Expansive homeomorphisms of the plane

This article tackles the problem of the classification of expansive homeomorphisms of the plane. Necessary and sufficient conditions for a homeomorphism to be conjugate to a linear hyperbolic automorphism will be presented. The techniques involve topological and metric aspects of the plane. The use of a Lyapunov metric function which defines the same topology as the one induced by the usual metric but that, in general, is not equivalent to it is an example of such techniques. The discovery of a hypothesis about the behavior of Lyapunov functions at infinity allows us to generalize some results that are valid in the compact context. Additional local properties allow us to obtain another classification theorem.Comment: 29 pages, 22 figure

Fleming-Viot selects the minimal quasi-stationary distribution: The Galton-Watson case

Consider N particles moving independently, each one according to a subcritical continuous-time Galton-Watson process unless it hits 0, at which time it jumps instantaneously to the position of one of the other particles chosen uniformly at random. The resulting dynamics is called Fleming-Viot process. We show that for each N there exists a unique invariant measure for the Fleming-Viot process, and that its stationary empirical distribution converges, as N goes to infinity, to the minimal quasi-stationary distribution of the Galton-Watson process conditioned on non-extinction.Comment: 25 page

Promoter and Riboswitch Control of the Mg2+ Transporter MgtA from Salmonella enterica

The MgtA protein from Salmonella enterica serovar Typhimurium mediates Mg(2+) uptake from the periplasm into the cytoplasm. Here we report that the PhoP/PhoQ two-component regulatory system, which responds to periplasmic Mg(2+), governs mgtA transcription initiation at all investigated Mg(2+) concentrations and that the Mg(2+)-sensing 5′ leader region of the mgtA gene controls transcription elongation into the mgtA coding region when Salmonella is grown in media with <50 μM Mg(2+). Overexpression of the Mg(2+) transporter CorA, which is believed to increase cytoplasmic Mg(2+) levels, decreased mgtA transcription in a manner dependent on a functional mgtA 5′ leader

Solitary vortex couples in viscoelastic Couette flow

We report experimental observation of a localized structure, which is of a new type for dissipative systems. It appears as a solitary vortex couple ("diwhirl") in Couette flow with highly elastic polymer solutions. A unique property of the diwhirls is that they are stationary, in contrast to the usual localized wave structures in both Hamiltonian and dissipative systems which are stabilized by wave dispersion. It is also a new object in fluid dynamics - a couple of vortices that build a single entity somewhat similar to a magnetic dipole. The diwhirls arise as a result of a purely elastic instability through a hysteretic transition at negligible Reynolds numbers. It is suggested that the vortex flow is driven by the same forces that cause the Weissenberg effect. The diwhirls have a striking asymmetry between the inflow and outflow, which is also an essential feature of the suggested elastic instability mechanism.Comment: 9 pages (LaTeX), 5 Postscript figures, submitte

Single polymer dynamics: coil-stretch transition in a random flow

By quantitative studies of statistics of polymer stretching in a random flow and of a flow field we demonstrate that the stretching of polymer molecules in a 3D random flow occurs rather sharply via the coil-stretch transition at the value of the criterion close to theoretically predicted.Comment: 4 pages, 5 figure

Efficient Mixing at low Reynolds numbers using polymer additives

Mixing in fluids is a rapidly developing field of fluid mechanics \cite{Sreen,Shr,War}, being an important industrial and environmental problem. The mixing of liquids at low Reynolds numbers is usually quite weak in simple flows, and it requires special devices to be efficient. Recently, the problem of mixing was solved analytically for a simple case of random flow, known as the Batchelor regime \cite{Bat,Kraich,Fal,Sig,Fouxon}. Here we demonstrate experimentally that very viscous liquids at low Reynolds number, $Re$. Here we show that very viscous liquids containing a small amount of high molecular weight polymers can be mixed quite efficiently at very low Reynolds numbers, for a simple flow in a curved channel. A polymer concentration of only 0.001% suffices. The presence of the polymers leads to an elastic instability \cite{LMS} and to irregular flow \cite{Ours}, with velocity spectra corresponding to the Batchelor regime \cite{Bat,Kraich,Fal,Sig,Fouxon}. Our detailed observations of the mixing in this regime enable us to confirm sevearl important theoretical predictions: the probability distributions of the concentration exhibit exponential tails \cite{Fal,Fouxon}, moments of the distribution decay exponentially along the flow \cite{Fouxon}, and the spatial correlation function of concentration decays logarithmically.Comment: 11 pages, 5 figure

Elastic turbulence in curvilinear flows of polymer solutions

Following our first report (A. Groisman and V. Steinberg, \sl Nature $\bf 405$, 53 (2000)) we present an extended account of experimental observations of elasticity induced turbulence in three different systems: a swirling flow between two plates, a Couette-Taylor (CT) flow between two cylinders, and a flow in a curvilinear channel (Dean flow). All three set-ups had high ratio of width of the region available for flow to radius of curvature of the streamlines. The experiments were carried out with dilute solutions of high molecular weight polyacrylamide in concentrated sugar syrups. High polymer relaxation time and solution viscosity ensured prevalence of non-linear elastic effects over inertial non-linearity, and development of purely elastic instabilities at low Reynolds number (Re) in all three flows. Above the elastic instability threshold, flows in all three systems exhibit features of developed turbulence. Those include: (i)randomly fluctuating fluid motion excited in a broad range of spatial and temporal scales; (ii) significant increase in the rates of momentum and mass transfer (compared to those expected for a steady flow with a smooth velocity profile). Phenomenology, driving mechanisms, and parameter dependence of the elastic turbulence are compared with those of the conventional high Re hydrodynamic turbulence in Newtonian fluids.Comment: 23 pages, 26 figure

Quasi-stationary distributions and Fleming-Viot processes in finite spaces

Consider a continuous time Markov chain with rates Q in the state space \Lambda\cup\{0\} with 0 as an absorbing state. In the associated Fleming-Viot process N particles evolve independently in \Lambda with rates Q until one of them attempts to jump to the absorbing state 0. At this moment the particle comes back to \Lambda instantaneously, by jumping to one of the positions of the other particles, chosen uniformly at random. When \Lambda is finite, we show that the empirical distribution of the particles at a fixed time converges as N\to\infty to the distribution of a single particle at the same time conditioned on non absorption. Furthermore, the empirical profile of the unique invariant measure for the Fleming-Viot process with N particles converges as N\to\infty to the unique quasi-stationary distribution of the one-particle motion. A key element of the approach is to show that the two-particle correlations is of order 1/N.Comment: 11 pages, changed title, added typos, references remove

Solitary coherent structures in viscoelastic shear flow: computation and mechanism

Starting from stationary bifurcations in Couette-Dean flow, we compute nontrivial stationary solutions in inertialess viscoelastic circular Couette flow. These solutions are strongly localized vortex pairs, exist at arbitrarily large wavelengths, and show hysteresis in the Weissenberg number, similar to experimentally observed ``diwhirl'' patterns. Based on the computed velocity and stress fields, we elucidate a heuristic, fully nonlinear mechanism for these flows. We propose that these localized, fully nonlinear structures comprise fundamental building blocks for complex spatiotemporal dynamics in the flow of elastic liquids.Comment: 5 pages text and 4 figures. Submitted to Physical Review Letter