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## No planar billiard possesses an open set of quadrilateral trajectories

The article is devoted to a particular case of Ivrǐ's conjecture on periodic orbits of billiards. The general conjecture states that the set of periodic orbits of the billiard in a domain with smooth boundary in the Euclidean space has measure zero. In this article we prove that for any domain with piecewise C 4-smooth boundary in the plane the set of quadrilateral trajectories of the corresponding billiard has measure zero.

We study regular global attractors of the dynamical systems corresponding to dissipative evolution equations and their nonautonomous perturbations. We prove that the nonautonomous dynamical system (process) resulting from a small nonautonomous perturbation of an autonomous dynamical system (semigroup) having a regular attractor has a regular nonautonomous attractor as well. Moreover, the symmetric Hausdorff deviation of the perturbed attractors from the unperturbed ones is bounded above by O(ε^κ), where ε is a perturbation parameter and 0 < κ < 1. We apply the obtained results to weakly dissipative wave equations in a bounded domain in the three-dimensional space perturbed by timedependent external forces.

We summarize some of the recent works, devoted to the study of one-dimensional (pseudo)group actions and codimension one foliations. We state a conjectural alternative for such actions (generalizing the already obtained results) and describe the properties in both alternative cases. We also discuss the generalizations for holomorphic one-dimensional actions. Finally, we state some open questions that seem to be already within the reach.

The present paper is devoted to the research into the topical questions of network evolution modeling considering the constant changes in the data environment as well as the data exchange rate. A two-level approach to the network community analysis based on the division into macro- and micro-levels of monitoring is suggested. Functionality of both levels is described. Suggestions for investigation and modeling of data flows in a network represented by a dynamical system of message senders and recipients are presented.

We discuss the construction of inverse Couchy problem by using characteristics.

Nonlinear differential dynamic model of the relation between the branches of production was proposed. Mathematically, this model is expressed as a system of first-order ODE. Dynamic variables of the model – the value of the output of each branch of production. Each differential equation of the system includes independent growth and diminution of finished goods; growth and decline of production related to the production of allied industries. Two models were proposed: a model with Malthusian products growth (model with no restrictions on the amount of product), the model with the Verhulst limiting of the growth of output. The equilibrium points of dynamical systems, system stability were determined as well as the qualitative analysis of dynamic systems was made.

The volume is dedicated to Stephen Smale on the occasion of his 80th birthday. Besides his startling 1960 result of the proof of the Poincaré conjecture for all dimensions greater than or equal to five, Smale’s ground breaking contributions in various fields in Mathematics have marked the second part of the 20th century and beyond. Stephen Smale has done pioneering work in differential topology, global analysis, dynamical systems, nonlinear functional analysis, numerical analysis, theory of computation and machine learning as well as applications in the physical and biological sciences and economics. In sum, Stephen Smale has manifestly broken the barriers among the different fields of mathematics and dispelled some remaining prejudices. He is indeed a universal mathematician. Smale has been honored with several prizes and honorary degrees including, among others, the Fields Medal(1966), The Veblen Prize (1966), the National Medal of Science (1996) and theWolf Prize (2006/2007).

The famous conjecture of V.Ya.Ivrii (1978) says that in every billiard with infinitely-smooth boundary in a Euclidean space the set of periodic orbits has measure zero. In the present paper we study the complex version of Ivrii's conjecture for odd-periodic orbits in planar billiards, with reflections from complex analytic curves. We prove positive answer in the following cases: 1) triangular orbits; 2) odd-periodic orbits in the case, when the mirrors are algebraic curves avoiding two special points at infinity, the so-called isotropic points. We provide immediate applications to the partial classification of k-reflective real analytic pseudo-billiards with odd k, the real piecewise-algebraic Ivrii's conjecture and its analogue in the invisibility theory: Plakhov's invisibility conjecture.

Conference covers both fundamental problems ofthe theory, and application to research of complex organizational and technical systems.