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

We prove that the family of all connected *n*-dimensional real Lie groups is uniformly Jordan for every n. This

implies that all algebraic (not necessarily affine) groups over fields of characteristic zero and some

transformation groups of complex spaces and Riemannian manifolds are Jordan.

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.

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.

The following topics about subgroups of the Cremona groups are discussed: (1) maximal tori; (2) conjugacy and classification of diagonalizable subgroups of codimensions 0 and 1; (3) conjugacy of finite abelian subgroups; (4) algebraicity of normalizers of diagonalizable subgroups; (5) torsion primes.

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.

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.