### Article

## Dispersion of the Arnold’s Asymptotic Ergodic Hopf Invariant and a Formula for Its Calculation

The main result of the paper is the formula that calculates the dispersion of the asymptotic Hopf invariant of a magnetic field. The paper contain examples, which describe magnetic fields in a conductive medium.

The nonlinear modes of coherent structure development in the Atmospheric boundary layer are investigated. Two-scale model of Atmospheric boundary layer is used in the calculation. The velocity field splits into large-scale profile of horizontal wind velocity and threescale velocity field. The former depends only on the vertical coordinate. The latter is connected with roll circulation and is subject to the vertical coordinate and the coordinate perpendicular to the roll direction. The influence of turbulence is parameterized by turbulent viscosity. The modification of wind profile by rolls is taken into account. Depending on the Reynolds number, different types of the hydrodynamic instabilities specific to the Atmospheric boundary layer occurred. This appears at the relative orientation of the arising geostrophic wind and roll circulation, and also at the scales and space periods of the structures. As the Reynolds number grows, the mean energy and helicity increase. Within the range of the Reynolds number between 200÷300 the dependence is close to linear, which points to the possibility of utilizing weakly nonlinear theory methods, where perturbation amplitudes increase as Re1/2. The rise of the roll asymmetry followed by remarkable growing of the extreme amplitude of a longitudinal velocity component in the direction opposite to geostrophic wind compared to the amplitudes along the lines of geostrophic wind is detected. Increase of the positive component of helicity by contrast to the negative ones is observed simultaneously. A qualitative comparison between the modeling findings and the measured characteristics of the coherent structures observed in the Atmospheric boundary layer is carried out. In July 2007, these structures were measured by acoustic sounding methods in Kalmykia, where asymmetry in the distribution of longitudinal velocity component was observed as well. An apparent pattern of roll circulation begins to reproduce in the mesoscale atmospheric model RAMS under grid size about 500 meters. Reasonably correspondence between numerical simulation findings and observable vortex with centers lying about 1200÷1300 meters high is received. The values of turbulent viscosity and effective Reynolds number are typical for unstable stratification conditions.

The possibility of the growth of a small-scale dynamo in a random flow with a weak mirror asymmetry under subcritical generation conditions is investigated. We show that the appearance of an asymmetry in a flow with nonzero helicity leads to the generation of small-scale periodic perturbations. These perturbations, in turn, support active field generation on small scales. We describe the properties of this support, the generation rate, and the characteristic scales dependent on the velocity of the random flow and the degree of its helicity. Transferring the problem to the spectral domain, we demonstrate the formation of a two-component small- and large-scale spectrum that consists of a peak localized on large scales and power laws at large and small k. The possibility of directly checking the spectral properties being described in subcritical laboratory MHD conditions is specially noted.

Building on previous results on the quadratic helicity in magnetohydrodynamics (MHD) we investigate particular minimum helicity states. These are eigenfunctions of the curl operator and are shown to constitute solutions of the quasi-stationary incompressible ideal MHD equations. We then show that these states have indeed minimum quadratic helicity.

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.