### Book

## "Тесты по курсу ТВМС" "Экзаменационные тесты по ТВМС"

The test assignments are given to the main topics of the textbook Probability theory and mathematical statistics for engineering and technical areas (Moscow, 2017, Yurayt ed.) In two parts: tests - tasks and tests - theoretical exam questions. The tests of each part are presented in two forms: complete (with solutions and answers) and - for working with students (without solutions and answers) with four possible answers in each test.

This paper looks at the way of teaching academic writing skills to first-year students in General English classes. It describes the stages, the exercises and the material. Every stage is assessed and recommendations are sometimes given. At the end the analysis of the course is offered.

The manual was designed for first-year students of economics. It aims to develop their reading comprehension skills. As reading is viewed here as a means and a target of teaching, other skills (writing, listening, speaking) are being taught at the same time.

The paper covers a case of teaching critical reading and speaking to students of Economics, ICT and Mechanics at senior levels at National Research University Higher School of Economics. Relevant tested teaching materials aimed at the development of required competencies in reading and speaking are presented as well as the methodical principles used in elaboration of these materials.

The manual was designed for first-year students of economics. It aims to develop their reading comprehension skills. As reading is viewed here as a means and a target of teaching, other skills (writing, listening, speaking) are being taught at the same time.

Two-person games and cost/surplus sharing problems are worth for studying because they are the base for their extending to the classes of such problems with variable population with the help of very powerful consistency properties. In the paper a family of cost-sharing methods for cost sharing problems with two agents is extended to a class of solutions for two-person cooperative games that are larger than both cost-sharing and surplus-sharing problems, since cooperative games have no restrictions on positivity of costs and surpluses. The tool of the extension is a new invariance axiom - self covariance - that can be applied both to cost-sharing methods and to cooperative game solutions. In particular, this axiom replaces the Lower composition axiom which is not applicable to methods for profit sharing problems.

The objective of this paper is to determine what semantic components in the meaning of a word facilitate its lexicalization as prosodically marked and aid its focalization in an utterance. The paper demonstrates that prosodic and communicative properties of a word correlate with its semantic properties. In particular, a case study of different senses of the words *tol’ko *‘only’, *pravda *‘true’, *eshche *‘still, more’, *voobshche *‘in principle, generally’, *po krajnej mere *‘at least’ and some others reveals that focalization and prosodic marking in a word are triggered by the semantics of contrast, high degree, and addition. On the other hand, semantics of concession in the meaning of a word limits its ability for accentual marking and focalization. The observed correlations between semantics and prosody are confirmed by the multimedia corpus data.

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.