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## Unbounded probability theory compatible with the probability theory of numbers

We study the case where the values of random variables increase without bound.

We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z) diagram by the second sheet.

We have compared the results of our mathematical conception with the data of recent experiments concerning gas and thermodynamics. We introduce new notions (variables): the temperature *T*, the number of degrees of freedom D, the chemical potential mu, or its analog in economics, the nominal credit rate R.

The passage from a fluctuation cluster to a cluster as a distinguishable object is considered as the passage to the appearance of the Boltzmann statistics, or the phenomenon of dissipation, viscosity. We consider the economic analogy: to the chemical potential corresponds, in economics, minus the nominal credit rate. The examples of such nonstandard situation are given.

We single out the main features of the mathematical theory of noble gases. It is proved that the points of degeneracy of the Bose gas fractal dimension in momentum space coincide with the critical points of noble gases, while the jumps of the critical indices and the Maxwell rule are related to tunnel quantization in thermodynamics. We consider semiclassical methods for tunnel quantization in thermodynamics as well as those for second and ultrasecond quantization (the creation and annihilation operators for pairs of particles). Each noble gas is associated with a new critical point of the limit negative pressure. The negative pressure is equivalent to covering the (P,Z)- diagram by the second sheet.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

By using superconducting quantum interference device (SQUID) magnetometry, we investigated anisotropic high-field (H less than or similar to 7T) low-temperature (10 K) magnetization response of inhomogeneous nanoisland FeNi films grown by rf sputtering deposition on Sitall (TiO2) glass substrates. In the grown FeNi films, the FeNi layer nominal thickness varied from 0.6 to 2.5 nm, across the percolation transition at the d(c) similar or equal to 1.8 nm. We discovered that, beyond conventional spin-magnetism of Fe21Ni79 permalloy, the extracted out-of-plane magnetization response of the nanoisland FeNi films is not saturated in the range of investigated magnetic fields and exhibits paramagnetic-like behavior. We found that the anomalous out-of-plane magnetization response exhibits an escalating slope with increase in the nominal film thickness from 0.6 to 1.1 nm, however, it decreases with further increase in the film thickness, and then practically vanishes on approaching the FeNi film percolation threshold. At the same time, the in-plane response demonstrates saturation behavior above 1.5-2T, competing with anomalously large diamagnetic-like response, which becomes pronounced at high magnetic fields. It is possible that the supported-metal interaction leads to the creation of a thin charge-transfer (CT) layer and a Schottky barrier at the FeNi film/Sitall (TiO2) interface. Then, in the system with nanoscale circular domains, the observed anomalous paramagnetic-like magnetization response can be associated with a large orbital moment of the localized electrons. In addition, the inhomogeneous nanoisland FeNi films can possess spontaneous ordering of toroidal moments, which can be either of orbital or spin origin. The system with toroidal inhomogeneity can lead to anomalously strong diamagnetic-like response. The observed magnetization response is determined by the interplay between the paramagnetic-and diamagnetic-like contributions.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are 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.

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.