and relate key pathophysiological processes, basing on clinically available data evaluate non-measurable parameters that are important for the diagnosis of diseases, predict the result of a therapeutic or surgical intervention. The use of models in clinical practice imposes additional restrictions: clinicians require model validation on clinical cases, the speed and automation of the entire calculated technological chain, from processing input data to obtaining a result. Limitations on the simulation time, determined by the time of making a medical decision (of the order of several minutes), imply the use of reduction methods that correctly describe the processes under study within the framework of reduced models or machine learning tools.

Personalization of models requires patient-oriented parameters, personalized geometry of a computational domain and generation of a computational mesh. Model parameters are estimated by direct measurements, or methods of solving inverse problems, or methods of machine learning. The requirement of personalization imposes severe restrictions on the number of fitted parameters that can be measured under standard clinical conditions. In addition to parameters, the model operates with boundary conditions that must take into account the patient’s characteristics. Methods for setting personalized boundary conditions significantly depend on the clinical setting of the problem and clinical data. Building a personalized computational domain through segmentation of medical images and generation of the computational grid, as a rule, takes a lot of time and effort due to manual or semi-automatic operations. Development of automated methods for setting personalized boundary conditions and segmentation of medical images with the subsequent construction of a computational grid is the key to the widespread use of mathematical modeling in clinical practice.

The aim of this work is to review our solutions for personalization of mathematical models within the framework of three tasks of clinical cardiology: virtual assessment of hemodynamic significance of coronary artery stenosis, calculation of global blood flow after hemodynamic correction of complex heart defects, calculating characteristics of coaptation of reconstructed aortic valve.

The purpose of this study is to develop a methodology for change points detection in time series, including financial data. The theoretical basis of the study is based on the pieces of research devoted to the analysis of structural changes in financial markets, description of the proposed algorithms for detecting change points and peculiarities of building classical and deep machine learning models for solving this type of problems. The development of such tools is of interest to investors and other stakeholders, providing them with additional approaches to the effective analysis of financial markets and interpretation of available data.

To address the research objective, a neural network was trained. In the course of the study several ways of training sample formation were considered, differing in the nature of statistical parameters. In order to improve the quality of training and obtain more accurate results, a methodology for feature generation was developed for the formation of features that serve as input data for the neural network. These features, in turn, were derived from an analysis of mathematical expectations and standard deviations of time series data over specific intervals. The potential for combining these features to achieve more stable results is also under investigation.

The results of model experiments were analyzed to compare the effectiveness of the proposed model with other existing changepoint detection algorithms that have gained widespread usage in practical applications. A specially generated dataset, developed using proprietary methods, was utilized as both training and testing data. Furthermore, the model, trained on various features, was tested on daily data from the S&P 500 index to assess its effectiveness in a real financial context.

As the principles of the model’s operation are described, possibilities for its further improvement are considered, including the modernization of the proposed model’s structure, optimization of training data generation, and feature formation. Additionally, the authors are tasked with advancing existing concepts for real-time changepoint detection.

The article describes a model of a human in the informational economy and demonstrates the multicriteria optimizational approach to the metric analysis of model-generated data. The traditional approach using the identification and study involves the model’s identification by time series and its further prediction. However, this is not possible when some variables are not explicitly observed and only some typical borders or population features are known, which is often the case in the social sciences, making some models pure theoretical. To avoid this problem, we propose a method of metric data analysis (MMDA) for identification and study of such models, based on the construction and analysis of the Kolmogorov – Shannon metric nets of the general population in a multidimensional space of social characteristics. Using this method, the coefficients of the model are identified and the features of its phase trajectories are studied. In this paper, we are describing human according to his role in information processing, considering his awareness and cognitive abilities. We construct two lifetime indices of human capital: creative individual (generalizing cognitive abilities) and productive (generalizing the amount of information mastered by a person) and formulate the problem of their multi-criteria (two-criteria) optimization taking into account life expectancy. This approach allows us to identify and economically justify the new requirements for the education system and the information environment of human existence. It is shown that the Pareto-frontier exists in the optimization problem, and its type depends on the mortality rates: at high life expectancy there is one dominant solution, while for lower life expectancy there are different types of Paretofrontier. In particular, the Pareto-principle applies to Russia: a significant increase in the creative human capital of an individual (summarizing his cognitive abilities) is possible due to a small decrease in the creative human capital (summarizing awareness). It is shown that the increase in life expectancy makes competence approach (focused on the development of cognitive abilities) being optimal, while for low life expectancy the knowledge approach is preferable.

