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The purpose of research was to develop a mathematical model for pulsating blood flow in the part of aorta with their branches. Since the deformation of this most solid part of the aorta is small during the passage of the pulse wave, the blood vessels were considered as non-deformable curved cylinders. The article describes the internal structure of blood and some internal structural effects. This analysis shows that the blood, which is essentially a suspension, can only be regarded as a non-Newtonian fluid. In addition, the blood can be considered as a liquid only in the blood vessels, diameter of which is much higher than the characteristic size of blood cells and their aggregate formations. As a non-Newtonian fluid the viscous liquid with the power law of the relationship of stress with shift velocity was chosen. This law can describe the behaviour not only of liquids but also dispersions. When setting the boundary conditions at the entrance into aorta, reflecting the pulsating nature of the flow of blood, it was decided not to restrict the assignment of the total blood flow, which makes no assumptions about the spatial velocity distribution in a cross section. In this regard, it was proposed to model the surface envelope of this spatial distribution by a part of a paraboloid of rotation with a fixed base radius and height, which varies in time from zero to maximum speed value. The special attention was paid to the interaction of blood with the walls of the vessels. Having regard to the nature of this interaction, the so-called semi-slip condition was formulated as the boundary condition. At the outer ends of the aorta and its branches the amounts of pressure were given. To perform calculations the tetrahedral computer network for geometric model of the aorta with branches has been built. The total number of meshes is 9810. The calculations were performed with use of the software package ABACUS, which has also powerful tools for creating geometry of the model and visualization of calculations. The result is a distribution of velocities and pressure at each time step. In areas of branching vessels was discovered temporary presence of eddies and reverse currents. They were born via 0.47 s from the beginning of the pulse cycle and disappeared after 0.14 s.

In real problems of exploration seismology we deal with a heterogeneity of the nature of elastic waves interaction with the surface of a fracture by the propagation through it. The fracture is a complex heterogeneous structure. In some locations the surfaces of fractures are placed some distance apart and are separated by filling fluid or emptiness, in some places we can observe the gluing of surfaces, when under the action of pressure forces the fracture surfaces are closely adjoined to each other. In addition, fractures can be classified by the nature of saturation: fluid or gas. Obviously, for such a large variety in the structure of fractures, one cannot use only one model that satisfies all cases.

This article is concerned with description of developed mathematical fracture models which can be used for numerical solution of exploration seismology problems using the grid-characteristic method on unstructured triangular (in 2D-case) and tetrahedral (in 3D-case) meshes. The basis of the developed models is the concept of an infinitely thin fracture, whose aperture does not influence the wave processes in the fracture area. These fractures are represented by bound areas and contact boundaries with different conditions on contact and boundary surfaces. Such an approach significantly reduces the consumption of computer resources since there is no need to define the mesh inside the fracture. On the other side, it allows the fractures to be given discretely in the integration domain, therefore, one can observe qualitatively new effects, such as formation of diffractive waves and multiphase wave front due to multiple reflections between the surfaces of neighbor fractures, which cannot be observed by using effective fracture models actively used in computational seismology.

The computational modeling of seismic waves propagation through layers of mesofractures was produced using developed fracture models. The results were compared with the results of physical modeling in problems in the same statements.

This paper considers the inverse problem of seismic exploration — determining the structure of the media based on the recorded wave response from it. Large cracks are considered as target objects, whose size and position are to be determined.

he direct problem is solved using the grid-characteristic method. The method allows using physically based algorithms for calculating outer boundaries of the region and contact boundaries inside the region. The crack is assumed to be thin, a special condition on the crack borders is used to describe the crack.

The inverse problem is solved using convolutional neural networks. The input data of the neural network are seismograms interpreted as images. The output data are masks describing the medium on a structured grid. Each element of such a grid belongs to one of two classes — either an element of a continuous geological massif, or an element through which a crack passes. This approach allows us to consider a medium with an unknown number of cracks.

The neural network is trained using only samples with one crack. The final testing of the trained network is performed using additional samples with several cracks. These samples are not involved in the training process. The purpose of testing under such conditions is to verify that the trained network has sufficient generality, recognizes signs of a crack in the signal, and does not suffer from overtraining on samples with a single crack in the media.

