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## Взаимодействие коротких однокомпонентных векторных солитонов в средах со смещенной дисперсией (адиабатическое приближение)

The interaction of short single-component vector solitons in the frame of the coupled third–order nonlinear Schrodinger equations taking into account third–order linear dispersion, self–stepping, self–stimulated Ramanscattering, cross–stepping and cross–stimulated Raman-scattering terms is considered. Conditions of reflection and propagation of the solitons through each other and also the condition of oscillator interaction (vector breather) are obtained.

The derivation of the polarization multipliers for improving the pulse** **characteristic method for aperture antennas field analysis is presented. Numerical electrodynamic modeling was performed using the FIT method. The results of the analytically calculated signals are compared with the simulation results, and the average error between the methods for various polarization multipliers was determined.

The complex phenomena of the individual creative activities as well as the historical development of scientific knowledge are under consideration from the point of view of the theory of self-organization (synergetics) in the book. Synergetics is characterized as a new research programme in a wide philosophical, cultural and historical context. The synergetical reinterpretations of some peculiarities of the creative thinking, such as the alternative ways and the scenarios, the latent attitudes and the predeterminations, the self-completing of whole images, are proposed here. The synergetical view of historical development of scientific knowledge is compiled in the book from the notions of the principal nonlinearity and cyclic character of science development,the inertia of the paradigmal consciousness in science, the value of marginal and archaic elements in science. For readers who are interested in evolutionary epistemology and the philosophical problems of synergetics.

The effect of dielectric supports on the slowing factor and dispersion of the helical line in TWT is considered. A method for the calculation of the slowing down in the helical line with the complicated configuration of the dielectric supports is proposed. A procedure for the experimental study of dispersion in the helical slow_wave system is presented. The calculated results are compared with the experimental data.

A TWT model formed by a meta-magnetic plate and a metal screen is offered and analyzed. The dispersion equation of the model in the presence of a homogeneous electron beam filling the space between the plate and the screen is derived and solved. The coupling and depression coefficients are calculated by the method of differentiation the dispersion equation. The calculated characteristics are compared with the TWT models on a dielectric plate and an “impedance” comb.

Propagation of the short vector envelope solitons in a inhomogeneous medium with linear potential in coupled third–order nonlinear Shrodinger equations frame is considered. Explicit vector soliton solution is obtained. The explicit solution for the solitons trajectories is studied. In particular cases this solitons solution can be reduced as to the short scalar soliton solution on linear inhomogeneity profile, as to well – known Chen soliton solution.

The evolutionary model elaborated by Sergei P. Kurdyumov is considered in the article. Some key ideas put forward by him constitute a basis for development of the methodology of sudy of complex selforganizing systems, called also synergetics. Four important theoretical notions form a fundament of this evolutionary model: connection between space and time, complexity and its nature, blow-up regimes, in which self-organization and rapid, avalanche-like growth of complexity occur, evolutionary cycles and switching of different regimes as a necessary mechanism for maintenance of “life” of complex structures. The methodology allows to understand the nature of innovative shifts in nature and society and to show a possibility of management of innovative processes and of construction of desirable future. Some approaches for possible application of this model for understanding of dynamics of complex social, demographic and geopolitical system are discussed.

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.