### Article

## GLOBAL ASYMPTOTICS OF THE FILTRATION PROBLEM IN A POROUS MEDIUM

Filtration of the suspension in a porous medium is important when strengthening the soil and creating

watertight partitions for the construction of tunnels and underground structures. A model of deep bed filtration

with variable porosity and fractional flow, and a size-exclusion mechanism of particle retention are considered.

A global asymptotic solution is constructed in the entire domain in which the filtering process takes place. The

obtained asymptotics is close to the numerical solution.

In the article "Secondary electronic emission and methods of its study" by author D.P Nikolaev physical and mathematical process models of secondary electronic emission of secondary electrons, wrested from hurled when irradiating a surface of metal by the flow of primary electrons are considered . Analysis of processes, running in the efficient emitters at the condition of irradiating by scanning bunch and choice necessary hardware and technical facilities has been offered. It has been considered an inaccuracy of measurement of secondary electronic emission factor under different checking. The theoretical motivations for the building mathematical models of leaving the secondary electrons from the solid-state under the action of falling on it flow of primary electrons are considered. On mathematical models, based on the result of deciding a kinetic equation, an algorithm of calculation of true secondary electronic emission factor depending on energy of primary electrons has been designed. A structured scheme of automated software-technical complex, allowing measure a factor of secondary electronic emission of different material with the high degree of validity are stated.

By analyzing the logs of corporate e-mail networks we found a number of patterns, showing how the size of ego-networks of individual employees changes on a day by day basis. We proposed a simple model that adequately describes the observed time dependence of an employee's "social circle". Comparison of experimental data with the theoretical model showed that employees are divided into two groups - with fast and slow changes in their social circles, respectively. We believe that the presence of these groups reflects both project-type and process-type of employees' activities. Comparison of data obtained before and during the global economic crisis has shown that the crisis led to an actual reduction in project-type activities.

The considered model of the failure rate of CMOS VHSIC design proposed in the article Piskun G.A., Alekseev V.F., "Improvement of mathematical models calculating of CMOS VLSIC taking into account features of impact of electrostatic discharge", published in the first issue of the journal "Technologies of electromagnetic compatibility" for the year 2016. It is shown that the authors claim that this model "...will more accurately assess the reliability of CMOS VHSIC design" is fundamentally flawed and its application will inevitably lead to inadequate results. Alternatively, the proposed model of the failure rate of CMOS VHSIC design, which also allows to take into account the views of ESD, but based on the use of resistance characteristics of CMOS VHSIC to the effects of ESD.

Simulating principles of proposed attribute (A) and object-attribute (OA) architectures of computer systems (CS) that implement the dataflow execution model, and the results of a dataflow-supercomputer system simulation are described. A new formalism of "Attribute Nets" (A-nets) is proposed for mathematical modeling of dataflow-CS that differs significantly from *apparatus* of *Petri Nets*. This formalism lays foundation for the OA-programming&simulation environment of a dataflow-CS which is used in development programming and test of the OA-supercomputer system.

Filtration problems in porous media are important for studying the movement of groundwater in porous formations and the spreading of liquid concrete injected into porous soil. Deep bed filtration of a monodisperse suspension in a homogeneous porous medium with two simultaneously acting particle capture mechanisms is considered. A mathematical model of suspension flow through porous medium with pore blocking by size-exclusion and arched bridging is developed. Exact solutions are obtained on the concentration front and at the porous medium inlet. For the linear filtration function, exact and asymptotic solutions are constructed.

The volume contains articles of scientific staff and faculty of the Department of Computer Science and Applied Mathematics and Scientific-Educational Center of computer modeling of unique buildings and complexes of Moscow State University of Civil Engineering (National Research University), devoted to actual problems of applied mathematics and computational mechanics.

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.