The paper shows that a convolutional network trained on samples with a single crack can be used to process data with multiple cracks. The networks detects fairly small cracks at great depths if they are sufficiently spatially separated from each other. In this case their wave responses are clearly distinguishable on the seismogram and can be interpreted by the neural network. If the cracks are close to each other, artifacts and interpretation errors may occur. This is due to the fact that on the seismogram the wave responses of close cracks merge. This cause the network to interpret several cracks located nearby as one. It should be noted that a similar error would most likely be made by a human during manual interpretation of the data. The paper provides examples of some such artifacts, distortions and recognition errors.

The biological structure is considered as an open nonequilibrium system which properties can be described on the basis of kinetic equations. New problems with nonequilibrium boundary conditions are introduced. The nonequilibrium distribution tends gradually to an equilibrium state. The region of spatial inhomogeneity has a scale depending on the rate of mass transfer in the open system and the characteristic time of metabolism. In the proposed approximation, the internal energy of the motion of molecules is much less than the energy of translational motion. Or in other terms we can state that the kinetic energy of the average blood velocity is substantially higher than the energy of chaotic motion of the same particles. We state that the relaxation problem models a living system. The flow of entropy to the system decreases in downstream, this corresponds to Shrödinger’s general ideas that the living system “feeds on” negentropy. We introduce a quantity that determines the complexity of the biosystem, more precisely, this is the difference between the nonequilibrium kinetic entropy and the equilibrium entropy at each spatial point integrated over the entire spatial region. Solutions to the problems of spatial relaxation allow us to estimate the size of biosystems as regions of nonequilibrium. The results are compared with empirical data, in particular, for mammals we conclude that the larger the size of animals, the smaller the specific energy of metabolism. This feature is reproduced in our model since the span of the nonequilibrium region is larger in the system where the reaction rate is shorter, or in terms of the kinetic approach, the longer the relaxation time of the interaction between the molecules. The approach is also used for estimation of a part of a living system, namely a green leaf. The problems of aging as degradation of an open nonequilibrium system are considered. The analogy is related to the structure, namely, for a closed system, the equilibrium of the structure is attained for the same molecules while in the open system, a transition occurs to the equilibrium of different particles, which change due to metabolism. Two essentially different time scales are distinguished, the ratio of which is approximately constant for various animal species. Under the assumption of the existence of these two time scales the kinetic equation splits in two equations, describing the metabolic (stationary) and “degradative” (nonstationary) parts of the process.

The article deals with the nonlinear boundary-value problem of hydrogen permeability corresponding to the following experiment. A membrane made of the target structural material heated to a sufficiently high temperature serves as the partition in the vacuum chamber. Degassing is performed in advance. A constant pressure of gaseous (molecular) hydrogen is built up at the inlet side. The penetrating flux is determined by mass-spectrometry in the vacuum maintained at the outlet side.

A linear model of dependence on concentration is adopted for the coefficient of dissolved atomic hydrogen diffusion in the bulk. The temperature dependence conforms to the Arrhenius law. The surface processes of dissolution and sorptiondesorption are taken into account in the form of nonlinear dynamic boundary conditions (differential equations for the dynamics of surface concentrations of atomic hydrogen). The characteristic mathematical feature of the boundary-value problem is that concentration time derivatives are included both in the diffusion equation and in the boundary conditions with quadratic nonlinearity. In terms of the general theory of functional differential equations, this leads to the so-called neutral type equations and requires a more complex mathematical apparatus. An iterative computational algorithm of second-(higher- )order accuracy is suggested for solving the corresponding nonlinear boundary-value problem based on explicit-implicit difference schemes. To avoid solving the nonlinear system of equations at every time step, we apply the explicit component of difference scheme to slower sub-processes.

The results of numerical modeling are presented to confirm the fitness of the model to experimental data. The degrees of impact of variations in hydrogen permeability parameters (“derivatives”) on the penetrating flux and the concentration distribution of H atoms through the sample thickness are determined. This knowledge is important, in particular, when designing protective structures against hydrogen embrittlement or membrane technologies for producing high-purity hydrogen. The computational algorithm enables using the model in the analysis of extreme regimes for structural materials (pressure drops, high temperatures, unsteady heating), identifying the limiting factors under specific operating conditions, and saving on costly experiments (especially in deuterium-tritium investigations).

