### Working paper

## Оценка распределения влияния в Европейском парламенте в 1979–2014гг.

The article contains an analysis of the Directive of the European Parliament and of the Council on services in the internal market. The study discusses the main amendments to the original Directive proposal; defines the content of the country of origin principle; determines the scope of the Directive and of the freedom to provide services. A particular attention is paid to the article 16 of the Directive. The author examines the principles, that Member States have to respect in order to impose requirements with regard to the provision of a service activity by providers, established within the territory of another Member State. Based on this research, the author makes an attempt to appraise the importance of this Directive for the liberalization process of trade in services sector.

We propose a novel method to estimate the level of interconnectedness of a financial institution or system, as the measures currently suggested in the literature do not fully take into consideration an important aspect of interconnectedness — group interactions of agents. Our approach is based on the power index and centrality analysis and is employed to find a key borrower in a loan market. It has three distinctive features: it considers long-range interactions among agents, agents’ attributes and a possibility of an agent to be affected by a group of other agents. This approach allows us to identify systemically important elements which cannot be detected by classical centrality measures or other indices. The proposed method is employed to analyze the banking foreign claims as of 1Q 2015. Using our approach, we detect two types of key borrowers (a) major players with high ratings and positive credit history; (b) intermediary players, which have a great scale of financial activities through the organization of favorable investment conditions and positive business climate.

This paper demonstrates that most existing voting schemes represent or can be rewritten as weighted games. However, axiomatics for power indices defined on simple games are not directly applied to weighted games, since related operations become ill-posed. The author shows that the majority of axiomatics can be adapted to weighted games. Finally, a series of examples are provided.

At present, it is difficult to talk about the growth of the EP's influence. A number of referendums in European countries show the lack of confidence of citizens in the institutions of the European Union. At the same time, in the EP itself, we see an increase in the influence of euro-skeptical parties.

This publication is devoted to issues related to religious policy of the European Parliament as well as the history of the European Union. The paper was requested by the EP.

We offer a general approach to describing power indices that account for preferences as suggested by F. Aleskerov. We construct two axiomatizations of these indices. Our construction generalizes the Laruelle-Valenciano axioms for Banzhaf (Penrose) and Shapley-Shubik indices. We obtain new sets of axioms for these indices, in particular, sets without the anonymity axiom.

At calculation of the power indices, both well-known (Banzhaf, Shapley-Shubik and others and new (depending on the agent preferences) indices, one generally has to enumerate almost all coalitions, that is, the subsets of the set of players, which makes calculations impossible if the number of players exceeds fifty. Yet, if all players have an integer number of votes, there are players with the same number of votes, many coalitions have equal total number of votes or the sum of votes of all players is small, then the algorithms based on calculations using the generating functions become efficient. But these algorithms works only for classical power indices and some particular types of the power indices based on agents’ preferences. In this paper we consider an important specific case when all players have the same number of votes. For classical power indices in this case all players have the same power. However, it is not the case for the indices which allow preferences of agents. We introduce effective algorithms for calculation of the latter indices for most types of these indices.

We consider certain spaces of functions on the circle, which naturally appear in harmonic analysis, and superposition operators on these spaces. We study the following question: which functions have the property that each their superposition with a homeomorphism of the circle belongs to a given space? We also study the multidimensional case.

We consider the spaces of functions on the m-dimensional torus, whose Fourier transform is p -summable. We obtain estimates for the norms of the exponential functions deformed by a C1 -smooth phase. The results generalize to the multidimensional case the one-dimensional results obtained by the author earlier in “Quantitative estimates in the Beurling—Helson theorem”, Sbornik: Mathematics, 201:12 (2010), 1811 – 1836.

We consider the spaces of function on the circle whose Fourier transform is p-summable. We obtain estimates for the norms of exponential functions deformed by a C1 -smooth phase.

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.