5,515 research outputs found
Inverse scattering results for manifolds hyperbolic near infinity
We study the inverse resonance problem for conformally compact manifolds
which are hyperbolic outside a compact set. Our results include compactness of
isoresonant metrics in dimension two and of isophasal negatively curved metrics
in dimension three. In dimensions four or higher we prove topological
finiteness theorems under the negative curvature assumption.Comment: 25 pages. v3: Minor corrections, references adde
The Miura Map on the Line
The Miura map (introduced by Miura) is a nonlinear map between function
spaces which transforms smooth solutions of the modified Korteweg - de Vries
equation (mKdV) to solutions of the Korteweg - de Vries equation (KdV). In this
paper we study relations between the Miura map and Schroedinger operators with
real-valued distributional potentials (possibly not decaying at infinity) from
various spaces. We also investigate mapping properties of the Miura map in
these spaces. As an application we prove existence of solutions of the Korteweg
- de Vries equation in the negative Sobolev space H^{-1} for the initial data
in the range of the Miura map.Comment: 33 page
CR-Invariants and the Scattering Operator for Complex Manifolds with Boundary
The purpose of this paper is to describe certain CR-covariant differential
operators on a strictly pseudoconvex CR manifold as residues of the
scattering operator for the Laplacian on an ambient complex K\"{a}hler manifold
having as a `CR-infinity.' We also characterize the CR -curvature in
terms of the scattering operator. Our results parallel earlier results of
Graham and Zworski \cite{GZ:2003}, who showed that if is an asymptotically
hyperbolic manifold carrying a Poincar\'{e}-Einstein metric, the -curvature
and certain conformally covariant differential operators on the `conformal
infinity' of can be recovered from the scattering operator on . The
results in this paper were announced in \cite{HPT:2006}.Comment: 32 page
A Spectral Approach to Consecutive Pattern-Avoiding Permutations
We consider the problem of enumerating permutations in the symmetric group on
elements which avoid a given set of consecutive pattern , and in
particular computing asymptotics as tends to infinity. We develop a general
method which solves this enumeration problem using the spectral theory of
integral operators on , where the patterns in has length
. Kre\u{\i}n and Rutman's generalization of the Perron--Frobenius theory
of non-negative matrices plays a central role. Our methods give detailed
asymptotic expansions and allow for explicit computation of leading terms in
many cases. As a corollary to our results, we settle a conjecture of Warlimont
on asymptotics for the number of permutations avoiding a consecutive pattern.Comment: a reference is added; corrected typos; to appear in Journal of
Combinatoric
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