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## Demonstrativeness in Mathematics and Physics

An approach to the physical language, pointed out by Poincare in a philosophical work, is to link probability theory to arithmetic and thermodynamics. We follow this program in the part concerning a generalization of probability theory.

An approach to the physical language, pointed out by Poincare in a philosophical work, is to link probability theory to arithmetic and thermodynamics. We follow this program in the part concerning a generalization of probability theory.

Developing a new approach to the problem of optimal control in the open dynamical model of a three-sector economy

The book includes the abstracts of communications submitted to the XXXI International Seminar on Stability Problems for Stochastic Models (ISSPSM'2013), associated VII International Workshop Applied Problems in Theory of Probabilities and Mathematical Statistics Related to Modeling of Information Systems (APTP + MS'2013) (Spring Session) and International Workshop Applied Probability Theory and Theoretical Informatics.

Book of abstracts

The functionals related to the quality of the system control are obtained

in the analytic form. The statement that the optimal strategy of controlling

the system is a deterministic strategy is proved. Analytic form representation

for the function the absolute extremum of which is determined as the optimal

control strategy is obtained also.

The purpose of this paper is the presentation of the ideas and concepts that

form the basis of the concept of mathematical model control some processes

occurring in the Russian market of cereals. The estimated model must have a

stochastic nature, i.e. constitute some random process. Indeed, in a free market

there are objectively random factors that cannot be described by deterministic.

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.