### Article

## Исследование усиления в полосах пропускания и запирания замедляющих систем мощных ламп бегущей волны

In this study with using of the small-signal theory of discrete electron-wave interaction in the passbands and stopbands resonator slow-wave systems (SWS) of power traveling-wave tubes (TWT), obtain the characteristic equation for the propagation constants of the 4-electron waves produced in the interaction of the electron beam forward and backward electromagnetic waves of SWS. The analysis of solutions of this equation, which allowed to establish the specific characteristics of these waves are compared with the known properties of electron waves in a "smooth", such as helical SWS. On the basis of solving the boundary value problem for the SWS segments were simulated and found gain of multisection TWT with transparent section and stopsection, as well as, the distribution of fields and currents along the stopsection.

A linear theory of the discrete interaction of electron beams and electromagnetic waves in slow-wave structures (SWS) is developed. The theory is based on the finite_difference equations of SWS excitation.The local coupling impedance entering these equations characterizes the field intensity excited by the electron beam in interaction gaps and has a finite value at SWS cutoff frequencies. The theory uniformly describes the electron–wave interaction in SWS passbands and stopbands without using equivalent circuits, a circumstance that allows considering the processes in the vicinity of cutoff frequencies and switching from the Cerenkov mechanism of interaction in a traveling wave tube to the klystron mechanism when passing to SWS stopbands. The features of the equations of the discrete electron–wave interaction in pseudoperiodic SWSs are analyzed.

In the study presents an approach to the modeling power TWT with stopband sections on the basis of the theory of discrete electron-wave interaction. Designed TWT without the use of equivalent circuits of SWS and with use of local coupling impedance and the characteristic equation of degree four.

In the age of globalisation which broadly means international interaction the idea of global communication comes to the front. Communicating globally implies using intercultural links and involves cultural knowledge of business counterparts as an integral part of global interaction. Language media being an essential tool of global interaction facilitate the process of business communication provided that certain guidelines are taken into consideration.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.

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.