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## История и математика: Аспекты демографических и социально-экономических процессов

HIV first appeared in West-Central Africa, then spread to the South, East and West and, at the same time, practically did not reach North Africa. A possible explanation of this pattern could be in the role of Islam which pays particular attention to the prevention of extramarital sexual relations. In addition, one can mention that circumcised men suffer from HIV significantly less frequently than non-circumcised. Against such background, we had certain grounds to expect that Islamic societies would have lower levels of HIV prevalence than non-Islamic. Our cross-cultural tests have supported this hypothesis. The data have been analyzed with power-law regression. We have found a significant (p < .001) and really strong (r = -.747) negative power-law correlation between percentage of Muslims and the HIV prevalence in African countries. Of course, one should take into account that the stigma attached to HIV is also much higher among Muslims and so, Muslims tend to be tested, identified and monitored at lower numbers than those from other religious and cultural backgrounds, which implies that further in-depth research is necessary in order to detect the real relationship between variables in question.

Information in the SGEM 2017 Conference Proceedings is subject to change without notice. No part of this book may be reproduced or transmitted in any form or by any means, electronic or mechanical, for any purpose, without the express written permission of the International Scientific Council of SGEM.

The three already traditional volumes of the WDS Proceedings you are holding in the hands are composed of the contributions which have been presented during the 21st Annual Conference of Doctoral Students that was held in Prague, at Charles University, Faculty of Mathematics and Physics from May 29 to June 1, 2012. In this year, 100 student manuscripts were submitted to publishing and 88 were accepted after the review process.

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.

The author argues on expediency and mutual conditionality of evolutionary changes in the nature and in society. In the article three major factors of the evolution are allocated, namely: the accident, the factor of coincidence of circumstances and the factor of acceleration of social evolution.

This paper is devoted to the problem of cultural crisis and those points of view on this problem that were maintained by russian and western philosophers. It was written a lot of books concerning this subject. At the beginning of XX century many philosophers within different philosophical tradition and schools began to reason about the crisis of culture. For some of them it was important to stress religious aspect of crisis: the mankind has lost the belief in God — this is the reason of crisis. For others it was importatt to understand the social aspect of cultural crisis.

Cultural crisis is the crisis of values: human and freedom. In the first half of the XXth century the culture has not found answers for two questions: what is freedom and what is human?

In his article Vladimir Kantor explores the destiny of Russia intelligentsia within the context of cultural crisis that took place at the turn of XIX and XX centuries, analyzing the Vekhovs, a group of leading intellectuals who ran a collection of essays, titled "Vekhi", studying their relationship towards that Russian cultural phenomenon. To author, the intelligentsia is considered as a critical factor in the development of Russian history. Within a context of the struggle around the "Vekhi", by referring to famous philosophical and literature books, published in 1909, the author focuses on relationships between intelligentsia and ordinary people, their attractive and repulsive interaction, which represents the key theme of the Russian destiny. Any historical movement occurs through tragedy; heroes who move the history have to sacrifice themselves to provide that movement. Confirmation to that idea would be rejection and exclusion of the Russian intelligentsia from the country's mentality throughout a number of generations which ultimately led to its tragic being.

This article presents the results of a pilot study assessing the level of formation of a stochastic competence among teachers of mathematics. Besides, the indicators that reflect the competence of formation of stochastic students are identified and ranked in order of importance. Different instruments (questionnaires, tests, assignments) have been used to solve the problem under 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.

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.