7,835 research outputs found
On nonperturbative localization with quasi-periodic potential
The two main results of the article are concerned with Anderson Localization
for one-dimensional lattice Schroedinger operators with quasi-periodic
potentials with d frequencies. First, in the case d = 1 or 2, it is proved that
the spectrum is pure-point with exponentially decaying eigenfunctions for all
potentials (defined in terms of a trigonometric polynomial on the d-dimensional
torus) for which the Lyapounov exponents are strictly positive for all
frequencies and all energies. Second, for every non-constant real-analytic
potential and with a Diophantine set of d frequencies, a lower bound is given
for the Lyapounov exponents for the same potential rescaled by a sufficiently
large constant.Comment: 45 pages, published version, abstract added in migratio
On the Existence and Uniqueness of Global Solutions for the KdV Equation with Quasi-Periodic Initial Data
We consider the KdV equation with quasi-periodic initial data whose Fourier coefficients decay
exponentially and prove existence and uniqueness, in the class of functions
which have an expansion with exponentially decaying Fourier coefficients, of a
solution on a small interval of time, the length of which depends on the given
data and the frequency vector involved. For a Diophantine frequency vector and
for small quasi-periodic data (i.e., when the Fourier coefficients obey with sufficiently small,
depending on and the frequency vector), we prove global
existence and uniqueness of the solution. The latter result relies on our
recent work \cite{DG} on the inverse spectral problem for the quasi-periodic
Schr\"{o}dinger equation.Comment: 26 pages, to appear in J. Amer. Math. So
The Bayes Linear Programming Language [B/D]
Bayes linear methodology provides a quantitative structure for expressing our beliefs and systematic methods for revising these beliefs given observational data. Particular emphasis is placed upon interpretation of and diagnostics for the specification. The approach is similar in spirit to the standard Bayes analysis, but is constructed so as to avoid much of the burden of specification and computation of the full Bayes case. This report is the first of a series describing Bayes linear methods. In this document, we introduce some of the basic machinery of the theory. Examples, computational issues, detailed derivations of results and approaches to belief elicitation will be addressed in related reports.
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