### Working paper

## A criterion for left-orthogonality of an effective divisor on a surface

This edition presents abstracts of the reports of the Meeting and Youth Conference on Neutron Scattering and Synchrotron Radiationin Condensed Matte (NSSR-CM-2014)r

In his reasoning concerning the relationship between surface or visible superficies (understood as the boundary or the limit of a body) and color (*De sensu* 439a19–b17), Aristotle asserts that the Pythagoreans called the surface (ἐπιφάνεια) color (χροιά), i.e. that they made no terminological difference between the former and the latter. In the scholarship on early Pythagoreans, this passage has been usually used as an indirect proof for the inaccuracy of attribution to the early Pythagoreans (1) of the abstract notion of surface (as found in Plato and Euclid), and thereby (2) of various forms of “derivation theory”. We argue that the colour-surface-limit doctrine has great significance for the understanding of the early Pythagorean concept of a number, since they articulated it, in various ways, precisely through the notion of a limit.

Diffusion transport of material sputtered from the surface of the powered electrode in the asymmetric alternating current discharge is theoretically studied. It is shown that amplitudes of the non-stationary component of the sputtered atom (SA) flow densities at the electrodes depend on the discharge frequency and two dimensionless parameters, which are functions of the SA mass, its mean free path length in the background gas and the distance between the electrodes. It is found that diffusion damping of the time-varying component of the SA number density takes place in the discharge volume under certain conditions and their flows at the electrodes can be considered as time-independent.

The book is an introduction to the qualitative theory of dynamical systems on manifolds of low dimension (on the circle and on surfaces). Along with classical results, it reflects the most significant achevements in this area obtained in recent times. The reader of this book need to be familiar only with basic courses in differential equations and smooth manifolds.

Main regularities of the influence of the air adsorbate on the interpretation of images of thin metal films were experimentally determined in the scanning tunneling microscopy (STM). Modification of the surface relief of a thin film of Pt was made in air.Effect of formation of surface structures of 50-100 nm, a cluster of polarized adsorbate molecules by a strong electric field in the electrode gap, was defined. Tunnel voltage and current threshold values of irreversible relief changes was obtained. Technique of local adsorbate removal from the test surface area was developed by pulse contactless interaction of STM electrodes.

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.