### Article

## On large deviations in the averaging principle for SDE’s with a ``full dependence’’, revisited

We use the characterization of distribution symmetry in terms of order statistics in order to obtain new tests of symmetry based on U-empirical distribution functions. We calculate their limiting distributions and large deviations and explore their local Bahadur efficiency against location alternatives which turns out to be rather high.

We propose new tests of exponentiality of integral and of Kolmogorov type based on a characterization of exponentiality proposed by Ahsanullah. Bahadur efficiency of new tests is computed, conditions of local asymptotic optimality are described.

We discuss some well-known facts about Clifford algebras: matrix representations, Cartan’s periodicity of 8, double coverings of orthogonal groups by spin groups, Dirac equation in different formalisms, spinors in <span data-mathml="nn dimensions, etc. We also present our point of view on some problems. Namely, we discuss the generalization of the Pauli theorem, the basic ideas of the method of averaging in Clifford algebras, the notion of quaternion type of Clifford algebra elements, the classification of Lie subalgebras of specific type in Clifford algebra, etc.

Symmetric random walks in $R^d$ and $Z^d$ are considered. It is assumed that the jump distribution density has moderate tails, i.e., several density moments are finite, including the second one. The global (for all $x$ and $t$) asymptotic behavior at infinity of the transition probability (fundamental solution of the corresponding parabolic convolution operator) is found. Front propagation of ecological waves in the corresponding population dynamics models is described.

In this paper, we consider the spectral problem for the magnetic Schrödinger operator on the 2-D plane (*x*1*, x*2) with the constant magnetic field normal to this plane and with the potential *V *having the form of a harmonic oscillator in the direction *x*1 and periodic with respect to variable *x*2. Such a potential can be used for modeling a long molecule. We assume that the magnetic field is quite large, this allows us to make the averaging and to reduce the original problem to a spectral problem for a 1-D Schrödinger operator with effective periodic potential. Then we use semiclassical analysis to construct the band spectrum of this reduced operator, as well as that of the original 2-D magnetic Schrödinger operator.

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.