A measure of non-convexity in the plane and the Minkowski sum

In this paper a measure of non-convexity for a simple polygonal region in the plane is introduced. It is proved that for "not far from convex" regions this measure does not decrease under the Minkowski sum operation, and guarantees that the Minkowski sum has no "holes".Comment: 5 figures; Discrete and Computational Geometry, 201

Gain properties of dye-doped polymer thin films

Hybrid pumping appears as a promising compromise in order to reach the much coveted goal of an electrically pumped organic laser. In such configuration the organic material is optically pumped by an electrically pumped inorganic device on chip. This engineering solution requires therefore an optimization of the organic gain medium under optical pumping. Here, we report a detailed study of the gain features of dye-doped polymer thin films. In particular we introduce the gain efficiency $K$, in order to facilitate comparison between different materials and experimental conditions. The gain efficiency was measured with various setups (pump-probe amplification, variable stripe length method, laser thresholds) in order to study several factors which modify the actual gain of a layer, namely the confinement factor, the pump polarization, the molecular anisotropy, and the re-absorption. For instance, for a 600 nm thick 5 wt\% DCM doped PMMA layer, the different experimental approaches give a consistent value $K\simeq$ 80 cm.MW$^{-1}$. On the contrary, the usual model predicting the gain from the characteristics of the material leads to an overestimation by two orders of magnitude, which raises a serious problem in the design of actual devices. In this context, we demonstrate the feasibility to infer the gain efficiency from the laser threshold of well-calibrated devices. Besides, temporal measurements at the picosecond scale were carried out to support the analysis.Comment: 15 pages, 17 figure

Monge's transport problem in the Heisenberg group

We prove the existence of solutions to Monge transport problem between two compactly supported Borel probability measures in the Heisenberg group equipped with its Carnot-Caratheodory distance assuming that the initial measure is absolutely continuous with respect to the Haar measure of the group

Symbolic approach and induction in the Heisenberg group

We associate a homomorphism in the Heisenberg group to each hyperbolic unimodular automorphism of the free group on two generators. We show that the first return-time of some flows in "good" sections, are conjugate to niltranslations, which have the property of being self-induced.Comment: 18 page

Buckling Instabilities of a Confined Colloid Crystal Layer

A model predicting the structure of repulsive, spherically symmetric, monodisperse particles confined between two walls is presented. We study the buckling transition of a single flat layer as the double layer state develops. Experimental realizations of this model are suspensions of stabilized colloidal particles squeezed between glass plates. By expanding the thermodynamic potential about a flat state of $N$ confined colloidal particles, we derive a free energy as a functional of in-plane and out-of-plane displacements. The wavevectors of these first buckling instabilities correspond to three different ordered structures. Landau theory predicts that the symmetry of these phases allows for second order phase transitions. This possibility exists even in the presence of gravity or plate asymmetry. These transitions lead to critical behavior and phases with the symmetry of the three-state and four-state Potts models, the X-Y model with 6-fold anisotropy, and the Heisenberg model with cubic interactions. Experimental detection of these structures is discussed.Comment: 24 pages, 8 figures on request. EF508

Sub-Riemannian Calculus on Hypersurfaces in Carnot Groups

We develope basic geometric quantities and properties of hypersurfaces in Carnot groups

Quenching dynamics in CdSe nanoparticles: surface-induced defects upon dilution

International audienceWe have analyzed the decays of the fluorescence of colloidal CdSe quantum dots (QDs) suspensions during dilution and titration by the ligands. A ligand shell made of a combination of trioctylphosphine (TOP), oleylamine (OA), and stearic acid (SA) stabilizes the as-synthesized QDs. The composition of the shell was analyzed and quantified using high resolution liquid state 1H nuclear magnetic resonance (NMR) spectroscopy. A quenching of the fluorescence of the QDs is observed upon removal of the ligands by diluting the stock solution of the QDs. The fluorescence is restored by the addition of TOP. We analyze the results by assuming a binomial distribution of quenchers among the QDs and predict a linear trend in the time-resolved fluorescence decays. We have used a nonparametric analysis to show that for our QDs, 3.0 0.1 quenching sites per QD on average are revealed by the removal of TOP. We moreover show that the quenching rates of the quenching sites add up. The decay per quenching site can be compared with the decay at saturation of the dilution effect. This provides a value of 2.88 0.02 for the number of quenchers per QD. We extract the quenching dynamics of one site. It appears to be a process with a distribution of rates that does not involve the ligands

Order and Frustration in Chiral Liquid Crystals

This paper reviews the complex ordered structures induced by chirality in liquid crystals. In general, chirality favors a twist in the orientation of liquid-crystal molecules. In some cases, as in the cholesteric phase, this favored twist can be achieved without any defects. More often, the favored twist competes with applied electric or magnetic fields or with geometric constraints, leading to frustration. In response to this frustration, the system develops ordered structures with periodic arrays of defects. The simplest example of such a structure is the lattice of domains and domain walls in a cholesteric phase under a magnetic field. More complex examples include defect structures formed in two-dimensional films of chiral liquid crystals. The same considerations of chirality and defects apply to three-dimensional structures, such as the twist-grain-boundary and moire phases.Comment: 39 pages, RevTeX, 14 included eps figure

Isometric group actions on Banach spaces and representations vanishing at infinity

Our main result is that the simple Lie group $G=Sp(n,1)$ acts properly isometrically on $L^p(G)$ if $p>4n+2$. To prove this, we introduce property ({\BP}_0^V), for $V$ be a Banach space: a locally compact group $G$ has property ({\BP}_0^V) if every affine isometric action of $G$ on $V$, such that the linear part is a $C_0$-representation of $G$, either has a fixed point or is metrically proper. We prove that solvable groups, connected Lie groups, and linear algebraic groups over a local field of characteristic zero, have property ({\BP}_0^V). As a consequence for unitary representations, we characterize those groups in the latter classes for which the first cohomology with respect to the left regular representation on $L^2(G)$ is non-zero; and we characterize uniform lattices in those groups for which the first $L^2$-Betti number is non-zero.Comment: 28 page

First and second variation formulae for the sub-Riemannian area in three-dimensional pseudo-hermitian manifolds

We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator for non-singular C^2 surfaces and another one for C2 (eventually singular) surfaces. Then we can obtain a necessary condition for the stability of a non-singular surface in a pseudo-hermitian 3-manifold in term of the pseudo-hermitian torsion and the Webster scalar curvature. Finally we classify complete stable surfaces in the roto-traslation group RT .Comment: 36 pages. Misprints corrected. Statement of Proposition 9.8 slightly changed and Remark 9.9 adde