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## Radiation due to homogeneously accelerating sources

The core of this work is an old and broadly discussed problem concerning electromagnetic radiation in the case of hyperbolic motion. We prove that radiation is nonzero in the lab (Minkowski) frame. Further, we attempt to understand this subject better by using comoving noninertial frames of reference, investigating other types of uniformly accelerated motion, and, finally, using scalar waves instead of pointlike particles as sources of radiation.

The paper deals with comparative analysis of two groups of polysemantic words in Russian and in English proving the idea of relativity of the structural scheme of the polysemantic word. The specificity of the lingual unit is always relative and depends on the background against which the comparison is performed.

This text is a translation of an article of B.L. Whorf “Language, mind and reality” (first published in 1941). The text was originally written for the journal Theosophist (India) during the last year of Whorf’s life. The article contains a formulation of the principle of linguistic relativity that relates to the idea of that the world picture of a user of a language depends on the grammar of the language she is using. The article also contains a critique of the Western science from Whorf’s theosophist perspective. The paper was translated in Russian by Andrey A. Veretennikov.

The article deals with the possibility of applying the systematic methodology in the process of investigating both the integral state-legal matter and the transitional one in terms of changing the social system on the example of the history of Russia of XX century**. Purpose**: to carry out the analysis of the existing system methodological tools for understanding the features of state-legal matter in transitional periods of Russian history. **Methods:** methodological basis of the research are system and historical methods to uncover quality content of the legal matter of the transitional period; to distinguish the principal difference between system and intersystem state-legal situation; to investigate the distinctive features of system changes in public and legal superstructure when moving from one social system to another; to review denying the possibility of topology of state and law of the transitional period. **Results:** during the investigation the basic approaches in system methodology were studied, including the possibility of its application in the process of studying the legal superstructure of the transitional period. The author agrees with the views of those scholars (A.I. Uyomov, D. G. Krasilnikov), who draws attention to such important feature of the system as relativity which allows to move away from characterizing the social systems and their subsystems only through the prizm of presence or absence of the integrity sign. Analysis of correlation of state-legal superstructure elements and strategic development of competing public systems allows us to provide both common and special typological features of state and law of the transitional period in the history of Russia. **Conclusions:** fundamental difference of approaches to systematic methodology concerns not the form but the content of its categorical framework. The categories of “system”, “nonsystem”, “intersystem” help carry out a more in-depth analysis of the legal superstructure in the process of changing social systems.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

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.