### Article

## Борис Васильевич Федосов (математический некролог)

Despite the fact that the range of players in telecommunication market is not large, the companies operate in continuously increasing competition on one side and slowing extensive growth of the industry on the other. This leads companies to an understanding the fact that the efforts of the company's management should be directed to the area of intensive growth. One of the factors that increase the intensification of a telecommunication company is planning a tariff policy strategy. This paper is devoted to forming the tariff policy of a company with regard to the preferences of subscribers.

In this paper, we present a purely algebraic construction of the normal factorization of multimode squeezed states and calculate their inner products. This procedure allows one to orthonormalize bases generated by squeezed states. We calculate several correct representations of the normalizing constant for the normal factorization, discuss an analog of the Maslov index for squeezed states, and show that the Jordan decomposition is a useful mathematical tool for problems with degenerate Hamiltonians. As an application of this theory, we consider a nontrivial class of squeezing problems which are solvable in any dimension.

This paper is aimed at applying and analyzing international active ageing indices in Russia, including the Active Ageing Index (AAI), developed by European Centre Vienna, and Global AgeWatch Index by HelpAge International, to provide the base for cross-national comparison and development of a comprehensive national policy on active ageing. Our research was motivated by the following questions (1) to what extent can the international approaches to measure active ageing be applied to the Russian context and data? (2) to what extent a country’s position in the ranking is sensitive to the index methodology and data used? (3) whether and under what conditions Russia can improve its positions in the active ageing indices? To answer these questions, we estimated the AAI for Russia based on eight data sources and recalculated some of the AgeWatch Index results based on reliable data. The methodology of both indices and the quality and adequacy of the data used are discussed in detail in the paper. The results show that ranking of Russia according to these indices varies considerably from the 65th place out of 96 countries by the Global AgeWatch Index to the 18th place among 29 countries (28 EU countries plus Russia) by the AAI. Nevertheless, both indices draw rather similar pictures of active ageing potential in Russia. We provide some recommendations on how the indicators can be modified to capture some peculiarities of the ageing context in Russia and other countries with similar demographic, economic and social context.

In the last years native RDF stores made enormous progress in closing the performance gap compared to RDBMS. This albeit smaller gap, however, still prevents adoption of RDF stores in scenarios with high requirements on responsiveness. We try to bridge the gap and present a native RDF store “OntoQuad” and its fundamental design principles. Basing on previous researches, we develop a vector database schema for quadruples, its realization on index data structures, and ways to efficiently implement the joining of two and more data sets simultaneously. We also offer approaches to optimizing the SPARQL query execution plan which is based on its heuristic transformations. The query performance efficiency is checked and proved on BSBM tests. The study results can be taken into consideration during the development of RDF DBMS’s suitable for storing large volumes of Semantic Web data, as well as for the creation of large-scale repositories of semantic data.

Methodology of automated forming of weak-formalized manufacturing documents for support the production processes of an enterprise on the basis of the cybernetic methods is considered. A model of elaboration of automaed documents preparation technology with use of automated lexicological synthesis is presented. The technology being proposed allows substantially reduce the man-hours at making of full-text documents.

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.