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

Efficiency of Developing Renewable Energy Market in Russia

The goal of this study is to systematize and provide a quantitative and qualitative assessment of potential positive economic and non-economic effects of the implementation of a new mechanism for supporting renewable energy in Russia. It should result in achieving the national medium-term objective to increase the share of renewable energy in the wholesale electricity and capacity market to 2.5 % by 2024. The introduction examines the mechanism for supporting the generators of renewable energy by capacity charge in the wholesale electricity and capacity market. It is assumed that the main positive effects from implementing this mechanism will be the replacement of hydrocarbon fuels burned for generating the electricity in traditional coal or gas power plants; improvement in the trade balance; multiplier effects from the development of RES in related industrial sectors, new value added and jobs in the sectors producing the generating and auxiliary equipment for generators; reduction of carbon dioxide emissions; decrease in average prices in the wholesale electricity market; reduction of expenditure on environmental activities and measures to protect the health in the territories with traditional power plants; additional scal charges. As a result, the quantification of these effects amounts to 47.77 billion rubles in 2024. The authors rely on the experience of foreign countries, expert estimates, forecasts by the Russian Ministry of Energy and the Ministry of Economic Development, research by the Russian Energy Agency, International Energy Agency, International Renewable Energy Agency, Community for Renewable Energy Policy in the 21st Century (REN21), statistics of Russian Federal State Statistics Service.The article has been supported by the Russian Foundation for Basic Research as part of the project 14–06–00075 “National Economic and Environmental Security: Threats, Effects, and Scenario for Interaction of Economy and Environment.

Using mathematical research methods for analyzing environmental and economic prospects of the Russian arctic oil and gas producing shelf projects

Objective: to study the ecological and economic prospects of Russian Arctic oil and gas offshore projects by the method of functional analysis.Methods: mathematical modeling, comparative analysis, empirical analysis.Results: a method is proposed for assessing the financial provision of measures to eliminate the alleged environmental damage caused by hydrocarbon spills in Arctic offshore oil and gas production projects; models of oil spills in the “Prirazlomnoye” and “Pobeda” Arctic offshore fields are considered; the cost of measures to eliminate these incidents is estimated.Scientific novelty: for the first time, the paper investigates the function of financial provision for measures to eliminate the alleged environmental damage due to hydrocarbons spills at the “Prirazlomnoye” and “Pobeda” Arctic offshore oil and gas fields with the mathematical method of functional analysis, calculation of the maximum costs, as well as the time from the start of the spill, which accounts for these functional values. The assumption of the mathematical model is the hypothesis of the constancy of the well flow rate (the top estimate is the characteristic flow rate of the per second production of the entire field), as well as the neglect of changes in the physical properties of liquid hydrocarbons (viscosity, density) due to the characteristic annual temperature dynamics of the Arctic regions and consideration of the dynamics of the oil spill spread on the sea surface in the inert phase without taking into account the gravitationally viscous and surface modes.Practical significance: the mathematical modeling data on spills can be used by insurance companies in the calculation of insurance premiums, by Rosprirodnadzor Agency in the calculation of fines, and by other supervisory agencies in the underwriting of oil and gas Arctic projects

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

Philosophical, medical and legal aspects of reproductive genetics

The development of modern biology and medical genetics raises the question of the philosophical interpretation of the results of testing for genetic susceptibility. The medical and legal implications of the information obtained and its use in clinical practice have not been fully studied, normative legal acts regulating the examination process for genetic markers of diseases are not perfect. At the same time, genetic testing is the most promising area of modern medical science and health care. The article sets out the views of the author on these problems.В статье рассматриваются современные взгляды на возможную роль различных генетических сетей в патогенезе больших акушерских синдромов. Рассмотрены гены тромбофилии, фолатного цикла, системы детоксикации, ренинангиотензин-альдостероновой системы, иммунологического ответа, энергетического обмена и «дисфункции» эндотелия. Обсуждаются дискуссионные вопросы фенотипической реализации генетических полиморфизмов

Special framed Morse functions on surfaces

Let $M$ be a smooth closed orientable surface. Let $F$ be the space of Morse functions on $M$, and $\mathbb{F}^1$ the space of framed Morse functions, both endowed with $C^\infty$-topology. The space $\mathbb{F}^0$ of special framed Morse functions is defined. We prove that the inclusion mapping $\mathbb{F}^0\hookrightarrow\mathbb{F}^1$ is a homotopy equivalence. In the case when at least $\chi(M)+1$ critical points of each function of $F$ are labeled, homotopy equivalences $\mathbb{\widetilde K}\sim\widetilde{\cal M}$ and $F\sim\mathbb{F}^0\sim{\mathscr D}^0\times\mathbb{\widetilde K}$ are proved, where $\mathbb{\widetilde K}$ is the complex of framed Morse functions, $\widetilde{\cal M}\approx\mathbb{F}^1/{\mathscr D}^0$ is the universal moduli space of framed Morse functions, ${\mathscr D}^0$ is the group of self-diffeomorphisms of $M$ homotopic to the identity.Comment: 8 pages, in Russia

Results of analyzing the passport database ‘Spring bread wheat landraces in the VIR collection’

Background. Recently, an increased interest in crop landraces has been observed among scientists working with plant genetic resources (PGR). Institutes have been organizing collecting missions to explore old varieties still cultivated locally. Attention paid to landraces is also associated with the introduction of GIS technologies into PGR studies. The N. I. Vavilov All-Russian Institute of Plant Genetic Resources (VIR) holds a unique collection of spring bread wheat landraces. One of the Institute’s top priorities is to develop and maintain a passport database, using a set of fields specifically fit for landraces.Analysis of the passport database. The passport database ‘Spring Bread Wheat Landraces in the VIR Collection’ is analyzed here. The database contains information on 5,925 accessions and has 36 fields. Statistical data are presented for the main fields: origin, collection site, botanical variety, year of registration in the collection, donor institute or expedition, and collector’s name