### Article

## The Bohr--Pal Theorem and the Sobolev Space W_2^{1/2}

The well-known Bohr--Pal theorem

asserts that for every continuous real-valued function f on

the circle T there exists a change of variable, i.e.,

a homeomorphism h of T onto itself, such that the

Fourier series of the superposition foh converges

uniformly. Subsequent improvements of this result imply that

actually there exists a homeomorphism that brings f into the

Sobolev space W_2^{1/2}(T). This refined version of

the Bohr--Pal theorem does not extend to complex-valued

functions. We show that if \alpha<1/2, then there exists a

complex-valued f that satisfies the Lipschitz condition of

order \alpha and at the same time has the property that

foh is not in W_2^{1/2}(T) for every homeomorphism

h of T.

This paper presents an empirical analysis of the Russian market of mergers and acquisitions in 2003-2012. This analysis allowed for the conclusion that, to assess and forecast the integration activity of Russian companies, the most precise and appropriate models are seasonal autoregressive integrated moving average models built on weighted observations to eliminate the effect of the structural changes which are characteristic of developing economies. Forecasting the values of development of the market for corporate control may serve as “input” information to form a prompt regulation system for the mergers and acquisitions of holding companies, which meets current needs.

We consider domains D ⊆ ℝn with C1 boundary and study the following question: For what domains D does the Fourier transform 1D of the characteristic function 1D belong to Lp(ℝn)?

This paper presents an empirical analysis of the Russian market of mergers and acquisitions (the largest market for corporate control in Central and Eastern Europe) in 2003-2012 in terms of the total volume and value of the merger and acquisition deals of the holding companies. This analysis allowed for the conclusion that, to assess and forecast the integration activity of holding companies, the most precise and appropriate models are seasonal autoregressive integrated moving average *models* built on weighted observations to eliminate the effect of the structural changes which are characteristic of developing economies. Forecasting the values of development of the market for corporate control may serve as “input” information to form a prompt regulation system for the mergers and acquisitions of holding companies, which meets current needs. The presented analysis makes it possible to work out measures of public policy to increase the efficiency of the integration activity of holding companies.

The theme of the conference is related to the different areas of mathematics, especially harmonic analysis, functional analysis, operator theory, function theory, differential equations and fractional analysis, developed intensively last decade. The relevance of this topic is related to the study of complex multiparameter objects that require, in particular, to attract operators with variable parameters and functional spaces with fractional and even variable exponents.

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.