53,443 research outputs found
Strichartz Estimates for Charge Transfer Models
In this note, we prove Strichartz estimates for scattering states of scalar
charge transfer models in . Based on the idea of the proof of
Strichartz estimates which follows \cite{CM,RSS}, we also show the energy of
the whole evolution is bounded independently of time without using the phase
space method, for example, in \cite{Graf}. One can easily generalize our
argument to for . Finally, in the last section, we
discuss the extension of these results to matrix charge transfer models in
.Comment: 26 pages, this is a revised version based on the comments of an
anonymous refere
Multisolitons for the defocusing energy critical wave equation with potentials
We construct multisoliton solutions to the defocusing energy critical wave
equation with potentials in based on regular and reversed
Strichartz estimates developed in \cite{GC3} for wave equations with charge
transfer Hamiltonians. We also show the asymptotic stability of multisoliton
solutions. The multisoliton structures with both stable and unstable solitons
are covered. Since each soliton decays slowly with rate , the interactions among the solitons are strong. Some
reversed Strichartz estimates and local decay estimates for the charge transfer
model are established to handle strong interactions.Comment: 33 page
Non-divergence Parabolic Equations of Second Order with Critical Drift in Morrey Spaces
We consider uniformly parabolic equations and inequalities of second order in
the non-divergence form with drift
in some domain .
We prove a variant of Aleksandrov-Bakelman-Pucci-Krylov-Tso estimate with
norm of the inhomogeneous term for some number . Based on it, we
derive the growth theorems and the interior Harnack inequality. In this paper,
we will only assume the drift is in certain Morrey spaces defined below
which are critical under the parabolic scaling but not necessarily to be
bounded. This is a continuation of the work in \cite{GC}.Comment: 23 pages. arXiv admin note: substantial text overlap with
arXiv:1511.0121
Non-divergence Parabolic Equations of Second Order with Critical Drift in Lebesgue Spaces
We consider uniformly parabolic equations and inequalities of second order in
the non-divergence form with drift
in some domain . We prove growth theorems and the
interior Harnack inequality as the main results. In this paper, we will only
assume the drift is in certain Lebesgue spaces which are critical under the
parabolic scaling but not necessarily to be bounded. In the last section, some
applications of the interior Harnack inequality are presented. In particular,
we show there is a "universal" spectral gap for the associated elliptic
operator. The counterpart for uniformly elliptic equations of second order in
non-divergence form is shown in \cite{S10}.Comment: 30 pages, the introduction is revise
Cosmological constraints on ultra-light axion fields
Ultra-light axions (ULAs) with mass less than 10^-20 eV have interesting
behaviors that may contribute to either dark energy or dark matter at different
epochs of the Universe. Its properties can be explored by cosmological
observations, such as expansion history of the Universe, cosmic large-scale
structure, cosmic microwave background, etc. In this work, we study the ULAs
with a mass around 10^-33 eV, which means the ULA field still rolls slowly at
present with the equation of state w=-1 as dark energy. In order to investigate
the mass and other properties of this kind of ULA field, we adopt the
measurements of Type Ia supernova (SN Ia), baryon acoustic oscillation (BAO),
and Hubble parameter H(z). The Markov Chain Monte Carlo (MCMC) technique is
employed to perform the constraints on the parameters. Finally, by exploring
four cases of the model, we find that the mass of this ULA field is about
3x10^-33 eV if assuming the initial axion field phi_i=M_pl. We also investigate
a general case by assuming phi_i< M_pl and find that the fitting results of
phi_i/M_pl are consistent with or close to 1 for the datasets we use.Comment: 10 pages, 4 figures, 5 tables. Accepted by Research in Astronomy and
Astrophysic
A Low-rank Tensor Dictionary Learning Method for Multi-spectral Images Denoising
As a 3-order tensor, a multi-spectral image (MSI) has dozens of spectral
bands, which can deliver more information for real scenes. However, real MSIs
are often corrupted by noises in the sensing process, which will further
deteriorate the performance of higher-level classification and recognition
tasks. In this paper, we propose a Low-rank Tensor Dictionary Learning (LTDL)
method for MSI denoising. Firstly, we extract blocks from the MSI and cluster
them into groups. Then instead of using the exactly low-rank model, we consider
a nearly low-rank approximation, which is closer to the latent low-rank
structure of the clean groups of real MSIs. In addition, we propose to learn an
spatial dictionary and an spectral dictionary, which contain the spatial
features and spectral features respectively of the whole MSI and are shared
among different groups. Hence the LTDL method utilizes both the latent low-rank
prior of each group and the correlation of different groups via the shared
dictionaries. Experiments on synthetic data validate the effectiveness of
dictionary learning by the LTDL. Experiments on real MSIs demonstrate the
superior denoising performance of the proposed method in comparison to
state-of-the-art methods
On Finite Block-Length Quantization Distortion
We investigate the upper and lower bounds on the quantization distortions for
independent and identically distributed sources in the finite block-length
regime. Based on the convex optimization framework of the rate-distortion
theory, we derive a lower bound on the quantization distortion under finite
block-length, which is shown to be greater than the asymptotic distortion given
by the rate-distortion theory. We also derive two upper bounds on the
quantization distortion based on random quantization codebooks, which can
achieve any distortion above the asymptotic one. Moreover, we apply the new
upper and lower bounds to two types of sources, the discrete binary symmetric
source and the continuous Gaussian source. For the binary symmetric source, we
obtain the closed-form expressions of the upper and lower bounds. For the
Gaussian source, we propose a computational tractable method to numerically
compute the upper and lower bounds, for both bounded and unbounded quantization
codebooks.Numerical results show that the gap between the upper and lower
bounds is small for reasonable block length and hence the bounds are tight
The limit on for tachyon dark energy
We get the same degeneracy relation between and for the tachyon
fields as for quintessence and phantom fields. Our results show that the
dynamics of scalar fields with different origins becomes indistinguishable when
the equation of state parameter does not deviate too far away from -1, and
the time variation satisfies the same bound for the same class of models.
For the tachyon fields, a limit on exists due to the Hubble damping and we
derived the generic bounds on for different classes of models. We may
distinguish different models in the phase plane of . The current
constraints on and are consistent with all classes of models. We need
to improve the constraint on by 50% to distinguish different models.Comment: use revtex, 2 figure
On Second Order Elliptic and Parabolic Equations of Mixed Type
It is known that solutions to second order uniformly elliptic and parabolic
equations, either in divergence or nondivergence (general) form, are H\"{o}lder
continuous and satisfy the interior Harnack inequality. We show that even in
the one-dimensional case (), these properties are not preserved for
equations of mixed divergence-nondivergence structure: for elliptic equations.
\begin{equation*}
D_i(a^1_{ij}D_ju)+a^2_{ij}D_{ij}u=0, \end{equation*} and parabolic equations
\begin{equation*} p\partial_t u=D_i(a_{ij}D_ju), \end{equation*} where
is a bounded strictly positive function. The H\"{o}lder continuity
and Harnack inequality are known if does not depend either on or on
. We essentially use homogenization techniques in our construction.
Bibliography: 23 titles.Comment: 16 page
Long-time asymptotics of the modified KdV equation in weighted Sobolev spaces
The long time behavior of solutions to the defocusing modified Korteweg-de
vries (MKdV) equation is established for initial conditions in some weighted
Sobolev spaces. Our approach is based on the nonlinear steepest descent method
of Deift and Zhou and its reformulation by Dieng and McLaughlin through
-derivatives. To extend the asymptotics to solutions with
initial data in lower regularity spaces, we apply a global approximation via
PDE techniques.Comment: 51 page
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