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## Об исследовании информационных характеристик естественных языков

The tasks associated with finding some information characteristics (entropy, the distribution of m-grams, syllables and phrases), the using of which allows to build the predictive models for the development of natural languages, are considered. The texts in Russian, English, German, French and Georgian languages, as classified by time intervals and styles, are experimental studied.

One of the most important criteria for selection of partners for the enterprise is the solvency of the entity. Insolvent Corporation does not attract any contractors or investors, as the interaction with them risk to lose their own money and resources. Although the insolvency does not always lead to bankruptcy or liquidation of the enterprise, but always serves as a signal about the adverse financial position for the company's partners. The insolvency of the enterprise can develop according to different scenarios that can be traced to the nature of the changes of financial indicators in dynamics. However, most existing prediction models solvency ignore this feature and do not take into account the retrospective dynamics data, which significantly reduces their accuracy. Therefore, the problem of forecasting of solvency taking into account dynamics of change of financial indicators is relevant.

Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet. An effective algorithm for calculating the entropy of an invariant Erdos measure is proposed. It is shown that, for certain values of the Bernulli parameter, the algorithm gives the Hausdorff dimension of an Erdos measure to 15 decimal places.

In the paper an approach to forecasting of probability of companies bankruptcy is proposed. The approach is based on retrospective dynamics of changes in financial position of a company. A model for estimation of bankruptcy probability is developed. Typical scenarios of changes in financial position during few years before bankruptcy are described. Time horizons of bankruptcy probability forecasting, depending on the scenario of changes in financial position, are defined.

Properties of Erdos measure and the invariant Erdos measure for the golden ratio and all values of the Bernoulli parameter are studies. It is proved that a shift on the two-sided Fibonacci compact set with invariant Erdos measure is isomorphic to the integral automorphism for a Bernoulli shift with countable alphabet.

In this paper we propose a new machine learning concept called randomized machine learning, in which model parameters are assumed random and data are assumed to contain random errors. Distinction of this approach from "classical" machine learning is that optimal estimation deals with the probability density functions of random parameters and the "worst" probability density of random data errors. As the optimality criterion of estimation, randomized machine learning employs the generalized information entropy maximized on a set described by the system of empirical balances. We apply this approach to text classification and dynamic regression problems. The results illustrate capabilities of the approach.

A new approach to network decomposition problems (and, hence, to classification problems, presented in network form) is suggested. Opposite to the conventional approach, consisting in construction of one, “the most correct” decomposition (classification), the suggested approach is focused on construction of a family of classifications. Basing on this family, two numerical indices are introduced and calculated. The suggested indices describe the complexity of the initial classification problem as whole. The expedience and applicability of the elaborated approach are illustrated by two well-known and important cases: political voting body and stock market. In both cases the presented results cannot be obtained by other known methods. It confirms the perspectives of the suggested approach.

The present article continues the investigation of the Soqotri verbal system undertaken by the Russian-Soqotri fieldwork team. The article focuses on the so-called “weak” and “geminated” roots in the basic stem. The investigation is based on the analysis of full paradigms (perfect, imperfect and jussive) of more than 170 “weak” and “geminated” Soqotri verbs.

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.