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## Exact field-theoretical description of passive scalar convection in an N-dimensional long-range velocity field

We describe a new functional integral method for the computation of averages containing chronological exponentials of random matrices of arbitrary dimension. We apply these results to the rigorous study of the statistics of a passive scalar advected by a large-scale *N*-dimensional flow. In the delta-correlated case the statistics of the rate of line stretching appears to be exactly Gaussian at all times and we explicitly compute the dependence of the mean value and variance of the stretching rate on the space dimension *N*. The probability distribution function of the passive scalar is also exactly computed. Further applications of our functional integral method are suggested.

The properties of extreme wave storms in the Darss Sill area, SW Baltic Sea, are analysed based on waverider data for 1991-2010 and long-term numerical simulations. The long-term significant wave height is HS ~0.7 m and the most frequent wave periods 2-4 s. The largest measured HS is 4.46 m. The typical measured and modelled wave periods differ by up to 2 s. The annual maximum HS has notched behaviour, with an increase for 1958-1990 and since 1993, and a drastic decrease in 1991-1992. The measured annual average and maximum HS have changed insignificantly in 1991-2010 but the threshold for the top 1% of waves has considerably decreased.

In the context of globalization and liberalization of financial markets, the mutual relations between the national stock markets become more relevant. Herewith decisions depend, and they were made regarding to the global diversification of the investment portfolio. The research aims to study the nature (asymmetries and powers) of the mutual relationships of the Russian stock market with foreign stock markets. To achieve this goal, I have researched the parameters of the copula functions of the joint distribution of returns of indexes of the Russian and foreign stock markets and assessed the quality of approximation of functions of the joint distribution of the copula functions under study. To meet these challenges, I consider the model of mixed copulas (which is a function of making the transition from private distributions of random variables to their joint distribution). An estimation of the parameters using the mixed copulas is performed by the method of pseudo-maximum likelihood. The private functions of distribution of returns of stock markets are set empirically. The study confirmed the changeable nature of the relationship of the Russian stock market with foreign stock markets of developed and developing countries. From January 2000 to May 2008, the relationship of the Russian stock market with most of the foreign stock markets has seen a left-handed bias. The period from June 2008 to December 2010 is characterized by increased tightness of the relationship in both “tails” of the joint distribution of returns of stock markets. The third period (January 2011 to March 2014) was characterized by the predominance of right-handed asymmetry in the Russian stock market relationship with the majority of the foreign stock markets. Mixed copulas in most cases have shown a better approximation to the function of the joint distribution of returns pairs of stock markets compared to simple copulas. The results suggest that mixed copula functions are more efficient modeling of the relationship of the stock markets with regard to the simple copulas. Mixed copulas may be applied when assessing the risks of investing in foreign stock, as well as to determine the optimal hedge ratio while hedging currency risks.

A model of scalar turbulent advection in compressible flow is analytically investigated. It is shown that, depending on the dimensionality d of space and the degree of compressibility of the smooth advecting velocity field, the cascade of the scalar is direct or inverse. If d>4 the cascade is always direct. For a small enough degree of compressibility, the cascade is direct again. Otherwise it is inverse; i.e., very large scales are excited. The dynamical hint for the direction of the cascade is the sign of the Lyapunov exponent for particles separation. Positive Lyapunov exponents are associated to direct cascade and Gaussianity at small scales. Negative Lyapunov exponents lead to inverse cascade, Gaussianity at large scales, and strong intermittency at small scales.

The dynamics of a two-component Davydov-Scott (DS) soliton with a small mismatch of the initial location or velocity of the high-frequency (HF) component was investigated within the framework of the Zakharov-type system of two coupled equations for the HF and low-frequency (LF) fields. In this system, the HF field is described by the linear Schrödinger equation with the potential generated by the LF component varying in time and space. The LF component in this system is described by the Korteweg-de Vries equation with a term of quadratic influence of the HF field on the LF field. The frequency of the DS soliton`s component oscillation was found analytically using the balance equation. The perturbed DS soliton was shown to be stable. The analytical results were confirmed by numerical simulations.

Radiation conditions are described for various space regions, radiation-induced effects in spacecraft materials and equipment components are considered and information on theoretical, computational, and experimental methods for studying radiation effects are presented. The peculiarities of radiation effects on nanostructures and some problems related to modeling and radiation testing of such structures are considered.

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.