The work presents an experimental study of the tree structure that occurs during a medical examination. At each meeting with a medical specialist, the patient receives a certain number of areas for consulting other specialists or for tests. A tree of directions arises, each branch of which the patient should pass. Depending on the branching of the tree, it can be as final — and in this case the examination can be completed — and endless when the patient’s goal cannot be achieved. In the work both experimentally and theoretically studied the critical properties of the transition of the system from the forest of the final trees to the forest endless, depending on the probabilistic characteristics of the tree.

For the description, a model is proposed in which a discrete function of the probability of the number of branches on the node repeats the dynamics of a continuous gaussian distribution. The characteristics of the distribution of the Gauss (mathematical expectation of $x_0$, the average quadratic deviation of $\sigma$) are model parameters. In the selected setting, the task refers to the problems of branching random processes (BRP) in the heterogeneous model of Galton – Watson.

Experimental study is carried out by numerical modeling on the final grilles. A phase diagram was built, the boundaries of areas of various phases are determined. A comparison was made with the phase diagram obtained from theoretical criteria for macrosystems, and an adequate correspondence was established. It is shown that on the final grilles the transition is blurry.

The description of the blurry phase transition was carried out using two approaches. In the first, standard approach, the transition is described using the so-called inclusion function, which makes the meaning of the share of one of the phases in the general set. It was established that such an approach in this system is ineffective, since the found position of the conditional boundary of the blurred transition is determined only by the size of the chosen experimental lattice and does not bear objective meaning.

The second, original approach is proposed, based on the introduction of an parameter of order equal to the reverse average tree height, and the analysis of its behavior. It was established that the dynamics of such an order parameter in the $\sigma = \text{const}$ section with very small differences has the type of distribution of Fermi – Dirac ($\sigma$ performs the same function as the temperature for the distribution of Fermi – Dirac, $x_0$ — energy function). An empirical expression has been selected for the order parameter, an analogue of the chemical potential is introduced and calculated, which makes sense of the characteristic scale of the order parameter — that is, the values of $x_0$, in which the order can be considered a disorder. This criterion is the basis for determining the boundary of the conditional transition in this approach. It was established that this boundary corresponds to the average height of a tree equal to two generations. Based on the found properties, recommendations for medical institutions are proposed to control the provision of limb of the path of patients.

The model discussed and its description using conditionally-infinite trees have applications to many hierarchical systems. These systems include: internet routing networks, bureaucratic networks, trade and logistics networks, citation networks, game strategies, population dynamics problems, and others.

Approbation of calculation of borders of water quality classes for the purpose of ecological diagnosis and standardization by data of the Rybinsk reservoir is carried out. For bioindication indicators of phytoplankton fluorescence and the contents of pigments of phytoplankton are used. Chesnokov's importance coefficient proved to be the most preferred measure of connection for analyzing the effects of environmental factors on indicators. The factors important for environmental condition are identified. Comparison of borders between quality classes “valid” and “invalid” of factors values and boundaries of the classifications of water quality.

The paper presents a physical and numerical analysis of the dynamics and radiation of explosion products formed during the Russian-American experiment in the ionosphere using an explosive generator based on hexogen (RDX) and trinitrotoluene (TNT). The main attention is paid to the radiation of the perturbed region and the dynamics of the products of explosion (PE). The detailed chemical composition of the explosion products is analyzed and the initial concentrations of the most important molecules capable of emitting in the infrared range of the spectrum are determined, and their radiative constants are given. The initial temperature of the explosion products and the adiabatic exponent are determined. The nature of the interpenetration of atoms and molecules of a highly rarefied ionosphere into a spherically expanding cloud of products is analyzed. An approximate mathematical model of the dynamics of explosion products under conditions of mixing rarefied ionospheric air with them has been developed and the main thermodynamic characteristics of the system have been calculated. It is shown that for a time of 0,3–3 sec there is a significant increase in the temperature of the scattering mixture as a result of its deceleration. In the problem under consideration the explosion products and the background gas are separated by a contact boundary. To solve this two-region gas dynamic problem a numerical algorithm based on the Lagrangian approach was developed. It was necessary to fulfill special conditions at the contact boundary during its movement in a stationary gas. In this case there are certain difficulties in describing the parameters of the explosion products near the contact boundary which is associated with a large difference in the size of the mass cells of the explosion products and the background due to a density difference of 13 orders of magnitude. To reduce the calculation time of this problem an irregular calculation grid was used in the area of explosion products. Calculations were performed with different adiabatic exponents. The most important result is temperature. It is in good agreement with the results obtained by the method that approximately takes into account interpenetration. The time behavior of the IR emission coefficients of active molecules in a wide range of the spectrum is obtained. This behavior is qualitatively consistent with experiments for the IR glow of flying explosion products.

