Constructive Approximation in de Branges–Rovnyak Spaces

In most classical holomorphic function spaces on the unit disk in which the polynomials are dense, a function f can be approximated in norm by its dilates \(f_r(z):=f(rz)~(r<1)\). We show that this is not the case for the de Branges–Rovnyak spaces \(\mathcal{H}(b)\). More precisely, we exhibit a space \(\mathcal{H}(b)\) in which the polynomials are dense and a function \(f\in \mathcal{H}(b)\) such that \(\lim _{r\rightarrow 1^-}\Vert f_r\Vert _{\mathcal{H}(b)}=\infty \). On the positive side, we prove the following approximation theorem for Toeplitz operators on general de Branges–Rovnyak spaces \(\mathcal{H}(b)\). If \((h_n)\) is a sequence in \(H^\infty \) such that \(\Vert h_n\Vert _{H^\infty }\le 1\) and \(h_n(0)\rightarrow 1\), then \(\Vert T_{\overline{h}_n}f-f\Vert _{\mathcal{H}(b)}\rightarrow 0\) for all \(f\in \mathcal{H}(b)\). Using this result, we give the first constructive proof that, if b is a nonextreme point of the unit ball of \(H^\infty \), then the polynomials are dense in \(\mathcal{H}(b)\).     

Thickness fluctuations in turbulent soap films

Rapidly flowing soap films provide a simple and attractive system to study two-dimensional hydrodynamics and turbulence. By measuring the rapid fluctuations of the thickness of the film in the turbulent regime, we find that the statistics of these fluctuations closely resemble those of a passive scalar field in a turbulent flow. The scalar spectra are well described by Kolmogorov-like scaling while the high-order moments show clear deviations from regular scaling just like dye or temperature fluctuations in 3D turbulent flows.     

Conformation statistics of a deformable material line in two-dimensional turbulence

We have observed the fluctuations of the centerline position of a thin column of water injected into a turbulent soap film. As the turbulence intensity increases, the second order structure function of the centerline position increments undergoes a change between different power law scaling regimes. These scalings indicate that the centerline position makes a continuous transition from an analytic to a non-analytic function. Cusps and other singular configurations of the column occur at higher turbulent intensities. Since the column supports random wave patterns, we propose that this experiment could serve as a test bed for theories of wave turbulence in one dimension.     

Observations of the collapse of dilute lyotropic lamellar phases under shear flow

Experimental evidence of the collapse of dilute lamellar phases due to shear flow is presented. Two systems are used: one composed of brine and an ionic surfactant, and another composed of water, a nonionic surfactant, and cosurfactant. We observe this transition for a range of lamellar spacings and brine salinity. The results are in reasonable agreement with recent theory in which the suppression of fluctuations by shear plays an important role.     

Coupling between flow and structure for a lamellar surfactant phase

The flow-structure relation of lamellar phases is studied using rheometry and cross-polarized microscopy under flow. The equilibrium phases show different defects. Low salinities lead to very viscous, "onion" phases, whereas at high salinity, a low viscosity plane lamellar phase is found. Under shear, the latter shows a sudden transition to a viscoelastic gel, with a texture and viscosity very similar to that of the onions. Gelation occurs after a certain delay time, increasing rapidly with salinity, by the nucleation of onions. This allows one to relate the delay time to the defect energy.     

Cyclicity in the Dirichlet space

Let be the Dirichlet space, namely the space of holomorphic functions on the unit disk whose derivative is square-integrable. We give a new sufficient condition, not far from the known necessary condition, for a function f∈ to be cyclic, i.e. for {pf: p is a polynomial} to be dense in . The proof is based on the notion of Bergman–Smirnov exceptional set introduced by Hedenmalm and Shields. Our methods yield the first known examples of such sets that are uncountable. One of the principal ingredients of the proof is a new converse to the strong-type inequality for capacity.     

Capillarylike fluctuations at the interface of falling granular jets

We study the interface fluctuations of a granular jet falling under gravity and show that for small scales they are the analog of the thermally induced capillary waves. Experimental results from radial height and velocity fluctuations, static correlation functions and capillary ripple velocities allow us to estimate a granular surface tension. The ultralow interfacial tensions measured (of the order of 100 microN/m) can be rationalized using a simple model.     

Where surface physics and fluid dynamics meet: rupture of an amphiphile layer by fluid flow

We investigate the fluctuating pattern created by a jet of fluid impingent upon an amphiphile-covered surface. This microscopically thin layer is initially covered with 50 microm floating particles so that the layer can be visualized. A vertical jet of water located below the surface and directed upward drives a hole in this layer. The hole is particle-free and is surrounded by the particle-laden amphiphile region. The jet ruptures the amphiphile layer creating a particle-free region that is surrounded by the particle-covered surface. The aim of the experiment is to understand the (fluctuating) shape of the ramified interface between the particle-laden and particle-free regions.