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## Численное исследование локализованного возмущения температуры в модели фазового поля в случае слияния свободных границ

In this paper software package for numerical modeling of transformation and propagation of internal gravity waves (IGW) in the World Ocean is presented. Short overview of implemented numerical models is given. They are: extended nonlinear evolutionary equation of Korteveg-de-Vries type with combined nonlinearity with variable coefficients (Gardner equation) and ray model reproducing the effect of refraction in an IGW package. The developed software package is unique and topical for this class of geophysical applications. Description of user interface and main working modes of the software are presented.

This book is a collection of research papers selected for presentation at the International Conference on Smart Computational Methods in Continuum Mechanics 2021, organized by Moscow Institute of Physics and Technology and the Institute for Computer Aided Design of Russian Academy of Sciences. The work is presented in two volumes. The primary objective of the book is to report the state-of-the-art on smart computational paradigms in continuum mechanics and explore the use of artificial intelligence paradigms such as neural nets and machine learning for improving the performance of the designed engineering systems. The book includes up-to-date smart computational methods which are used to solve problems in continuum mechanics, engineering, seismic prospecting, non-destructive testing, and so on. The main features of the book are the research papers on the application of novel smart methods including neural nets and machine learning, computational algorithms, smart software systems, and high-performance computer systems for solving complex engineering problems. The case studies pertaining to the real-world applications in the above fields are included. The book presents a collection of best research papers in English language from some of the world leaders in the field of smart system modelling and design of engineering systems.

In this paper we demonstrate a new (probably unknown) effect of soliton type temperature perturbation which occurs in a free boundary confluence in the frames of phase field system model.

A new mathematical model of heat transfer in silicon field emission pointed cathode of small dimensions is constructed which permits taking its partial melting into account. This mathematical model is based on the phase field system, i.e., on a contemporary generalization of Stefan-type problems. The approach used by the authors is not purely mathematical but is based on the understanding of the solution structure (construction and study of asymptotic solutions) and computer calculations. The book presents an algorithm for numerical solution of the equations of the obtained mathematical model including its parallel implementation. The results of numerical simulation conclude the book.

The book is intended for specialists in the field of heat transfer and field emission processes and can be useful for senior students and postgraduates.

The physical-mathematical model of the sensors block of space radiation fluxes parameters monitoring module has been developed. The simulation of the sensors block output has been carried out using the series of the spectra representing space radiation spectra at different spaceship orbits in different phases of the solar activity cycle. The optimisation of the sensors block of space radiation fluxes parameters monitoring module has been carried out based on the simulation results.

**Purpose:** Numerical modeling of internal baroclinic disturbances of different shapes in a model lake with variable depth, analysis of velocity field of wave-induced current, especially in the near-bed layer.

**Approach:** The study is carried out with the use of numerical full nonlinear nonhydrostatic model for stratified fluid.

**Findings:** The full nonlinear numerical modeling of internal wave dynamics in a stratified lake is carried out. The calculated distributions of near-bed velocities are analyzed; the significance of 3D effects for the velocity fields is emphasized; the regions of maximal (where internal waves are the main driving factor for sediment resuspension and erosion processes on the bed) and minimal velocities are marked out.

**Originality:** The results are new and can have practical application for many applied problems, especially ecological and economical, concerned with the processes of propagation of natural and anthropogenic pollutions in natural basins and the investigation of water quality, as well as with influence upon engineering structures and sediment transport.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.