To understand the relative importance of natural selection and random genetic drift in finite but growing populations, the asymptotic behavior of a class of generalized Polya urns is studied using the method of ordinary differential equation (ODE). Of particular interest is the replicator process: two balls (individuals) are chosen from an urn (the population) at random with replacement and bal...

The purpose of this article is to point out a mistake in the published paper Graphs hyperbolic groups and limit set intersection theorem- Proc AMS, vol 146, no 5, pp 1859--1871, which subsequently weakens main theorem that paper. We state prove weaker result note.

We show that for the periodic 2D Navier-Stokes equations (NSE) the set of initial data for which the solution exists for all negative times and has exponential growth is rather rich. We study this set and show that it is dense in the phase space of the NSE. This yields to a positive answer to a question in [BT]. After an appropriate resealing the large Reynolds limit dynamics on this set is Eul...

We describe a one-joint planar arm which repeatedly throws and catches parts on its surface, and we demonstrate that proper choice of the throw velocity and arm geometry guarantees that the part will enter a unique recurrent motion pattern from a large set of initial configurations. The resulting system resembles an open-loop stable juggler of polygonal parts.