### Working paper

## Topological classification of Ω-stable flows on surfaces by means of effectively distinguishable multigraphs

At the particular article we provide a methodological approach to selection of companies for horizontal cooperation in procurement logistics. In context of modern logistics (globalization, high customer expectations, high transportation costs), and changes regarding Russia’s plans to join WTO, this topic is highly relevant from a practical point of view. The purpose of this article is to provide single methodological approach to selection of companies for horizontal cooperation.

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.

1. Description of the problem. Instrumental analysis makes it possible to find the arguments of adjudication on the bounders and structure of corpus delicti, its correlation to criminal and filling-up legislation. 2. Initial theses. Corpus delicti is regarded as that expressed in criminal law doctrine result of reorganization of orders of criminal law into other practically necessary form. That happens in the process of theory and practical experience accumulation. The construction of corpus delicti is transformed for practical needs, textually expressed system of features, regulated by criminal law and characterizing deeds as a crime of a definite type. Correlation of construction of corpus delicti with law and doctrine. Corpus delicti, its algorithm. Transition from law regulations to corpus delicti can be done: 1) prog-nostically; 2) within constant analysis of law; 3) in the process of law application. 3. Stages of instrumental building of corpus delicti: prognostic, doctrinal, law applicatory. Instrumental approach to corpus delicti includes within each stage: 1) based on criminal law decision of classification of corpus delicti and its borders; 2) objective description of a factual model; 3) acception of meaning correlated with legal notions and constructions; 4) choice of the construction of the corpus delicti and disposal of characteristics; 5) verification of legitimacy, necessity and adequacy of foundation. 4. Instrumental analysis of disputable questions of understanding and application of constructions of corpus delicti. A. Functions and purposes of application of construction of corpus delicti. Functions of corpus delicti: a) modeling; b) communicative; c) identificatory; d) technological. B. Contents of corpus delicti. Contents of corpus delicti as it is traditionally regarded does not correspond to indications of crime, does not characterize features of social danger; sign of danger of penalty also does go into corpus delicti. Two variants are proposed for the discussion: widening of the borders of corpus delicti by means of introduction of signs of social danger and signs, defining individualization of penalty and to limitate corpus delicti by characteristic of criminally punished act, separating it from contents of guilt and contents of social danger. C. Structure of corpus delicti. There are two problems: division of elements of crime seems to be extremely harsh and inadequate - it is expedient to include signs of special and time limits of act, causal links, crossing signs of objective and subjective sides, first of all consequences and an object of crime, into the structure of corpus delicti. Forms of committing a criminally punished act is a crime commitment in complicity, ideal system, not finished crime.

We present an approach based on a two-stage ltration of the set of feasible solutions for the multiprocessor job-shop scheduling problem. On the rst stage we use extensive dominance relations, whereas on the second stage we use lower bounds. We show that several lower bounds can eciently be obtained and implemented.

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.