Regularity properties of the cubic nonlinear Schr\"odinger equation on the half line

In this paper we study the local and global regularity properties of the cubic nonlinear Schr\"odinger equation (NLS) on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results and the behavior of higher order Sobolev norms of the solutions. In particular, we prove that the nonlinear part of the cubic NLS on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic cubic NLS \cite{et2} and the cubic NLS on the line \cite{erin}. We also prove that in the defocusing case the norm of the solution grows at most polynomially-in-time while in the focusing case it grows exponentially-in-time. As a byproduct of our analysis we provide a different proof of an almost sharp local wellposedness in $H^s(\R^+)$. Sharp $L^2$ local wellposedness was obtained in \cite{holmer} and \cite{bonaetal}. Our methods simplify some ideas in the wellposedness theory of initial and boundary value problems that were developed in \cite{collianderkenig, holmer,holmer1,bonaetal}.Comment: 30 pages. Minor revisions. To appear in Journal of Functional Analysi

A weighted dispersive estimate for Schr\"{o}dinger operators in dimension two

Let $H=-\Delta+V$, where $V$ is a real valued potential on $\R^2$ satisfying |V(x)|\les \la x\ra^{-3-}. We prove that if zero is a regular point of the spectrum of $H=-\Delta+V$, then \|w^{-1} e^{itH}P_{ac}f\|_{L^\infty(\R^2)}\les \f1{|t|\log^2(|t|)} \|w f\|_{L^1(\R^2)}, |t| >2, with $w(x)=\log^2(2+|x|)$. This decay rate was obtained by Murata in the setting of weighted $L^2$ spaces with polynomially growing weights.Comment: 23 page

Aerial Vehicle Tracking by Adaptive Fusion of Hyperspectral Likelihood Maps

Hyperspectral cameras can provide unique spectral signatures for consistently distinguishing materials that can be used to solve surveillance tasks. In this paper, we propose a novel real-time hyperspectral likelihood maps-aided tracking method (HLT) inspired by an adaptive hyperspectral sensor. A moving object tracking system generally consists of registration, object detection, and tracking modules. We focus on the target detection part and remove the necessity to build any offline classifiers and tune a large amount of hyperparameters, instead learning a generative target model in an online manner for hyperspectral channels ranging from visible to infrared wavelengths. The key idea is that, our adaptive fusion method can combine likelihood maps from multiple bands of hyperspectral imagery into one single more distinctive representation increasing the margin between mean value of foreground and background pixels in the fused map. Experimental results show that the HLT not only outperforms all established fusion methods but is on par with the current state-of-the-art hyperspectral target tracking frameworks.Comment: Accepted at the International Conference on Computer Vision and Pattern Recognition Workshops, 201

First principles study of electronic and structural properties of CuO

We investigate the electronic and structural properties of CuO, which shows significant deviations from the trends obeyed by other transition-metal monoxides. Using an extended Hubbard corrective functional, we uncover an orbitally ordered insulating ground state for the cubic phase of this material, which was expected but never found before. This insulating state results from a fine balance between the tendency of Cu to complete its d-shell and Hund's rule magnetism. Starting from the ground state for the cubic phase, we also study tetragonal distortions of the unit cell (recently reported in experiments), the consequent electronic reorganizations and identify the equilibrium structure. Our calculations reveal an unexpected richness of possible magnetic and orbital orders, relatively close in energy to the ground state, whose stability depends on the sign and entity of distortion.Comment: 9 pages, 9 figure

Fractal solutions of linear and nonlinear dispersive partial differential equations

In this paper we study fractal solutions of linear and nonlinear dispersive PDE on the torus. In the first part we answer some open questions on the fractal solutions of linear Schr\"odinger equation and equations with higher order dispersion. We also discuss applications to their nonlinear counterparts like the cubic Schr\"odinger equation (NLS) and the Korteweg-de Vries equation (KdV). In the second part, we study fractal solutions of the vortex filament equation and the associated Schr\"odinger map equation (SM). In particular, we construct global strong solutions of the SM in $H^s$ for $s>\frac32$ for which the evolution of the curvature is given by a periodic nonlinear Schr\"odinger evolution. We also construct unique weak solutions in the energy level. Our analysis follows the frame construction of Chang {\em et al.} \cite{csu} and Nahmod {\em et al.} \cite{nsvz}.Comment: 28 page