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## Метод графов для решения задач перечислительной комбинаторики

The possibilities of research of combinatorial circuits of particles in cells based on graphs of stochastic processes in the respective schemes poedinichnom adding particles with a specific numbering at each step to organize easily calculable probabilities. Such information enables precise probabilistic analysis of interest layouts. The essence of the method consists in constructing a graph of a random process with poedinichnom adding particles in the combinatorial circuit in all possible ways with certain distinct discipline their numbering in the corresponding graph states. Number of process steps defined in the schema specify the total number of particles placed. We are interested in the list of all the states, and, hence, their number on, the last step. If on, the edges of the graph indicate the probability of all transitions with standing in state at any step of the process, given its properties, the probability of all outcomes scheme calculated by the formulas of addition and multiplication of probabilities and give full information about the process, allowing to conduct further analysis of the scheme. Therefore, the immediate goal of research of combinatorial circuits is to get all of their probability distributions explicitly listed outcomes. A first problem will be solved enumerative combinatorics for all outcomes of interest combinatorial circuits.

The primal problem of research consists in finding of number of outcomes of the studied scheme and their transfer in an explicit form. The form of representation of outcomes is for this purpose discussed. On the basis of comparison with the similar scheme with indiscernible particles studied in [1] with the found number of outcomes of S the apparent formula for number of outcomes of N these schemes expressed through S is removed.

Along with rather laborious analytical calculation of number of outcomes of the scheme N, the assuming preliminary finding of number S on [1], is offered a numerical method of definition of N at any fixed values of parameters of the scheme (n – numbers of cells and r – number of particles) by creation of a state graph of casual process of poyedinichny serial placement of particles on cells to r-oho step with particular discipline of numbering of the states describing scheme outcomes in the taken form. On the same count with the indication of easily calculated probabilities of transitions from a state to a state procedure of calculation of probability distribution of outcomes of the scheme after placement of the fixed number of particles is offered. Thus, we receive the complete list of all outcomes of the scheme and their probability distribution. Besides, the numerical method of approximate calculation of number of outcomes of the scheme is given by method of stochastic model operation. Questions of model operation of outcomes of the scheme are also considered.

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.

The paper discusses the specific features and presents the procedure and the results of studying a wide variety of specific combinatorial schemes in the pre-asymptotic region of their parameters change. It is suggested that the schemes are analyzed by an unconventional quality analysis of their outcomes, the result of which include quantitative characteristics.

A model for organizing cargo transportation between two node stations connected by a railway line which contains a certain number of intermediate stations is considered. The movement of cargo is in one direction. Such a situation may occur, for example, if one of the node stations is located in a region which produce raw material for manufacturing industry located in another region, and there is another node station. The organization of freight traﬃc is performed by means of a number of technologies. These technologies determine the rules for taking on cargo at the initial node station, the rules of interaction between neighboring stations, as well as the rule of distribution of cargo to the ﬁnal node stations. The process of cargo transportation is followed by the set rule of control. For such a model, one must determine possible modes of cargo transportation and describe their properties. This model is described by a ﬁnite-dimensional system of diﬀerential equations with nonlocal linear restrictions. The class of the solution satisfying nonlocal linear restrictions is extremely narrow. It results in the need for the “correct” extension of solutions of a system of diﬀerential equations to a class of quasi-solutions having the distinctive feature of gaps in a countable number of points. It was possible numerically using the Runge–Kutta method of the fourth order to build these quasi-solutions and determine their rate of growth. Let us note that in the technical plan the main complexity consisted in obtaining quasi-solutions satisfying the nonlocal linear restrictions. Furthermore, we investigated the dependence of quasi-solutions and, in particular, sizes of gaps (jumps) of solutions on a number of parameters of the model characterizing a rule of control, technologies for transportation of cargo and intensity of giving of cargo on a node station.

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.