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## Внутренние полиэдры Клейна

The convex hull of all integer points of a noncompact polyhedron is closed and is a generalized polyhedron only under certain conditions. It is proved that if only the integer points in the interior of the polyhedron are taken, then most of the conditions can be dropped. Moreover, the object thus obtained has properties resembling those of a Klein polyhedron, and it is a Klein polyhedron in the case of an irrational simplicial cone.

It has been shown recently that the normalized median Genocchi numbers are equal to the Euler characteristics of the degenerate flag varieties. The q-analogues of the Genocchi numbers can be naturally defined as the Poincare polynomials of the degenerate flag varieties. We prove that the generating function of the Poincare polynomials can be written as a simple continued fraction. As an application we prove that the Poincare polynomials coincide with the q-version of the normalized median Genocchi numbers introduced by Han and Zeng.

This is continuation of our article [4]. When F and G in [4] are constant sequences, we obtain continued fraction for zeta(3) parametrized by some family of points (F,G) on projective line. This family of points can be obtained if from full projective line would be removed some no more than countable nowhere dense exeptional set of finite points. A countable nowhere dense set, which contains the above exeptional set of finite points, is specified also.

The paper presents a brief review on interpretation of continued fractions as a chain of superposition operations relative to one unary and one binary or two binary logical functions. Such construction makes it possible to dene continued fractions with partial quotients from the values of many-valued logic.

We found a series of continued fractions for zeta(3), parametrized by some family of pairs of sequences F,G. Two members of this series are present here; they are different from Apery-Nesterenko continued fraction.

Full papers (articles) of 2nd Stochastic Modeling Techniques and Data Analysis (SMTDA-2012) International Conference are represented in the proceedings. This conference took place from 5 June by 8 June 2012 in Chania, Crete, Greece.

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.