### Article

## On combinatorial properties of a higher asymptotic ergodic invariant of magnetic lines

We investigate combinatorial properties of a higher invariant of magnetic lines. Assume that a 3-component link L is modeled by a magnetic eld B, which is represented by 3 closed magnetic lines. Main Theorem relates the integral invariant M(B) and a combinatorial invariant ~M (L), dened from the Conway polynomial. As a corollary of Main Theorem, asymptotic properties for combinatorial links are proposed. The combinatorial invariant ~M satises these asymptotic properties.

Spontaneous emission spectra of two initially excited closely spaced identical atoms are very sensitive to the strength and the direction of the applied magnetic field. We consider the relevant schemes that ensure the determination of the mutual spatial orientation of the atoms and the distance between them by entirely optical means. A corresponding theoretical description is given accounting for the dipole-dipole interaction between the two atoms in the presence of a magnetic field and for polarizations of the quantum field interacting with magnetic sublevels of the two-atom system.

A method is proposed for the creation of an entangled metastable (subradiance) excited state in a system of two closely spaced identical atoms. The system of unexcited atoms is first placed in a magnetic field that is directed at a magic angle of \alpha_0= arccos(1/ 3) ≈ 54.7° to the line connecting the atoms and has a transverse gradient. The gradient of the field results in the detuning of frequencies of an optical transition of the atoms. Then, the resonant laser excitation of an atom with a higher transition frequency is performed with the subsequent adiabatic switching-off of the gradient of the magnetic field. It is shown that the excited atomic system in this case transits with overwhelming probability to an entangled subradiance state. Requirements on the spectroscopic parameters of the transitions and on the rate of varying the gradient of the magnetic field necessary for the implementation of this effect are analyzed.

The hypothesis stating that the distribution of sunspot groups versus their size ($\varphi$) follows a power law in the domain of small groups was recently highlighted but rejected in favor of a Weibull distribution. In this paper we re-consider this question, and are led to the opposite conclusion. We suggest a new definition of group size, namely the spatio-temporal ``volume'' ($V$) obtained as the sum of the observed daily areas instead of a single area associated with each group. With this new definition of ``size'', the width of the power-law part of the distribution increases from 1.5 to 2.5. The exponent of the power-law is close to 1. The width of the power-law part and its exponent are stable with respect to the different catalogues and computational procedures used to reduce errors in the data. The observed distribution is not fit adequately by a Weibull distribution. The existence of a wide power-law part of the distribution suggests that self-organized criticality underlies the generation and evolution of sunspot groups and that the mechanism responsible for it is scale-free over a large range of sizes.

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.

This proceedings publication is a compilation of selected contributions from the “Third International Conference on the Dynamics of Information Systems” which took place at the University of Florida, Gainesville, February 16–18, 2011. The purpose of this conference was to bring together scientists and engineers from industry, government, and academia in order to exchange new discoveries and results in a broad range of topics relevant to the theory and practice of dynamics of information systems. Dynamics of Information Systems: Mathematical Foundation presents state-of-the art research and is intended for graduate students and researchers interested in some of the most recent discoveries in information theory and dynamical systems. Scientists in other disciplines may also benefit from the applications of new developments to their own area of study.