### Book

## Струйные и нестационарные течения в газовой динамике

Moderately strong shocks arise naturally when two subclusters merge. For instance, when a smaller subcluster falls into the gravitational potential of a more massive cluster, a bow shock is formed and moves together with the subcluster. After pericentre passage, however, the subcluster is decelerated by the gravity of the main cluster, while the shock continues moving away from the cluster centre. These shocks are considered as promising candidates for powering radio relics found in many clusters. The aim of this paper is to explore the fate of such shocks when they travel to the cluster outskirts, far from the place where the shocks were initiated. In a uniform medium, such a ‘runaway’ shock should weaken with distance. However, as shocks move to large radii in galaxy clusters, the shock is moving down a steep density gradient that helps the shock to maintain its strength over a large distance. Observations and numerical simulations show that, beyond *R_{*500}, gas density profiles are as steep as, or steeper than, ∼*r^{*−3}, suggesting that there exists a ‘habitable zone’ for moderately strong shocks in cluster outskirts where the shock strength can be maintained or even amplified. A characteristic feature of runaway shocks is that the strong compression, relative to the initial state, is confined to a narrow region just behind the shock. Therefore, if such a shock runs over a region with a pre-existing population of relativistic particles, then the boost in radio emissivity, due to pure adiabatic compression, will also be confined to a narrow radial shell.

The application of mathematical modeling methods (with subsequent computer sales) to determine the parameters of accuracy geometry bands obtained with the new equipment and process the step deformation bands of hard alloys based on copper

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.

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.

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.