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## An algebraic formula for the index of a 1-form on a real quotient singularity

Let a finite abelian group G act (linearly) on the space R^n and thus on its complexification C^n. Let W be the real part of the quotient C^n/G (in general W \neq R^n/G). We give an algebraic formula for the radial index of a 1-form \omega on the real quotient W. It is shown that this index is equal to the signature of the restriction of the residue pairing to the G-invariant part \Omega^G_\omega of \Omega_\omega=\Omega^n_{R^n,0}/\omega \wedge \Omega^{n-1}_{R^n,0}. For a G-invariant function f, one has the so-called quantum cohomology group defined in the quantum singularity theory (FJRW-theory). We show that, for a real function f, the signature of the residue pairing on the real part of the quantum cohomology group is equal to the orbifold index of the 1-form df on the preimage \pi^{-1}(W) of W under the natural quotient map.

We summarize some of the recent works, devoted to the study of one-dimensional (pseudo)group actions and codimension one foliations. We state a conjectural alternative for such actions (generalizing the already obtained results) and describe the properties in both alternative cases. We also discuss the generalizations for holomorphic one-dimensional actions. Finally, we state some open questions that seem to be already within the reach.

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.

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.