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## Олимпиада как одно из средств мониторинга уровня профессиональной компетентности учителей математики

The problem of finding ways to ensure the quality of education at all levels of the educational system is particularly important. So currently, in accordance with the analytical report of the Federal Institute for educational measurement by results of EGE on the mathematician in 2012 (http://www.fipi.ru/binaries/1354/2.1.pdf only 16% of the graduates can plan for obtaining technical education. One of the reasons for insufficient school of mathematical preparation is inefficient work of teachers of mathematics. Professional level of school teachers of mathematics does not always correspond necessary to ensure a high level of teaching students. Even the Moscow teachers make the same common mistakes.

Institutions affect investment decisions, including investments in human capital. Hence institutions are relevant for the allocation of talent. Good market-supporting institutions attract talent to productive value-creating activities, whereas poor ones raise the appeal of rent-seeking. We propose a theoretical model that predicts that more talented individuals are particularly sensitive in their career choices to the quality of institutions, and test these predictions on a sample of around 95 countries of the world. We find a strong positive association between the quality of institutions and graduation of college and university students in science, and an even stronger negative correlation with graduation in law. Our findings are robust to various specifications of empirical models, including smaller samples of former colonies and transition countries. The quality of human capital makes the distinction between educational choices under strong and weak institutions particularly sharp. We show that the allocation of talent is an important link between institutions and growth.

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.