The Liouville-type theorem for integrable Hamiltonian systems with incomplete flows

For integrable Hamiltonian systems with two degrees of freedom whose Hamiltonian vector fields have incomplete flows, an analogue of the Liouville theorem is established. A canonical Liouville fibration is defined by means of an "exact" 2-parameter family of flat polygons equipped with certain pairing of sides. For the integrable Hamiltonian systems given by the vector field $v=(-\partial f/\partial w, \partial f/\partial z)$ on ${\mathbb C}^2$ where $f=f(z,w)$ is a complex polynomial in 2 variables, geometric properties of Liouville fibrations are described.Comment: 6 page

Connected components of spaces of Morse functions with fixed critical points

Let $M$ be a smooth closed orientable surface and $F=F_{p,q,r}$ be the space of Morse functions on $M$ having exactly $p$ critical points of local minima, $q\ge1$ saddle critical points, and $r$ critical points of local maxima, moreover all the points are fixed. Let $F_f$ be the connected component of a function $f\in F$ in $F$. By means of the winding number introduced by Reinhart (1960), a surjection $\pi_0(F)\to{\mathbb Z}^{p+r-1}$ is constructed. In particular, $|\pi_0(F)|=\infty$, and the Dehn twist about the boundary of any disk containing exactly two critical points, exactly one of which is a saddle point, does not preserve $F_f$. Let $\mathscr D$ be the group of orientation preserving diffeomorphisms of $M$ leaving fixed the critical points, ${\mathscr D}^0$ be the connected component of ${\rm id}_M$ in $\mathscr D$, and ${\mathscr D}_f\subset{\mathscr D}$ the set of diffeomorphisms preserving $F_f$. Let ${\mathscr H}_f$ be the subgroup of ${\mathscr D}_f$ generated by ${\mathscr D}^0$ and all diffeomorphisms $h\in{\mathscr D}$ which preserve some functions $f_1\in F_f$, and let ${\mathscr H}_f^{\rm abs}$ be its subgroup generated ${\mathscr D}^0$ and the Dehn twists about the components of level curves of functions $f_1\in F_f$. We prove that ${\mathscr H}_f^{\rm abs}\subsetneq{\mathscr D}_f$ if $q\ge2$, and construct an epimorphism ${\mathscr D}_f/{\mathscr H}_f^{\rm abs}\to{\mathbb Z}_2^{q-1}$, by means of the winding number. A finite polyhedral complex $K=K_{p,q,r}$ associated to the space $F$ is defined. An epimorphism $\mu:\pi_1(K)\to{\mathscr D}_f/{\mathscr H}_f$ and finite generating sets for the groups ${\mathscr D}_f/{\mathscr D}^0$ and ${\mathscr D}_f/{\mathscr H}_f$ in terms of the 2-skeleton of the complex $K$ are constructed.Comment: 12 pages with 2 figures, in Russian, to be published in Vestnik Moskov. Univ., a typo in theorem 1 is correcte

BUSINESS DECISION TRENDS IN THE INTELLECTUAL PRODUCTION DEVELOPMENT

The main trends have been described and the system characteristics of intellectual production have been presented: customization of products, integration of the product and the service it provides, the transition to integrated solutions and services, mass personification while reducing the share of costs per unit of output and customer waiting time for an order. The key components of the transformation of business management models in the fourth industrial revolution have been identified. Cross-correlation and regression analysis for indicators of high-tech production have been made. A basic, optimistic and pessimistic forecast of changes in the share of high-tech and technology products in the gross domestic product, depending on the growth of high-performance jobs, has been built. The analysis of the development of intellectual production presented in the article, as well as the results of modeling can be used in the construction of forecast trends of the economy in the transition to the fourth industrial revolution

Genetic structure analysis of leaf rust resistant triticale accessions from the VIR collection using gliadin patterns

The gliadin banding patterns of important accessions from the collection of the N. I. Vavilov All‑Russian Institute of Plant Genetic Resources (VIR) registered in the form of “protein formulas” provide reliable information for the preparation of a “protein passport” for each accession and is convenient for storage and computer processing. It helps to control originality and integrity of accessions during regeneration and their use in breeding. The study involved 17 triticale accessions resistant to leaf rust. The analysis was carried out on single grains of the original accession (a sample of 13–26 kernels) according to the standard protocol adopted by VIR and approved by the International Seed Testing Association (ISTA). The gliadin electrophoretic banding patterns of triticale accessions were registered in the form of “protein formulas”; polymorphism of each accession and genetic diversity within the collection were estimated, and genetic structure of accessions was identified based on the marker protein components. A large variety of the revealed genotypes opens a possibility to identify accessions that combine resistance with other useful traits. Stable and polymorphic accessions including from 2 to 7 biotypes were found. The discovery of interbiotype hybrids and recombinant genotypes in the composition of some polymorphic accessions indicates the instability of their genetic structure and the ongoing formation process. This is due to the heterogeneity of the original parental forms, the tendency to cross‑pollination and insufficiently thorough selection. The data on the triticale genotypic structure can be used in introgressive breeding to control the transfer of rye genetic material to wheat varieties in order to increase their immunity and resistance to adverse factors

Topology of the spaces of Morse functions on surfaces

Let $M$ be a smooth closed orientable surface, and let $F$ be the space of Morse functions on $M$ such that at least $\chi(M)+1$ critical points of each function of $F$ are labeled by different labels (enumerated). Endow the space $F$ with $C^\infty$-topology. We prove the homotopy equivalence $F\sim R\times{\widetilde{\cal M}}$ where $R$ is one of the manifolds ${\mathbb R}P^3$, $S^1\times S^1$ and the point in dependence on the sign of $\chi(M)$, and ${\widetilde{\cal M}}$ is the universal moduli space of framed Morse functions, which is a smooth stratified manifold. Morse inequalities for the Betti numbers of the space $F$ are obtained.Comment: 15 pages, in Russia