### Working paper

## The Jordan property for Lie groups and automorphism groups of complex spaces

Discusses the problems of a museum object as a historical source in the system of information resources of historical science and space of “museum history” in the structure of actual historical knowledge. It also reveals the impact of the transformation of social and cultural situations, social change request to the historical knowledge and, accordingly, modification of the relationship between the “museum history” and historical knowledge in general. Examines the impact of the transformation of the epistemological situation to engage in scientific circulation of visual material as historical sources – museum objects. Analyzed the museum exhibition as a form of representation of history in the historical context of the relation of science and socially oriented research.

Formation of democratic societies of the Western type presupposes appearance on the historical scene of a new strong actor - the bourgeois class: "No bourgeoisie, no democracy" (Barrington Moore). The articulation and defense of vital interests of that class creates a new social space - "the bourgeois public sphere" which helps to make up "counterbalance" to absolutism of a corporate state - a civil society, the core of which is composed by public opinion. In the confrontation between the authorities and society one of the most important roles is played by the press that provides free debate and discussion of generally valid problems, especially economic and political. The recognition of the mass media role was stamped in its characterization in XIII century as "the fourth power". Technological development of the media incredibly expanded its functions, turning journalists into creating informational analogue of reality, saturating daily life with new meanings. Methods of the representation of reality, the specific nature of political influence of journalists - key members of the reflexive elites (Helmut Shelski), are the themes of this article.

Публичная сфера, журналистика, четвертая власть, порядки знания, Повседневность, научное и повседневное знание, экспертиза, Репрезентация, public sphere, journalism, fourth estate, orders of knowledge, Everyday life, scientific and everyday knowledge, Expertise, representation

In this paper a unified method for studying foliations with transversal parabolic geometry of rank one is presented.

Ideas of Fraces' paper on parabolic geometry of rank one and of works of the author on conformal foliations

are developed.

We present a new method of investigation of G-structures on orbifolds. This method is founded on the consideration of a G-structure on an n-dimensional orbifold as the corresponding transversal structure of an associated foliation. For a given orbifold, there are different associated foliations. We construct and apply a compact associated foliation (M,F) on a compact manifold M for a compact orbifold. If an orbifold admits a G-structure, we construct and use a foliated G-bundle for the compact associated foliation. Using our method we prove the following statement.

Theorem 1. On a compact orbifold N the group of all automorphisms of an elliptic G-structure is a Lie group, this group is equipped with the compact-open topology, and its Lie group structure is defined uniquely.

By the analogy with manifolds we define the notion of an almost complex structure on orbifolds and get the following statement.

Theorem 2. The automorphism group of an almost complex structure on a compact orbifold is a Lie group, its topology is compact-open and its Lie group structure is defined uniquely.

For manifolds, the statements of Theorems 1 – 2 are classical results. Theorem 1 for manifolds was proved by Ochiai. In particular, in the case of flat elliptic G-structures on manifolds, Theorem 1 was proved by Guillemin and Sternberg and also by Ruh. Theorem 2 for manifolds was proved by Boothby, Kobayashi, Wang.

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.

This is a review article on Artemy Magun’s book «Unity and loneliness» (Moscow: New Literary Review, 2011). The main research principles of the author are critically analysed. His book is viewed as an important contribution to the Russian political philosophy, however, there are some places in the book that should be discussed or rewritten more thoroughly.

Th e article is devoted to the secondary nomination. The essence of the act of nomination is to fi x the communication of the subject and name, the phenomenon and its designation, the structures of the consciousness and its object. Man picks the right means of nomination when forms a notion of an object or phenomenon. The results showed that one of the types of secondary nomination is a semantic transposition, which does not change the material appearance of rethoughtful unit and leads to formation of its new value, i.e. for multiple purposes, namely, to metaphor, in particular, an anthropomorphic comparison.

In this work, we study the optimal risk sharing problem for an insurer between himself and a reinsurer in a dynamical insurance model known as the Kramer–Lundberg risk process, which, unlike known models, models not per claim reinsurance but rather periodic reinsurance of damages over a given time interval. Here we take into account a natural upper bound on the risk taken by the reinsurer. We solve optimal control problems on an infinite time interval for mean-variance optimality criteria: a linear utility functional and a stationary variation coefficient. We show that optimal reinsurance belongs to the class of total risk reinsurances. We establish that the most profitable reinsurance is the stop-loss reinsurance with an upper limit. We find equations for the values of parameters in optimal reinsurance strategies.

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.