Upper bounds for the number of orbital topological types of planar polynomial vector fields "modulo limit cycles"

The paper deals with planar polynomial vector fields. We aim to estimate the number of orbital topological equivalence classes for the fields of degree n. An evident obstacle for this is the second part of Hilbert's 16th problem. To circumvent this obstacle we introduce the notion of equivalence modulo limit cycles. This paper is the continuation of the author's paper in [Mosc. Math. J. 1 (2001), no. 4] where the lower bound of the form 2^{cn^2} has been obtained. Here we obtain the upper bound of the same form. We also associate an equipped planar graph to every planar polynomial vector field, this graph is a complete invariant for orbital topological classification of such fields.Comment: 23 pages, 5 figure

On the multiplicity of the hyperelliptic integrals

Let $I(t)= \oint_{\delta(t)} \omega$ be an Abelian integral, where $H=y^2-x^{n+1}+P(x)$ is a hyperelliptic polynomial of Morse type, $\delta(t)$ a horizontal family of cycles in the curves $\{H=t\}$, and $\omega$ a polynomial 1-form in the variables $x$ and $y$. We provide an upper bound on the multiplicity of $I(t)$, away from the critical values of $H$. Namely: $ord\ I(t) \leq n-1+\frac{n(n-1)}{2}$if$\deg \omega <\deg H=n+1$. The reasoning goes as follows: we consider the analytic curve parameterized by the integrals along$\delta(t)$of the$n$``Petrov'' forms of$H$(polynomial 1-forms that freely generate the module of relative cohomology of$H$), and interpret the multiplicity of$I(t)$as the order of contact of$\gamma(t)$and a linear hyperplane of$\textbf C^ n$. Using the Picard-Fuchs system satisfied by$\gamma(t)$, we establish an algebraic identity involving the wronskian determinant of the integrals of the original form$\omega$along a basis of the homology of the generic fiber of$H$. The latter wronskian is analyzed through this identity, which yields the estimate on the multiplicity of$I(t)$. Still, in some cases, related to the geometry at infinity of the curves$\{H=t\} \subseteq \textbf C^2$, the wronskian occurs to be zero identically. In this alternative we show how to adapt the argument to a system of smaller rank, and get a nontrivial wronskian. For a form$\omega$of arbitrary degree, we are led to estimating the order of contact between$\gamma(t)$and a suitable algebraic hypersurface in$\textbf C^{n+1}$. We observe that$ord I(t)$grows like an affine function with respect to$\deg \omega$.Comment: 18 page

On the Number of Zeros of Abelian Integrals: A Constructive Solution of the Infinitesimal Hilbert Sixteenth Problem

We prove that the number of limit cycles generated by a small non-conservative perturbation of a Hamiltonian polynomial vector field on the plane, is bounded by a double exponential of the degree of the fields. This solves the long-standing tangential Hilbert 16th problem. The proof uses only the fact that Abelian integrals of a given degree are horizontal sections of a regular flat meromorphic connection (Gauss-Manin connection) with a quasiunipotent monodromy group.Comment: Final revisio

Consumer behavior in the context of global economic transformations

Transition to alternative methods of service and omnichannelity allows the buyer to be more demanding and discerning. The current trends in the development of retail trade caused by globalization forcing the redistribution of consumer budget from a high-margin offline cart to a low-margin online cart that instigate not only the necessity of transformations in management, but also the study of consumer behavior. Dynamical development of trading innovative technologies allows the buyers not only to expand the range of their aspiration, but also to become participants in the cognitive and learning processes, where the acquirement of the new experience and emotions influences the traditional ways of making purchases on offline trading. To remain competitive, retailers should not only provide high-quality goods and services at the best prices, but also to contribute to their high-quality and get-to-able supply. Most purchasing decisions are made "by feel", that contradict to the theories of rational choice and theories using marketing concepts. The article gives views on the factors and criteria that determine consumers behavior in online and offline trade.peer-reviewe

Oscillation of linear ordinary differential equations: on a theorem by A. Grigoriev

We give a simplified proof and an improvement of a recent theorem by A. Grigoriev, placing an upper bound for the number of roots of linear combinations of solutions to systems of linear equations with polynomial or rational coefficients.Comment: 16 page

