Linearly implicit methods for nonlinear PDEs with linear dispersion and dissipation

The linear stability of IMEX (IMplicit–EXplicit) methods and exponential integrators for stiff systems of ODEs arising in the discrete solution of PDEs is examined for nonlinear PDEs with both linear dispersion and dissipation, and a clear method of visualization of the linear stability regions is proposed. Predictions are made based on these visualizations and are supported by a series of experiments on five PDEs including quasigeostrophic equations and stratified Boussinesq equations. The experiments, involving 24 IMEX and exponential methods of third and fourth order, confirm the predictions of the linear stability analysis, that the methods are typically limited by small eigenvalues of the linear term and by eigenvalues on or near the imaginary axis rather than by large eigenvalues near the negative real axis. The experiments also demonstrate that IMEX methods achieve comparable stability to exponential methods, and that exponential methods are significantly more accurate only when the problem is nearly linear. Novel IMEX predictor–corrector methods are also derived.     

Shape transition in artificial tumors: from smooth buckles to singular creases

Using swelling hydrogels, we study the evolution of a thin circular artificial tumor whose growth is confined at the periphery. When the volume of the outer proliferative ring increases, the tumor loses its initial symmetry and bifurcates towards an oscillatory shape. Depending on the geometrical and elastic parameters, we observe either a smooth large-wavelength undulation of the swelling layer or the formation of sharp creases at the free boundary. Our experimental results as well as previous observations from other studies are in very good agreement with a nonlinear poroelastic model.     

Inhibition, enhancement, and control of spontaneous emission in photonic nanowires

We experimentally investigate the spontaneous emission (SE) rates of single InAs quantum dots embedded in GaAs photonic nanowires. For a diameter leading to the optimal confinement of the fundamental guided mode HE11, the coupling to HE11 dominates the SE process and an increase of the SE rate by a factor of 1.5 is achieved. When the diameter is decreased, the coupling to this mode vanishes rapidly, thus allowing the coupling to the other radiation modes to be probed. In these conditions, a SE inhibition factor of 16, equivalent to the one obtained in state-of-the-art photonic crystals, is measured. These results, which are supported by fully vectorial calculations, confirm the potential of photonic nanowires for a nearly perfect, broadband SE control.     

Building designed granular towers one drop at a time

A dense granular suspension dripping on an imbibing surface is observed to give rise to slender mechanically stable structures that we call granular towers. Successive drops of grain-liquid mixtures are shown to solidify rapidly upon contact with a liquid absorbing substrate. A balance of excess liquid flux and drainage rate is found to capture the typical growth and height of the towers. The tower width is captured by the Weber number, which gives the relative importance of inertia and capillary forces. Various symmetric, smooth, corrugated, zigzag, and chiral structures are observed by varying the impact velocity and the flux rate from droplet to jetting regime.     

Controlling the transmission of ultrahigh frequency bulk acoustic waves in silicon by 45° mirrors

In this paper, we present a feasible microsystem in which the direction of localized ultrahigh frequency (∼1 GHz) bulk acoustic wave can be controlled in a silicon wafer. Deep etching technology on the silicon wafer makes it possible to achieve high aspect ratio etching patterns which can be used to control bulk acoustic wave to transmit in the directions parallel to the surface of the silicon wafer. Passive 45° mirror planes obtained by wet chemical etching were employed to reflect the bulk acoustic wave. Zinc oxide (ZnO) thin film transducers were deposited by radio frequency sputtering with a thickness of about 1 μm on the other side of the wafer, which act as emitter/receptor after aligned with the mirrors. Two opponent vertical mirrors were inserted between the 45° mirrors to guide the transmission of the acoustic waves. The propagation of the bulk acoustic wave was studied with simulations and the characterization of S₂₁ scattering parameters, indicating that the mirrors were efficient to guide bulk acoustic waves in the silicon wafer.     

Single resonance monolithic Fabry-Perot filters formed by volume Bragg gratings and multilayer dielectric mirrors

A new class of Fabry-Perot filters produced by a multilayer dielectric mirror deposited on top of a reflecting volume Bragg grating is described. The first fabricated prototype for the 852 nm region demonstrates a 30 pm bandwidth, 90+% transmission at resonance, and a good agreement with theoretical simulation.     

Fluoride glass Raman fiber laser at 2185 nm

We report on the first Raman laser based on a fluoride glass optical fiber. The Raman fiber laser was pumped by a 9.6?WTm3+:silica CW fiber laser operating at a wavelength of 1940?nm. A maximum output power of 580?mW was measured at 2185.1?nm, corresponding to a frequency shift of 579?cm(-1) (17.37?THz). We observed a threshold power of 3.8?W and a low power slope efficiency of 29% with respect to the launched pump power. Using those results and the known fiber parameters, we estimated a Raman gain peak value of 3.52*10(-14)?m/W, which is lower than the previously reported values.     

Generic representations of abelian groups and extreme amenability

If G is a Polish group and Γ is a countable group, denote by Hom(Γ, G) the space of all homomorphisms Γ → G. We study properties of the group $\overline {\pi (\Gamma )} $ for the generic π ∈ Hom(Γ, G), when Γ is abelian and G is one of the following three groups: the unitary group of an infinite-dimensional Hilbert space, the automorphism group of a standard probability space, and the isometry group of the Urysohn metric space. Under mild assumptions on Γ, we prove that in the first case, there is (up to isomorphism of topological groups) a unique generic $\overline {\pi (\Gamma )} $ ; in the other two, we show that the generic $\overline {\pi (\Gamma )} $ is extremely amenable. We also show that if Γ is torsionfree, the centralizer of the generic π is as small as possible, extending a result of Chacon and Schwartzbauer from ergodic theory.     

Greenberg Approximation and the Geometry of Arc Spaces

We study the differential properties of generalized arc schemes and geometric versions of Kolchin's Irreducibility Theorem over arbitrary base fields. As an intermediate step, we prove an approximation result for arcs by algebraic curves.     

Review of 5-V electrodes for Li-ion batteries: status and trends

Lithium-ion batteries have dominated the battery industry for the past several years in portable electronic devices due to their high volumetric and gravimetric energy densities. The success of these batteries in small-scale applications translates to large-scale applications, with an important impact in the future of the environment by improving energy efficiency and reduction of pollution. We present the progress that allows several lithium-intercalation compounds to become the active cathode element of a new generation of Li-ion batteries, namely the 5-V cathodes, which are promising to improve the technology of energy storage and electric transportation, and address the replacement of gasoline engine by meeting the increasing demand for green energy power sources. The compounds considered here include spinel LiNi_0.5Mn_1.5O₄ and its related doped-structures, olivine LiCoPO₄, inverse spinel LiNiVO₄ and fluorophosphate Li₂CoPO₄F. LiNi_0.5Mn_1.5O₄ thin films, nanoscale prepared materials and surface-modified cathode particles are also considered. Emphasis is placed on the quality control that is needed to guarantee the reliability and the optimum electrochemical performance of these materials as the active cathode element of Li-ion batteries. The route to increase the performance of Li-ion batteries with the other members of the family is also discussed.