### Article

## Topological conjugacy of gradient-like flows on surfaces

The class of C^1-smooth gradient-like flows (Morse flows) on closed surface is the subclass

of the Morse-Smale flows class, which are rough. Their non-wandering set consists of a finite number

of hyperbolic fixed points and a finite number of hyperbolic limit cycles, and they does not have

trajectories connecting saddle points. It is well known that the topological equivalence class of a Morse-

Smale flow on a surface can be described combinatorially, for example, by the directed Peixoto graph,

or by the Oshemkov-Sharko molecule. However, the description of the class of the topological conjugacy

of such a system already requires the introduction of continuous invariants (moduli), corresponding

to the periods of limit cycles at least. Thus, one class of the equivalence contains continuum classes

of the topological conjugacy. Gradient-like flows are Morse-Smale flows without limit cycles. In this

paper we prove that gradient-like flows on a closed surface are topologically conjugate iff they are

topologically equivalent.

The study aimed to test measurement invariance of the Russian-language EmIn questionnaire (by D. Lyusin) for emotional intelligence assessment in two samples, from Russia (n = 275) and Azerbaijan (n = 275). Exploratory factor analysis on pooled sample revealed a 4-factor structure with dimensions interpreted as understanding of one’s own emotions, management of one’s own emotions, understanding of others’ emotions, management of others’ emotions. Using confirmatory factor analysis, strong factorial invariance (equivalence of factor loadings and intercepts) was established, which allows to compare means scores in two cultures. Russians, compared to the Azerbaijani, report better understanding of one’s own emotions and management of one’s own emotions. Russian males report better management of their own emotions, compared to Russian females (in all age groups). Azerbaijani females report better understanding of others’ emotions, compared to Azerbaijani males (except for the senior age group). The results are interpreted based on existing knowledge of cross-cultural differences between Russian and Azerbaijan in cultural values, such as individualism and masculinity.

For gradient-likeflows without heteroclinic intersections of the stable and unstable manifolds of saddle periodic points all of whose saddle equilibrium states have Morse index 1 or n−1, the notion of consistent equivalence of energy functions is introduced. It is shown that the consistent equivalence of energy functions is necessary and sufficient for topological equivalence.

We classify up to conjugacy the subgroups of certain types in the full, in the affine, and in the special affine Cremona groups. We prove that the normalizers of these subgroups are algebraic. As an application, we obtain new results in the Linearization Problem generalizing to disconnected groups Bialynicki-Birula's results of 1966-67. We prove ``fusion theorems'' for n-dimensional tori in the affine and in the special affine Cremona groups of rank n. In the final section we introduce and discuss the notions of Jordan decomposition and torsion prime numbers for the Cremona groups.

We prove that simplest Morse-Smale systems can have locally flat and wildly embedded separatrices of saddle periodic point.

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.