330 research outputs found
Existence of the Stark-Wannier quantum resonances
In this paper we prove the existence of the Stark-Wannier quantum resonances
for one-dimensional Schrodinger operators with smooth periodic potential and
small external homogeneous electric field. Such a result extends the existence
result previously obtained in the case of periodic potentials with a finite
number of open gaps.Comment: 30 pages, 1 figur
TernausNetV2: Fully Convolutional Network for Instance Segmentation
The most common approaches to instance segmentation are complex and use
two-stage networks with object proposals, conditional random-fields, template
matching or recurrent neural networks. In this work we present TernausNetV2 - a
simple fully convolutional network that allows extracting objects from a
high-resolution satellite imagery on an instance level. The network has popular
encoder-decoder type of architecture with skip connections but has a few
essential modifications that allows using for semantic as well as for instance
segmentation tasks. This approach is universal and allows to extend any network
that has been successfully applied for semantic segmentation to perform
instance segmentation task. In addition, we generalize network encoder that was
pre-trained for RGB images to use additional input channels. It makes possible
to use transfer learning from visual to a wider spectral range. For
DeepGlobe-CVPR 2018 building detection sub-challenge, based on public
leaderboard score, our approach shows superior performance in comparison to
other methods. The source code corresponding pre-trained weights are publicly
available at https://github.com/ternaus/TernausNetV
On the spectral properties of L_{+-} in three dimensions
This paper is part of the radial asymptotic stability analysis of the ground
state soliton for either the cubic nonlinear Schrodinger or Klein-Gordon
equations in three dimensions. We demonstrate by a rigorous method that the
linearized scalar operators which arise in this setting, traditionally denoted
by L_{+-}, satisfy the gap property, at least over the radial functions. This
means that the interval (0,1] does not contain any eigenvalues of L_{+-} and
that the threshold 1 is neither an eigenvalue nor a resonance. The gap property
is required in order to prove scattering to the ground states for solutions
starting on the center-stable manifold associated with these states. This paper
therefore provides the final installment in the proof of this scattering
property for the cubic Klein-Gordon and Schrodinger equations in the radial
case, see the recent theory of Nakanishi and the third author, as well as the
earlier work of the third author and Beceanu on NLS. The method developed here
is quite general, and applicable to other spectral problems which arise in the
theory of nonlinear equations
Embedded Eigenvalues and the Nonlinear Schrodinger Equation
A common challenge to proving asymptotic stability of solitary waves is
understanding the spectrum of the operator associated with the linearized flow.
The existence of eigenvalues can inhibit the dispersive estimates key to
proving stability. Following the work of Marzuola & Simpson, we prove the
absence of embedded eigenvalues for a collection of nonlinear Schrodinger
equations, including some one and three dimensional supercritical equations,
and the three dimensional cubic-quintic equation. Our results also rule out
nonzero eigenvalues within the spectral gap and, in 3D, endpoint resonances.
The proof is computer assisted as it depends on the sign of certain inner
products which do not readily admit analytic representations. Our source code
is available for verification at
http://www.math.toronto.edu/simpson/files/spec_prop_asad_simpson_code.zip.Comment: 29 pages, 27 figures: fixed a typo in an equation from the previous
version, and added two equations to clarif
Conditional stability theorem for the one dimensional Klein-Gordon equation
This is the published version, also available here: http://dx.doi.org/10.1063/1.3660780.The paper addresses the conditional non-linear stability of the steady state solutions of the one-dimensional Klein-Gordon equation for large time. We explicitly construct the center-stable manifold for the steady state solutions using the modulation method of Soffer and Weinstein and Strichartz type estimates. The main difficulty in the one-dimensional case is that the required decay of the Klein-Gordon semigroup does not follow from Strichartz estimates alone. We resolve this issue by proving an additional weighted decay estimate and further refinement of the function spaces, which allows us to close the argument in spaces with very little time decay
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