Semilinear Schrödinger flows on hyperbolic spaces: scattering in <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>

We prove global well-posedness and scattering in H ¹ for the defocusing nonlinear Schrödinger equations

$\left\{\begin{array}{ll}(i\partial_t+\Delta_g)u=u|u|^{2\sigma};\\u(0)=\phi,\end{array}\right.$

on the hyperbolic spaces \({\mathbb{H}^d}\), d ≥ 2, for exponents \({\sigma \in (0, 2/(d-2))}\). The main unexpected conclusion is scattering to linear solutions in the case of small exponents σ; for comparison, on Euclidean spaces scattering in H ¹ is not known for any exponent \({\sigma \in (1/d, 2/d]}\) and is known to fail for \({\sigma \in (0, 1/d]}\). Our main ingredients are certain noneuclidean global in time Strichartz estimates and noneuclidean Morawetz inequalities.

     

On the Fourier spectrum of symmetric Boolean functions

We study the following questionWhat is the smallest t such that every symmetric boolean function on κ variables (which is not a constant or a parity function), has a non-zero Fourier coefficient of order at least 1 and at most t?We exclude the constant functions for which there is no such t and the parity functions for which t has to be κ. Let τ (κ) be the smallest such t. Our main result is that for large κ, τ (κ)≤4κ/logκ.The motivation for our work is to understand the complexity of learning symmetric juntas. A κ-junta is a boolean function of n variables that depends only on an unknown subset of κ variables. A symmetric κ-junta is a junta that is symmetric in the variables it depends on. Our result implies an algorithm to learn the class of symmetric κ-juntas, in the uniform PAC learning model, in time n ^o(κ) . This improves on a result of Mossel, O’Donnell and Servedio in [16], who show that symmetric κ-juntas can be learned in time n ^2κ/3.     

Perron Conditions for Pointwise and Global Exponential Dichotomy of Linear Skew-Product Flows

The aim of this paper is to give necessary and sufficient conditions for pointwise and uniform exponential dichotomy of linear skew-product flows. We shall obtain that the pointwise exponential dichotomy of a linear skew-product flow is equivalent to the pointwise admissibility of the pair As a consequence, we prove that a linear skew-product flow on is uniformly exponentially dichotomic if and only if the pair is uniformly admissible for .     

Exponential dichotomies of evolution operators in Banach spaces

This paper considers three dichotomy concepts (exponential dichotomy, uniform exponential dichotomy and strong exponential dichotomy) in the general context of non-invertible evolution operators in Banach spaces. Connections between these concepts are illustrated. Using the notion of Green function, we give necessary conditions and sufficient ones for strong exponential dichotomy. Some illustrative examples are presented to prove that the converse of some implication type theorems are not valid.     

Sufficient conditions of non‐uniqueness for the Coulomb friction problem

We consider the Signorini problem with Coulomb friction in elasticity. Sufficient conditions of non‐uniqueness are obtained for the continuous model. These conditions are linked to the existence of real eigenvalues of an operator in a Hilbert space. We prove that, under appropriate conditions, real eigenvalues exist for a non‐local Coulomb friction model. Finite element approximation of the eigenvalue problem is considered and numerical experiments are performed. Copyright © 2003 John Wiley & Sons, Ltd.     

Numerical solution of the steady incompressible Navier—Stokes equations for the flow past a sphere by a multigrid defect correction technique

A nested non-linear multigrid algorithm is developed to solve the Navier–Stokes equations which describe the steady incompressible flow past a sphere. The vorticity–streamfunction formulation of the Navier–Stokes equations is chosen. The continuous operators are discretized by an upwind finite difference scheme. Several algorithms are tested as smoothing steps. The multigrid method itself provides only a first-order-accurate solution. To obtain at least second-order accuracy, a defect correction iteration is used as outer iteration. Results are reported for Re = 50, 100, 400 and 1000.     

Surface plasmon polariton assisted optical pulling force

We demonstrate both analytically and numerically the existence of optical pulling forces acting on particles located near plasmonic interfaces. Two main factors contribute to the appearance of this negative recoil force. The interference between the incident and reflected waves induces a rotating dipole with an asymmetric scattering pattern, while the directional excitation of surface plasmon polaritons (SPPs) enhances the linear momentum of scattered light. The strongly asymmetric SPP excitation is determined by spin–orbit coupling of the rotating dipole and surface plasmon polariton. As a result of the total momentum conservation, the force acting on the particle points in a direction opposite to the incident wave propagation. We derive analytical expressions for the force acting on dipolar particles placed in the proximity of plasmonic surfaces. Analytical expressions for this pulling force are derived within the dipole approximation and are in excellent agreement with results of electromagnetic numerical calculations. The forces acting on larger particles are analyzed numerically, beyond the dipole approximation.

     

Green light radiation effects on free radicals inhibition in cellular and chemical systems

Free radicals generation is inhibited through green light (GL) irradiation in cellular systems and in chemical reactions. Standard melanocyte cultures were UV-irradiated and the induced cellular reactive oxygen species (ROS) were quantified by the fluorescence technique. The same cell cultures, previously protected by a 24h GL exposure, displayed a significantly lower ROS production. A simple chemical reaction is subsequently chosen, in which the production of free radicals is well defined. Paraffin wax and mineral oil were GL irradiated during thermal degradation and the oxidation products checked by chemiluminescence [CL] and Fourier transform infrared spectra [FT-IR]. The same clear inhibition of the radical oxidation of alkanes is recorded. A quantum chemistry modeling of these results is performed and a mechanism involving a new type of Rydberg macromolecular systems with implications for biology and medicine is suggested.     

The Structure of Multiplicative Tilings of the Real Line

Journal of Fourier Analysis and Applications - Suppose $$\Omega , A \subseteq \mathbb {R}\backslash {\left\{ {0}\right\} }$$ are two sets, both of mixed sign, that $$\Omega $$ is Lebesgue...