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## Solutions of Polynomial Equations in Subgroups of F_p^*

We present an upper bound on the number of solutions of an algebraic equation P(x,y)=0 where *x* and *y* belong to the union of cosets of some subgroup of the multiplicative group κ∗ of some field of positive characteristic. This bound generalizes the bound of Corvaja and Zannier (J Eur Math Soc 15(5):1927–1942, 2013) to the case of union of cosets. We also obtain the upper bounds on the generalization of additive energy.

This paper deals with different properties of polynomials in random elements: bounds for characteristic functionals of polynomials, a stochastic generalization of the Vinogradov mean value theorem, the characterization problem, and bounds for probabilities to hit the balls. These results cover the cases when the random elements take values in finite as well as infinite dimensional Hilbert spaces.

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.

A general theorem on the purity of invariant field extensions is proved. Using it, a criterion of rational triangulability of connected solvable affine algebraic subgroups of the Cremona groups is obtained. This criterion is applied for proving the existence of rationally nontriangulable subgroups of the above form and for proving their stable rational triangulability. The latter property answers in the affirmative Bass’ Trian gulability Problem in the stable range. A general construction of all rationally triangulable connected solvable affine algebraic subgroups of the Cremona groups is obtained. As an application, a classification of all ratio nally triangulable connected onedimensional unipotent affine algebraic subgroups of the Cremona groups up to conjugacy is given.

This chapter focuses on contradictions in the development of social anthropology curriculum in contemporary Russia. Ethnography as a predecessor to social anthropology has been developing in Russia for several centuries as an academic discipline and occupation with a strong focus on folk culture, ethnicity. In Soviet times, professional training of ethnographers was offered within the Departments of History at several universities. The Institute of Ethnology and Anthropology (previously The Institute of Enthography) is the oldest institution in Russia for studies of humanities, which sprang from the Kunstkamera (Cabinet of Curiosities) founded by Peter the Great. This long tradition of ethnography as a scholarly discipline is based on field research with emphasis on ethnic peculiarities and inter-ethnic conflicts. In the beginning of 1990s, the oldest academic institution, the Institute of Ethnography of the Russian Academy of Sciences (RAS) acquired a new name: the Institute of Ethnology and Anthropology of RAS, which signified a shift in self-identification of traditional ethnographers towards international recognition. A number of university-based and independent research centres were established in various Russian regions. The thematic scope of their research interests is wide and includes not only focus on past and present folk cultures, but also on issues of society, culture and diversity as seen in the programs of conferences and content of publications. The institutional resource for disciplinary and professional identity is a new Association of anthropologists and ethnographers that includes now more than a thousand members. The transformation of social anthropology curricula is explored on the national and local levels in relation to implications of the Bologna project and what makes social anthropology a distinctive area of professional training. The analysis shows that the characteristics of social anthropology education and training are defined as well as constrained by such structuring parameters as the conception of professionalism, highly ambivalent relations with contemporary post-socialist governments, the backgrounds of teachers and departments, a philosophy and ideology of diversity, reception of the notion of human rights and international exchange. Based on the results of analysing interviews and relevant documents, we will show contradictory processes in social anthropology curriculum in Russia.

Abstract—On the basis of logical-probabilistic approach proposed risk model successful attacks on social media on the Internet in terms of information warfare. We formulated and investigated the criterion of decision-making framework to achieve the goals of information warfare. Proposed algorithmic foundations of the developed criteria in Hadoop cluster.

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.