### Working paper

## Disquisitiones 235

Studying of color palette in watercolor painting is being reported in the following article. To create an expressive image, out of the real world colors the artist chooses those that match his thoughts, feelings and artistic solution of a painting. Examples of tasks are being offered which help with the studying of color palette and auspiciously influence development of creative abilities, creative thinking of students, pupils and scholars.

The ultimate goal of this study is to demonstrate the legitimacy of “Nekrasov’s byronism” as a research topic. For this purpose we are trying to prove that Byron’s poetry influenced the poetry of Nekrasov. In particular, Byron’s poems partially formed the structure of the Nekrasov’s poem “Who is Happy in Russia” (1863–1876). Byron’s poems are inclined to dramatic composition, separate scenes and include songs. “Komu na Rusi zhit’ khorosho” also tends to drama, consists of a chain of scenes, monologues, dialogues and contains inserts in the form of songs. Poet Grisha Dobrosklonov, one of the main Nekrasov’s poem characters, corresponds to Byronic romantic hero. Byron’s influence on the poetry of Nekrasov is confirmed by the fact that among his first books Nekrasov called the tragedy of Valerian Olin “The Corsaire”, the translation of the Byron’s poem. The ballet “The Corsaire” made after the poem by Lord Byron, twice delivered in St. Petersburg at the turn of the 1850s and 1860s, as well as increasing public interest in the figure of British classics could also revived Nekrasov’s long-standing impressions at that time.

We review the results about the accuracy of approximations for distributions of functionals of sums of independent random elements with values in a Hilbert space. Mainly we consider recent results for quadratic and almost quadratic forms motivated by asymptotic problems in mathematical statistics. Some of the results are optimal and could not be further improved without additional conditions.

Despite all the advantages brought by service-oriented architecture (SOA), experts argue that SOA introduces more complexity into information systems rather than resolving it. The problem of service integration challenges modern companies taking the risk of implementing SOA. One of important aspects of this problem relates to dynamic service composition, which has to take into account many types of information and restrictions existing in each enterprise. Moreover, all the changes in business logic should also be promptly reflected. This chapter proposes the approach to solution of the stated problem based on such concepts as model-driven architecture (MDA), ontology modelling and logical analysis. The approach consists of several steps of modelling and finite scope logical analysis for automated translation of business processes into the sequence of service invocations. Formal language of relational logic is proposed as a key element of the proposed approach which is responsible for logical analysis and service workflow generation. We present a logical theory to automatically specialize generic orchestration templates which are close to semantic specification of abstract services in OWL-S. The developed logical theory is described formally in terms of Relational Logic. Our approach is implemented and tested using MIT Alloy Analyzer software.

Plein-air in additional education studio organization is being discussed in the article. As a part of estetic students' education, open-air represents drawing and painting outside in open-air. Purpose of open-air working is to fasten and broaden knowledge and experience received during academic year, to develop abilities of its creative application in open space natural illumination conditions. Different activities are being described: drawing (doodle, sketch, long-tirmed drawing) and painting (short-termed and long-termed etude).

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.