29,746 research outputs found
Low regularity solutions of two fifth-order KdV type equations
The Kawahara and modified Kawahara equations are fifth-order KdV type
equations and have been derived to model many physical phenomena such as
gravity-capillary waves and magneto-sound propagation in plasmas. This paper
establishes the local well-posedness of the initial-value problem for Kawahara
equation in with and the local well-posedness
for the modified Kawahara equation in with .
To prove these results, we derive a fundamental estimate on dyadic blocks for
the Kawahara equation through the multiplier norm method of Tao
\cite{Tao2001} and use this to obtain new bilinear and trilinear estimates in
suitable Bourgain spaces.Comment: 17page
Incoherent dictionaries and the statistical restricted isometry property
In this article we present a statistical version of the Candes-Tao restricted
isometry property (SRIP for short) which holds in general for any incoherent
dictionary which is a disjoint union of orthonormal bases. In addition, under
appropriate normalization, the eigenvalues of the associated Gram matrix
fluctuate around 1 according to the Wigner semicircle distribution. The result
is then applied to various dictionaries that arise naturally in the setting of
finite harmonic analysis, giving, in particular, a better understanding on a
remark of Applebaum-Howard-Searle-Calderbank concerning RIP for the Heisenberg
dictionary of chirp like functions.Comment: Key words: Incoherent dictionaries, statistical version of Candes -
Tao RIP, Semi-Circle law, deterministic constructions, Heisenberg-Weil
representatio
Compressive Network Analysis
Modern data acquisition routinely produces massive amounts of network data.
Though many methods and models have been proposed to analyze such data, the
research of network data is largely disconnected with the classical theory of
statistical learning and signal processing. In this paper, we present a new
framework for modeling network data, which connects two seemingly different
areas: network data analysis and compressed sensing. From a nonparametric
perspective, we model an observed network using a large dictionary. In
particular, we consider the network clique detection problem and show
connections between our formulation with a new algebraic tool, namely Randon
basis pursuit in homogeneous spaces. Such a connection allows us to identify
rigorous recovery conditions for clique detection problems. Though this paper
is mainly conceptual, we also develop practical approximation algorithms for
solving empirical problems and demonstrate their usefulness on real-world
datasets
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