### Article

## Restoration of Randomized Model Characteristics under Small Amounts of Data: Entropy-Robust Estimation

restoration

We study the Gaussian and robust covariance estimation, assuming the true covariance matrix to be a Kronecker product of two lower dimensional square matrices. In both settings we define the estimators as solutions to the constrained maximum likelihood programs. In the robust case, we consider Tyler’s estimator defined as the maximum likelihood estimator of a certain distribution on a sphere. We develop tight sufficient conditions for the existence and uniqueness of the estimates and show that in the Gaussian scenario with the unknown mean, p/q+q/p+2 samples are almost surely enough to guarantee the existence and uniqueness, where p and q are the dimensions of the Kronecker product factors. In the robust case with the known mean, the corresponding sufficient number of samples is max[p/q,q/p]+1.

This study deals with the practice of human resource valuation in the professional command football during the implementation of sport club resource strategy. The choice of the direction of research is not random. Resource view in the analysis of strategy and strategy shaping (Barney, 1991) is an effective long term success tool. There are many costs in the process of resource allocation, in particular during resource valuation, in the market economy in connection, e.g. costs related to information collection and negotiating. Costs rise with the globalization of the economy. Therefore, it is important to create and maintain various institutions to control the level of costs and to study the most common practice of the resource strategy implementation. Particularly, it is important to study resource valuation in the context of institutional environment given the heterogeneity of markets. Human resource is the most important strategic resource on the market of professional command sport. At the same time, solving the problem of human resource allocation on this market leads to serious practical problems. Specifics of economic component of command sport results in the inevitability of labor market failure (Downward, Dawson, 2002) for sustainable development of the sector. Therefore, the need for institution which is to regulate the labor market occurs. Institute of transfer plays this role in some kinds of command sport, such as football. It is the system of concepts and rules, which regulate the transfer of athlete from one sport club to another prior to the expiration of the contract. The number of requirements exist due to the institute of transfer to the practice of athlete’s valuation, which determine the sum and the order of payment of compensation for prior sporting club management. The latter should correlate with the value of the athletes, which they are interest in, with the existing restrictions, which is the reason of the strongly pronounced heterogeneity of markets. Formalization of existing practices of pricing in different segments of professional athletes’ labor market is important for the support of decisions to invest in staff during resource strategy implementation. The current research deals with this problem. The model of linear regression with variable structure is used in the paper as the main instrument of formalization. The structure of the model reflects the structure of professional athletes’ labor market.

Smoking is a problem, bringing signifi cant social and economic costs to Russiansociety. However, ratifi cation of the World health organization Framework conventionon tobacco control makes it possible to improve Russian legislation accordingto the international standards. So, I describe some measures that should be taken bythe Russian authorities in the nearest future, and I examine their effi ciency. By studyingthe international evidence I analyze the impact of the smoke-free areas, advertisementand sponsorship bans, tax increases, etc. on the prevalence of smoking, cigaretteconsumption and some other indicators. I also investigate the obstacles confrontingthe Russian authorities when they introduce new policy measures and the public attitudetowards these measures. I conclude that there is a number of easy-to-implementanti-smoking activities that need no fi nancial resources but only a political will.

One of the most important indicators of company's success is the increase of its value. The article investigates traditional methods of company's value assessment and the evidence that the application of these methods is incorrect in the new stage of economy. So it is necessary to create a new method of valuation based on the new main sources of company's success that is its intellectual capital.

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.

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.

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.