An improved maximal inequality for 2D fractional order Schr\"{o}dinger operators

The local maximal inequality for the Schr\"{o}dinger operators of order \a>1 is shown to be bounded from $H^s(\R^2)$ to $L^2$ for any $s>\frac38$. This improves the previous result of Sj\"{o}lin on the regularity of solutions to fractional order Schr\"{o}dinger equations. Our method is inspired by Bourgain's argument in case of \a=2. The extension from \a=2 to general \a>1 confronts three essential obstacles: the lack of Lee's reduction lemma, the absence of the algebraic structure of the symbol and the inapplicable Galilean transformation in the deduction of the main theorem. We get around these difficulties by establishing a new reduction lemma at our disposal and analyzing all the possibilities in using the separateness of the segments to obtain the analogous bilinear $L^2-$estimates. To compensate the absence of Galilean invariance, we resort to Taylor's expansion for the phase function. The Bourgain-Guth inequality in \cite{ref Bourgain Guth} is also rebuilt to dominate the solution of fractional order Schr\"{o}dinger equations.Comment: Pages47, 3figures. To appear in Studia Mathematic

A Streaming Multi-GPU Implementation of Image Simulation Algorithms for Scanning Transmission Electron Microscopy

Simulation of atomic resolution image formation in scanning transmission electron microscopy can require significant computation times using traditional methods. A recently developed method, termed plane-wave reciprocal-space interpolated scattering matrix (PRISM), demonstrates potential for significant acceleration of such simulations with negligible loss of accuracy. Here we present a software package called Prismatic for parallelized simulation of image formation in scanning transmission electron microscopy (STEM) using both the PRISM and multislice methods. By distributing the workload between multiple CUDA-enabled GPUs and multicore processors, accelerations as high as 1000x for PRISM and 30x for multislice are achieved relative to traditional multislice implementations using a single 4-GPU machine. We demonstrate a potentially important application of Prismatic, using it to compute images for atomic electron tomography at sufficient speeds to include in the reconstruction pipeline. Prismatic is freely available both as an open-source CUDA/C++ package with a graphical user interface and as a Python package, PyPrismatic

Bilinear Kakeya-Nikodym averages of eigenfunctions on compact Riemannian surfaces

We obtain an improvement of the bilinear estimates of Burq, G\'erard and Tzvetkov in the spirit of the refined Kakeya-Nikodym estimates of Blair and the second author. We do this by using microlocal techniques and a bilinear version of H\"ormander's oscillatory integral theorem.Comment: 19 pages, 1 figure. Affiliation correcte