The paper presents the results of applying a scheme of very high accuracy and resolution to obtain numerical solutions of the Navier – Stokes equations of a compressible gas describing the occurrence and development of instability of a two-dimensional laminar boundary layer on a flat plate. The peculiarity of the conducted studies is the absence of commonly used artificial exciters of instability in the implementation of direct numerical modeling. The multioperator scheme used made it possible to observe the subtle effects of the birth of unstable modes and the complex nature of their development caused presumably by its small approximation errors. A brief description of the scheme design and its main properties is given. The formulation of the problem and the method of obtaining initial data are described, which makes it possible to observe the established non-stationary regime fairly quickly. A technique is given that allows detecting flow fluctuations with amplitudes many orders of magnitude smaller than its average values. A time-dependent picture of the appearance of packets of Tollmien – Schlichting waves with varying intensity in the vicinity of the leading edge of the plate and their downstream propagation is presented. The presented amplitude spectra with expanding peak values in the downstream regions indicate the excitation of new unstable modes other than those occurring in the vicinity of the leading edge. The analysis of the evolution of instability waves in time and space showed agreement with the main conclusions of the linear theory. The numerical solutions obtained seem to describe for the first time the complete scenario of the possible development of Tollmien – Schlichting instability, which often plays an essential role at the initial stage of the laminar-turbulent transition. They open up the possibilities of full-scale numerical modeling of this process, which is extremely important for practice, with a similar study of the spatial boundary layer.

Technological development determines the emergence of highly detailed data in time and space, which expands the possibilities of analysis, allowing us to consider consumer decisions and the competitive behavior of enterprises in all their diversity, taking into account the context of the territory and the characteristics of time periods. Despite the promise of such studies, they are currently limited in the scientific literature. This is due to the range of problems, the solution of which is considered in this paper. The article draws attention to the complexity of the analysis of depersonalized high-frequency data and the possibility of modeling consumption changes in time and space based on them. The features of the new type of data are considered on the example of real depersonalized data received from the fiscal data operator “First OFD” (JSC “Energy Systems and Communications”). It is shown that along with the spectrum of problems inherent in high-frequency data, there are disadvantages associated with the process of generating data on the side of the sellers, which requires a wider use of data mining tools. A series of statistical tests were carried out on the data under consideration, including a Unit-Root Test, test for unobserved individual effects, test for serial correlation and for cross-sectional dependence in panels, etc. The presence of spatial autocorrelation of the data was tested using modified tests of Lagrange multipliers. The tests carried out showed the presence of a consistent correlation and spatial dependence of the data, which determine the expediency of applying the methods of panel and spatial analysis in relation to high-frequency data accumulated by fiscal operators. The constructed models made it possible to substantiate the spatial relationship of sales growth and its dependence on the day of the week. The limitation for increasing the predictive ability of the constructed models and their subsequent complication, due to the inclusion of explanatory factors, was the lack of open access statistics grouped in the required detail in time and space, which determines the relevance of the formation of high-frequency geographically structured data bases.

When making decisions in various fields of human activity, it is often required to create text documents. Traditionally, the study of texts is engaged in linguistics, which in a broad sense can be understood as a part of semiotics — the science of signs and sign systems, while semiotic objects are of different types. The method of rank distributions is widely used for the quantitative study of sign systems. Rank distribution is a set of item names sorted in descending order by frequency of occurrence. For frequency-rank distributions, researchers often use the term «power-law distributions».

In this paper, the rank distribution method is used to analyze the Criminal Code of various countries. The general idea of the approach to solving this problem is to consider the code as a text document, in which the sign is the measure of punishment for certain crimes. The document is presented as a list of occurrences of a specific word (character) and its derivatives (word forms). The combination of all these signs characters forms a punishment dictionary, for which the occurrence frequency of each punishment in the code text is calculated. This allows us to transform the constructed dictionary into a frequency dictionary of punishments and conduct its further research using the V. P. Maslov approach, proposed to analyze the linguistics problems. This approach introduces the concept of the virtual frequency of crime occurrence, which is an assessment measure of the real harm to society and the consequences of the crime committed in various spheres of human life. On this path, the paper proposes a parametrization of the rank distribution to analyze the punishment dictionary of the Special Part of the Criminal Code of the Russian Federation concerning punishments for economic crimes. Various versions of the code are considered, and the constructed model was shown to reflect objectively undertaken over time by legislators its changes for the better. For the Criminal Codes in force in the Federal Republic of Germany and the People’s Republic of China, the texts including similar offenses and analogous to the Russian special section of the Special Part were studied. The rank distributions obtained in the article for the corresponding frequency dictionaries of codes coincide with those obtained by V. P. Maslov’s law, which essentially clarifies Zipf’s law. This allows us to conclude both the good text organization and the adequacy of the selected punishments for crimes.