3,670 research outputs found
Non-Weyl Resonance Asymptotics for Quantum Graphs
We consider the resonances of a quantum graph that consists of a
compact part with one or more infinite leads attached to it. We discuss the
leading term of the asymptotics of the number of resonances of in
a disc of a large radius. We call a \emph{Weyl graph} if the
coefficient in front of this leading term coincides with the volume of the
compact part of . We give an explicit topological criterion for a
graph to be Weyl. In the final section we analyze a particular example in some
detail to explain how the transition from the Weyl to the non-Weyl case occurs.Comment: 29 pages, 2 figure
Semiclassical bounds for spectra of biharmonic operators
We provide complementary semiclassical bounds for the Riesz means of
the eigenvalues of various biharmonic operators, with a second term in the
expected power of . The method we discuss makes use of the averaged
variational principle (AVP), and yields two-sided bounds for individual
eigenvalues, which are semiclassically sharp. The AVP also yields comparisons
with Riesz means of different operators, in particular Laplacians
A Noninformative Prior on a Space of Distribution Functions
In a given problem, the Bayesian statistical paradigm requires the
specification of a prior distribution that quantifies relevant information
about the unknowns of main interest external to the data. In cases where little
such information is available, the problem under study may possess an
invariance under a transformation group that encodes a lack of information,
leading to a unique prior---this idea was explored at length by E.T. Jaynes.
Previous successful examples have included location-scale invariance under
linear transformation, multiplicative invariance of the rate at which events in
a counting process are observed, and the derivation of the Haldane prior for a
Bernoulli success probability. In this paper we show that this method can be
extended, by generalizing Jaynes, in two ways: (1) to yield families of
approximately invariant priors, and (2) to the infinite-dimensional setting,
yielding families of priors on spaces of distribution functions. Our results
can be used to describe conditions under which a particular Dirichlet Process
posterior arises from an optimal Bayesian analysis, in the sense that
invariances in the prior and likelihood lead to one and only one posterior
distribution
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