Periodic shadowing and Ω\Omega-stability

We show that the following three properties of a diffeomorphism $f$ of a smooth closed manifold are equivalent: (i) $f$ belongs to the $C^1$-interior of the set of diffeomorphisms having periodic shadowing property; (ii) $f$ has Lipschitz periodic shadowing property; (iii) $f$ is $\Omega$-stable.Comment: 21 page

Holder Shadowing on Finite Intervals

For any $\theta, \omega > 1/2$ we prove that, if any $d$-pseudotrajectory of length $\sim 1/d^{\omega}$ of a diffeomorphism $f\in C^2$ can be $d^{\theta}$-shadowed by an exact trajectory, then $f$ is structurally stable. Previously it was conjectured by Hammel-Grebogi-Yorke that for $\theta = \omega = 1/2$ this property holds for a wide class of non-uniformly hyperbolic diffeomorphisms. In the proof we introduce the notion of sublinear growth property for inhomogenious linear equations and prove that it implies exponential dichotomy.Comment: 19 pages. Minor change

Relativistic runaway electron breakdown and Terrestrial Gamma ray Flashes: GEANT4 simulation

Numerical simulation of a relativistic runaway electron breakdown in the upper atmosphere is performed using GEANT4 simulation toolkit. General features of a relativistic runaway electron avalanche are reconstructed and properties of radiations accompanying breakdown are obtained. It is demonstrated dependance of the high energy branch of photon spectra with respect to an altitude and shown what at the reasonable parameters hard photons have spectral index close to the observed value.Comment: 17 pages, 9 figure

Vector Fields with the Oriented Shadowing Property

We give a description of the \Cone-interior (\Int^1(\OrientSh)) of the set of smooth vector fields on a smooth closed manifold that have the oriented shadowing property. A special class \Bb of vector fields that are not structurally stable is introduced. It is shown that the set \Int^1(\OrientSh\setminus\Bb) coincides with the set of structurally stable vector fields. An example of a field of the class \Bb belonging to \Int^1(\OrientSh) is given. Bibliography: 18 titles.Comment: 42 page

Optimal polymer slugs injection curves

In the paper we study miscible injection of liquids with decreasing viscosity in two-dimensional geometry. Polymer flooding of oil fields is our prototype model. Based on several one-dimensional approximate models we give rigorous formulation of the graded viscosity banks (GVB) technology which helps to save polymer by injecting several slugs with gradual decrease of viscosity instead of one slug. Limiting injection curves when number of slugs tends to infinity is calculated

Lipschitz shadowing implies structural stability

We show that the Lipschitz shadowing property of a diffeomorphism is equivalent to structural stability. As a corollary, we show that an expansive diffeomorphism having the Lipschitz shadowing property is Anosov.Comment: 11 page

THE MECHANISM OF INTRODUCTION OF CLUSTERS IN THE RUSSIAN FEDERATION

Purpose: The purpose of this article is to reveal distinctive features of clusters and determine whether those territorial economic entities that are being actively implemented in Russia today may be referred to as clusters. Methodology: Our analysis work made it possible to ascertain that clusters as networked (hybrid) entities have important distinctive features. The key to which we believe, along with the local base of formation, is to be considered the fact that being comprised of organizations of different types, they do not provide a single mechanism to coordinate interaction. Result: Study of the territorial entities originating in Russia shows that they cannot be referred to as clusters, mainly because of the administrative voluntarist mechanism of their formation. Applications: This research can be used for Students in mathematic and teachers. Novelty/Originality: In this research, the model of the mechanism of introduction of clusters in the Russian Federation is presented in a comprehensive and complete manner