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## The fourth-order correlation function of a randomly advected passive scalar

We study the joint exit probabilities of particles in the totally asymmetric simple exclusion process (TASEP) from space-time sets of a given form. We extend previous results on the space-time correlation functions of the TASEP, which correspond to exits from the sets bounded by straight vertical or horizontal lines. In particular, our approach allows us to remove ordering of time moments used in previous studies so that only a natural space-like ordering of particle coordinates remains. We consider sequences of general staircase-like boundaries going from the northeast to southwest in the space-time plane. The exit probabilities from the given sets are derived in the form of a Fredholm determinant defined on the boundaries of the sets. In the scaling limit, the staircase-like boundaries are treated as approximations of continuous differentiable curves. The exit probabilities with respect to points of these curves belonging to an arbitrary space-like path are shown to converge to the universal Airy 2 process.

Homogeneous and isotropic with respect to horizontal variables random fields are useful for study of geophysical (in particular, meteorological) functions of spatial-temporal variables. The following horizontal scale (30 — 3000 km), which is induced by the spatial scale of the observing grid for the Earth’s atmosphere and by the power of modern computers for solutions of the system of hydrothermodynamics equations, which included water phase transformations etc, is important for the weather forecast problems.

The correlation functions (CFs) of the random fields may be applied for the following goals:

1) For the optimal interpolation of the meteorological information from the points of observation into the points of a regular finite-difference grid, as well as (for the checking of some observations by other ones) into another point of the observation.

2) For the models’ testing, if a climate model simulates adequately not only mean fields, but the fields of the relative dispersions and CFs, too, then we should consider the climate model as a certain one.

The CFs are evaluated by the global checked archive of meteorological observations by meteorological sounds. A special regularization procedure provides the strong positive definiteness of the CFs. The areas in the Earth atmosphere, where the isotropy hypothesis is essentially not fulfilled, were localized by a special algorithm.

Let us consider an algorithm, which can construct atmospheric fronts that separate so named homogeneous synoptic atmospheric volumes. Then we can evaluate separately CFs for the ensemble of the pairs of points, which are in a unite volume and CFs for the ensemble of the pairs of points, which are in a various volumes. We can see the difference between the different CFs. The difference will be more for a better algorithm. So, we obtain a quality criterion for such algorithms. The statistical approach given possibility to optimize the algorithm with respect to a lot of numerical parameters. The optimal algorithm was exploited in the operative regime in Hydrometeorological Center of Russia. The similar algorithms of numerical construction of boundaries between homogeneous volumes by a discrete set of observations can be realized for various physical media.

In this study we investigate the properties of correlation function of the special type. Function is used in deriving basic integro - differential equation for the evolution of the averaged concentration of particles in the presence of the random force.

Our result confirms the previously used proposition, which is the core for basic equation deriving.

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.