The Demoiselle crane (Anthropoides virgo) population genetic structure in Russia

The Demoiselle crane (Anthropoides virgo Linneaus, 1758) is a widespread crane species of Eurasia distributed in the steppe and semi-desert zones from southeast Ukraine eastward to Northern China. The Demoiselle crane uses two wintering grounds in Africa and India corresponding to the European and Asian breeding parts of the range subdivided into several spatially separated breeding flocks. The first estimates of the genetic diversity and differentiation have been obtained from five of them: 1) Azov &amp; Black Sea, 2) Caspian, 3) Volga &amp; Ural, 4) South Siberian and 5) Eastern Asian sampled across the total breeding range in Russia using data from 10 microsatellite loci and the 1 003-bp control region of mitochondrial DNA. In total, the Demoiselle crane demonstrates high level of observed (HO = 0.638 ± 0.032) and expected (HE = 0.657 ± 0.023) hete-rozygosity and haplotype diversity (h = 0.960). Genetic dif­ferentiation among populations has shown to be weak for both the microsatellite loci (Wright’s FST = 0.052 or AMOVA estimate 0.016) and mtDNA (FST = 0.040). No evidence of significant population structuring of the Demoiselle crane has been found using the STRUCTURE analysis of multilo­cus microsatellite genotypes and the NETWORK grouping of control region haplotypes. Despite the haplotype diversity was high, the nucleotide diversity of the species was low (0.0033 ± 0.0003). Negative but non-significant Tajima’s and Fu’s tests did not suggest the recent population expansion in the Demoiselle crane evolutionary history which contrasts to other cranes of the Palearctic (the Eurasian crane Grus grus, and the Hooded crane G. monacha). These data indicate more stable conditions for the Demoiselle crane breeding groups in the steppe zone in Pleistocene as compared to boreal and subarctic breeding grounds of other crane species

Эндотоксикоз как содержание постреанимационной болезни при острых отравлениях

Objective: to study the specific features of the diagnosis and treatment of endothoxicosis as a manifestation of postresus-citative disease in acute exogenous poisoning.Subjects and methods. Clinical, laboratory, and statistical methods of the diagnosis of endotoxicosis complicating the course of acute poisoning by narcotics, psychopharmacological agents, and cauterants, as well as complex physicochemical detoxification were applied to 554 patients.Results. The study pathology has been ascertained to have 3 developmental stages — from functional (primary) to developed and terminal clinical and laboratory manifestations as multiple organ dysfunctions. The best therapeutic results are achieved by effective therapeutic measures just in early-stage endotoxicosis, by substantially reducing the rates of death and pneumonia and the time of the latter’s resolution. The major contribution to the reduction in the rate of death due acute poisoning is associated with physicochemical detoxification that considerably lessens the influence of the intoxication factors of postresuscitative disease.Conclusion. When diagnosing endotoxicosis in patients with acute endogenous intoxication, it is necessary to keep in mind a whole spectrum of typical changes in endotoxicosis markers and homeostatic parameters: hematologi-cal, blood rheological, lipid peroxidation/antioxidative systems, which should be timely corrected by the basic efferent artificial detoxification techniques (hemosorption, hemodiafiltration, hemofiltration), by compulsorily employing physio-and chemohemotherapy (laser-ultraviolet hemotherapy, sodium hypochlorite infusion). Цель исследования . Изучить особенности диагностики и лечения эндотоксикоза как проявления постреанимационной болезни при острых экзогенных отравлениях.Материалы и методы. Использование клинико-лаборатор-ных и статистических методов диагностики эндотоксикоза, осложнившего течение острых отравлений наркотиками, психофармакологическими средствами и прижигающими жидкостями, и применение комплексной физико-химической детоксикации для его лечения у 554 больных.Результаты. Установлено, что при изучаемой патологии наблюдается 3 стадии развития эндотоксикоза — от функциональных (начальных) до развитых и терминальных клинико-лабораторных проявлений в виде полиорганной недостаточности. Наилучшие лечебные результаты достигаются на фоне использования эффективных лечебных мероприятий уже в начальной стадии эндо-токсикоза, что позволяет существенно снизить летальность и частоту пневмоний, уменьшить сроки разрешения пневмоний. Основной вклад в сокращение частоты летальных исходов при острых отравлениях связан с физико-химической детоксикацией, значительно уменьшающей влияние интоксикационного фактора постреанимационной болезни.Заключение. При диагностике эндотоксикоза у больных с острыми экзогенными отравлениями необходимо учитывать весь комплекс типичных изменений маркеров эндотоксикоза и показателей гомеостаза: гематологических, иммунологических, гемореологических, показателей ПОЛ/АОС, которые необходимо своевременно корригировать с помощью основных эфферентных методов искусственной детоксикации (гемосорбция, гемодиафильтрация, гемофильтрация) с обязательным использованием физио- и химиогемотерапии (лазерно-ультрафиолетовая гемотерапия, инфузии гипохлорита натрия).

EndCap Module Production for the ATLAS Silicon Tracker (SCT) at CERN and the University of Geneva

This note describes the infrastructure, procedure and quality assurance for the construction of approximately one third of the EndCap modules for the ATLAS Semiconductor Tracker (SCT) by groups at the University of Geneva and CERN