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## Analytic number theory and disinformation

We introduce the notion of disinformation as an object annihilated by the corresponding information and analyze the relationships of this object with abstract analytic number theory and thermodynamics. For the entropy in this statistics, we present Nazaikinskii’s model, which contains only the statistics of Bose and Fermi gases (and excludes parastatistics).

This paper presents UD-statistics almost coinciding with Gentile statistics for negative chemical potentials, but having another extension in the case of positive chemical potentials. The physical meaning of the metamorphosis of the phase transition of the third kind to the phase transition of the first kind is explained. The coincidence of isochores and isotherms in the supercritical domain corresponding to UD statistics is shown.

In this paper, general questions concerning equilibrium and non-equilibrium states are discussed. Using the Van-der-Waals model, the relationship between Gentile statistics and non-ideal gas is demonstrated. The second virial coefficient is expressed in terms of collective degrees of freedom. The admissible cluster size at given temperature is determined.

The order statistics and the empirical mathematical expectation (also called the estimate of mathematical expectation in the literature) are considered in the case of infinitely increasing random variables. The Kolmogorov concept which he used in the theory of complexity and the relationship with thermodynamics which was pointed out already by Poincar\'e are considered. We compare the mathematical expectation (which is a generalization of the notion of arithmetical mean, and is generally equal to infinity for any increasing sequence of random variables) with the notion of temperature in thermodynamics similarly to nonstandard analysis. It is shown that there is a relationship with the Van-der-Waals law of corresponding states. A number of applications of this concept in economics, in internet information network, and self-teaching systems are also 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.

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.