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## Распределения многочленов на многомерных и бесконечномерных пространствах с мерами

We give a survey of results about distributions of polynomials on multidimensional spaces with measures.

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.

The book gives a detailed account of the theory of topological vector spaces and their applications.

Data from a field survey of the 2011 Tohoku-oki tsunami in the Sanriku area of Japan is used to plot the distribution function of runup heights along the coast. It is shown that the distribution function can be approximated by a theoretical log-normal curve. The characteristics of the distribution functions of the 2011 event are compared with data from two previous catastrophic tsunamis (1896 and 1933) that occurred in almost the same region. The number of observations during the last tsunami is very large, which provides an opportunity to revise the conception of the distribution of tsunami wave heights and the relationship between statistical characteristics and the number of observed runup heights suggested by Kajiura (1983) based on a small amount of data on previous tsunamis. The distribution function of the 2011 event demonstrates the sensitivity to the number of measurements (many of them cannot be considered independent measurements) and can be used to determine the characteristic scale of the coast, which corresponds to the statistical independence of observed wave heights.

The surface-active substances everywhere are presented on the water surface, they are exposed to waves and currents, and are involved in the processes of ocean-atmosphere exchange. They play a prominent role in the formation of small-scale part of the spectrum of wind waves and affect the manifestations of internal waves on the sea surface. In this paper, on the basis of experimental data, measurements of the surface-active agents "in situ", calculated the probability distributions of elastic films of surface-active substances in the slicks (smoothed areas of the sea surface) for the two oceanic regions (the Pacific Ocean near the island of San Diego (California ) and the central part of the Atlantic Ocean between the equator and the 35th parallel of northern latitude and 1° - 65° W). Obtained almost the same distribution function of elasticity of the films of surface-active substances (in normalized variables), which indicates the existence of a universal "climatic" of the distribution function of elasticity of the films of surfactants. The results can be used to assess the visibility of slicks on the sea surface by remote radio-physical methods.

This book gives a systematic presentation of modern measure theory as it has developed over the past century. It includes material for a standard graduate course, advanced material not covered by the standard course but necessary in order to read research literature in the area, and extensive additional information on the most diverse aspects of measure theory and its connections with other fields. Over 850 exercises with detailed hints or references are given. Bibliographical comments and an extensive bibliography with 2000 works covering more than a century are provided. The subject index includes more than 1000 items. The book is intended for graduate students, instructors of courses in measure and integration theory, and researchers in all fields of mathematics; it may serve as either a textbook, a source for a variety of advanced courses, or a reference work.

In the monography we consider theoretic and methodic origins of fundamental notions in the theory of functions of real variable. The text is designed for future and active school math teachers.

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.