238 research outputs found
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 to for any .
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 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
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