Global well-posedness,scattering and blow-up for the energy-critical focusing non-linear wave equation

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H ¹ than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.     

Ordering of SiO<Subscript>x</Subscript>H<Subscript>y</Subscript>C<Subscript>z</Subscript> islands deposited by atmospheric pressure microwave plasma torch on Si(100) substrates patterned by nanoindentation

SiO _x H _y C _z nanometric layers are deposited from hexamethyldisiloxane by atmospheric pressure microwave plasma torch on Si(100) substrates submitted to temperatures varying on the range [0 °C; 120 °C]. Atomic force microscopy (AFM) characterizations of samples grown at intermediate substrate temperatures (~30 °C) demonstrate a layer-by-layer growth (Frank van der Merwe growth) leading to smooth flat and compact films while films deposited at lower and higher substrates temperatures show an island-like growth (Volmer-Weber growth) generating a high surface roughness. Concomitantly, a detailed infrared spectroscopy analysis of the growing films evidences structural modifications due to changes in the bond types, Si-O-Si conformation and stoichiometry correlated with scanning electron microscopy and AFM characterizations. Then, deposition conditions and specific microstructure are selected with the aim of generating 3-dimensional SiO _x H _y C _z nanostructure arrays on nanoindented Si (100) templates. The first results are discussed.     

Beam-helicity asymmetry in photon and pion electroproduction in the Δ(1232)-resonance region at Q2 = 0.35(GeV/c)2

The beam-helicity asymmetry has been measured simultaneously for the reactions p→epγ and p→epπ ⁰ in the Δ(1232)-resonance region at Q ² = 0.35(GeV/c)². The experiment was performed at MAMI with a longitudinally polarized beam and an out-of-plane detection of the proton. The results are compared with calculations based on dispersion relations for virtual Compton scattering and with the MAID model for pion electroproduction. There is an overall good agreement between experiment and theoretical calculations. The remaining discrepancies may be ascribed to an imperfect parametrization of some γ ^(*) N→πN multipoles, mainly contributing to the non-resonant background. The beam-helicity asymmetry in both channels (γ and π ⁰) shows a good sensitivity to these multipoles and should allow future improvement in their parametrization.     

Two‐Photon Voltmeter for Measuring a Molecular Electric Field

We present a new approach for determining the strength of the dipolar solute‐induced reaction field, along with the ground‐ and excited‐state electrostatic dipole moments and polarizability of a solvated chromophore, using exclusively one‐photon and two‐photon absorption measurements. We verify the approach on two benchmark chromophores N,N‐dimethyl‐6‐propionyl‐2‐naphthylamine (prodan) and coumarin 153 (C153) in a series of toluene/dimethyl sulfoxide (DMSO) mixtures and find that the experimental values show good quantitative agreement with literature and our quantum‐chemical calculations. Our results indicate that the reaction field varies in a surprisingly broad range, 0–10⁷ V cm^?1, and that at close proximity, on the order of the chromophore radius, the effective dielectric constant of the solute–solvent system displays a unique functional dependence on the bulk dielectric constant, offering new insight into the close‐range molecular interaction.     

A cutting plane method from analytic centers for stochastic programming

The stochastic linear programming problem with recourse has a dual block-angular structure. It can thus be handled by Benders' decomposition or by Kelley's method of cutting planes; equivalently the dual problem has a primal block-angular structure and can be handled by Dantzig-Wolfe decomposition—the two approaches are in fact identical by duality. Here we shall investigate the use of the method of cutting planes from analytic centers applied to similar formulations. The only significant difference form the aforementioned methods is that new cutting planes (or columns, by duality) will be generated not from the optimum of the linear programming relaxation, but from the analytic center of the set of localization.This research has been supported by the Fonds National de la Recherche Scientifique Suisse (grant # 12-26434.89), NSERC-Canada and FCAR-Quebec.Corresponding author.     

Scattering losses from planar waveguides with material inhomogeneity

We consider the propagation of guided waves in a planar waveguide that has a finite-length segment of inhomogeneous media. Except for the inhomogeneous segment, which varies in length from 0 to 160λ, the waveguide is piecewise homogeneous everywhere. The inhomogeneity is modelled by two-dimensional random permittivity fluctuations that are numerically generated from an assumed Gaussian correlation function.

In 2D, the Maxwell equations are solved in the frequency domain for both TE and TM polarization by using modal expansion methods, perfectly matched absorbing boundary layers and the R-matrix transfer matrix algorithm. The guided waves are excited by a Gaussian beam incident on the waveguide aperture. For various waveguide design parameters, numerical results are given for waveguide power loss per unit length of waveguide inhomogeneity. The power loss curves are calculated as the average from numerous realizations of the random permittivity and the coefficient of variation is also given.     

Structural study of the Diels-Alder reaction between tetrachlorodimethoxycyclopentadiene and the nickelocene-dimethyl acetylenedicarboxylate adduct. Crystal structures of the products

The cyclopentadienylnickel complex (2) and the parent dimethyl norbornadienedicarboxylate (4) each react with 1,2,3,4-tetrachloro-5,5-dimethoxycyclopentadiene (3) on the less hindred exo-face of the dienophile to produce anti-sequinorbornane adducts 5 and 6. X-ray crystal structures of these adducts have been determined and refined to R values of 0.029 and 0.042 respectively.     

Band structure enhancement of indirect transitions

Indirect and direct transitions in Ga_xIn_1?xP alloys has been studied in modulation spectroscopy, between 10 and 300 K. Deviation from simple indirect absorption has been studied and shown to be consistant with a model which takes into account band structure enhancement of the absorption coefficient at alloy composition values where direct and indirect edges occur, at nearly the same energy. For the first time we observe structures of Δε₁ corresponding to band structure enhanced indirect transitions.