The paper gives a detailed description of the developed methods for simulating the turbulent flow around a helicopter rotor and calculating its aerodynamic characteristics. The system of Reynolds-averaged Navier – Stokes equations for a viscous compressible gas closed by the Spalart –Allmaras turbulence model is used as the basic mathematical model. The model is formulated in a non-inertial rotating coordinate system associated with a rotor. To set the boundary conditions on the surface of the rotor, wall functions are used.

The numerical solution of the resulting system of differential equations is carried out on mixed-element unstructured grids including prismatic layers near the surface of a streamlined body.The numerical method is based on the original vertex-centered finite-volume EBR schemes. A feature of these schemes is their higher accuracy which is achieved through the use of edge-based reconstruction of variables on extended quasi-onedimensional stencils, and a moderate computational cost which allows for serial computations. The methods of Roe and Lax – Friedrichs are used as approximate Riemann solvers. The Roe method is corrected in the case of low Mach flows. When dealing with discontinuities or solutions with large gradients, a quasi-one-dimensional WENO scheme or local switching to a quasi-one-dimensional TVD-type reconstruction is used. The time integration is carried out according to the implicit three-layer second-order scheme with Newton linearization of the system of difference equations. To solve the system of linear equations, the stabilized conjugate gradient method is used.

The numerical methods are implemented as a part of the in-house code NOISEtte according to the two-level MPI–OpenMP parallel model, which allows high-performance computations on meshes consisting of hundreds of millions of nodes, while involving hundreds of thousands of CPU cores of modern supercomputers.

Based on the results of numerical simulation, the aerodynamic characteristics of the helicopter rotor are calculated, namely, trust, torque and their dimensionless coefficients.

Validation of the developed technique is carried out by simulating the turbulent flow around the Caradonna – Tung two-blade rotor and the KNRTU-KAI four-blade model rotor in hover mode mode, tail rotor in duct, and rigid main rotor in oblique flow. The numerical results are compared with the available experimental data.

The technology of creating patterns from natural language words (concepts) based on text data in the bag of words model is considered. Patterns are used to reduce the dimension of the original space in the description of documents and search for semantically related words by topic. The process of dimensionality reduction is implemented through the formation of patterns of latent features. The variety of structures of document relations is investigated in order to divide them into themes in the latent space.

It is considered that a given set of documents (objects) is divided into two non-overlapping classes, for the analysis of which it is necessary to use a common dictionary. The belonging of words to a common vocabulary is initially unknown. Class objects are considered as opposition to each other. Quantitative parameters of oppositionality are determined through the values of the stability of each feature and generalized assessments of objects according to non-overlapping sets of features.

To calculate the stability, the feature values are divided into non-intersecting intervals, the optimal boundaries of which are determined by a special criterion. The maximum stability is achieved under the condition that the boundaries of each interval contain values of one of the two classes.

The composition of features in sets (patterns of words) is formed from a sequence ordered by stability values. The process of formation of patterns and latent features based on them is implemented according to the rules of hierarchical agglomerative grouping.

A set of latent features is used for cluster analysis of documents using metric grouping algorithms. The analysis applies the coefficient of content authenticity based on the data on the belonging of documents to classes. The coefficient is a numerical characteristic of the dominance of class representatives in groups.

To divide documents into topics, it is proposed to use the union of groups in relation to their centers. As patterns for each topic, a sequence of words ordered by frequency of occurrence from a common dictionary is considered.

The results of a computational experiment on collections of abstracts of scientific dissertations are presented. Sequences of words from the general dictionary on 4 topics are formed.