### Working paper

## On a weight system conjecturally related to sl2

We introduce a new series Rk, k = 2, 3, 4, ..., of integer valued weight systems. The value of the weight system Rk on a chord diagram is a signed number of cycles of even length 2k in the intersection graph of the diagram.Weshow that this value depends on the intersection graph only. We check that for small orders of the diagrams, the value of the weight system Rk on a diagram of order exactly 2k coincides with the coefficient of ck in the value of the sl2-weight system on the projection of the diagram to primitive elements.

The approaches based on applying of metamodeling and domain-specific languages are widely used in software engineering. There are many different tools for creating visual domain-specific modeling languages with a possibility of determining user’s graphical notations. However, these tools possess disadvantages. The article presents an approach to the development of language workbench that allows to eliminate some restrictions of existing DSM-platforms. The MetaLanguage system is designed for creation of visual dynamic adaptable domain-specific modeling languages and for models construction with these languages. It allows executing transformations of the created models in various textual and graphical notations. Basic metalanguage constructions of this system are described. The formal description of modeling languages metamodel used in MetaLanguage is given. The architecture of MetaLanguage toolkit is presented.

The paper proposes morphological models for the analysis of complex electronic systems quality criterion. The reason for resorting to morphological models is the need to increase attention to improving the quality and reliability of electronic systems in the early design stages. At the same time, many difficulties of mathematical modeling of the investigated heterogeneous physical processes occurring in electronic systems are solved. In this paper, the unification of the notation and the image of components based on analogies of mathematical descriptions. In linear synthesis morphological models in the form of connection of multipolar systems are offered. But they allow you to explore the electrical, mechanical and thermal processes only in linear approximation. In contrast to this representation, the paper also proposes unified view of morphological models in the form of a hypergraph, which will cover nonlinear cases.

This book constitutes the revised selected papers of the 43rd International Workshop on Graph-Theoretic Concepts in Computer Science, WG 2017, held in Eindhoven, The Netherlands, in June 2017.

The 31 full papers presented in this volume were carefully reviewed and selected from 71 submissions. They cover a wide range of areas, aiming at connecting theory and applications by demonstrating how graph-theoretic concepts can be applied in various areas of computer science. Another focus is on presenting recent results and on identifying and exploring promising directions of future research.

In this article we use the modular decomposition technique for exact solving the weighted maximum clique problem. Our algorithm takes the modular decomposition tree from the paper of Tedder et. al. and finds solution recursively. Also, we propose algorithms to construct graphs with modules. We show some interesting results, comparing our solution with Ostergards algorithm on DIMACS benchmarks and on generated graphs.

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.