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## О ПРОИСХОЖДЕНИИ РУССКИХ ЧИСЛИТЕЛЬНЫХ

In many languages of the world, the forms in the irrealis domain (subjunctive, conjunctive, conditional) are also used in complement clauses. The set of verbs that require subjunctive complementation is similar but not identical across languages. The paper identifies Russian verbs licensing subjunctive in complement clauses, either as the only option or as an alternative to the indicative. Basing on the Russian National Corpus, a list of these predicates is compiled, with relative frequencies of subjunctive vs. indicative for each predicate. The main result of the study is distinguishing two types of subjunctive complement clauses. Most predicates belong to the group which is similar to purpose clauses with *чтобы*, both semantically and syntactically. The subject of the main predicate is involved in the situation described by the subordinate clause by wishing it to be realized, by intention, or causal relations. The second, minor group includes epistemic uses of *чтобы* with e.g. *сомневаться* and other predicates in the context of negation, interrogation and other constructions expressing low probability.

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.

The paper discusses sociolinguistic implementations of statistical analysis of the spoken subcorpus of the Russian National Corpus. Given the considerable size of the corpus (about 10 mln tokens), an analysis of co-variation of various linguistic parameters with one of the few sociolinguistic parameters available – the speaker’s gender – may give rich and interesting results. One specific example of co-variation is considered in detail: the mean length of the utterance (in tokens). Comparing this parameter in public communication shows statistically significant difference between the speech of men and women (men talk more), while the same difference is absent in private communication. Another important parameter is the gender of the addressee. Again, co-variation is quite different in public and private discourse. In private communication, the utterances are longer when addressing someone of the same sex, the difference between men and women is not statistically significant. In public communication, the utterances are longer when addressing a woman, whether the speaker herself is a man or woman. These conclusions are consistent with the results of sociolinguistic gender studies obtained elsewhere and by other methods. Linguistic difference between men and women are not absolute but depend on the communicative situation (public vs. private). Public discourse is a playground for linguistic competition in which men are the winning party. In private discourse, competition dissolves.

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.