125,309 research outputs found
On the negative spectrum of the Robin Laplacian in corner domains
For a bounded corner domain , we consider the Robin Laplacian in
with large Robin parameter. Exploiting multiscale analysis and a
recursive procedure, we have a precise description of the mechanism giving the
ground state of the spectrum. It allows also the study of the bottom of the
essential spectrum on the associated tangent structures given by cones. Then we
obtain the asymptotic behavior of the principal eigenvalue for this singular
limit in any dimension, with remainder estimates. The same method works for the
Schr\"odinger operator in with a strong attractive
delta-interaction supported on . Applications to some Erhling's
type estimates and the analysis of the critical temperature of some
superconductors are also provided
An Algorithm for Computing the Limit Points of the Quasi-component of a Regular Chain
For a regular chain , we propose an algorithm which computes the
(non-trivial) limit points of the quasi-component of , that is, the set
. Our procedure relies on Puiseux series expansions
and does not require to compute a system of generators of the saturated ideal
of . We focus on the case where this saturated ideal has dimension one and
we discuss extensions of this work in higher dimensions. We provide
experimental results illustrating the benefits of our algorithms
Chains in CR geometry as geodesics of a Kropina metric
With the help of a generalization of the Fermat principle in general
relativity, we show that chains in CR geometry are geodesics of a certain
Kropina metric constructed from the CR structure. We study the projective
equivalence of Kropina metrics and show that if the kernel distributions of the
corresponding 1-forms are non-integrable then two projectively equivalent
metrics are trivially projectively equivalent. As an application, we show that
sufficiently many chains determine the CR structure up to conjugacy,
generalizing and reproving the main result of [J.-H. Cheng, 1988]. The
correspondence between geodesics of the Kropina metric and chains allows us to
use the methods of metric geometry and the calculus of variations to study
chains. We use these methods to re-prove the result of [H. Jacobowitz, 1985]
that locally any two points of a strictly pseudoconvex CR manifolds can be
joined by a chain. Finally, we generalize this result to the global setting by
showing that any two points of a connected compact strictly pseudoconvex CR
manifold which admits a pseudo-Einstein contact form with positive
Tanaka-Webster scalar curvature can be joined by a chain.Comment: are very welcom
Ground Energy of the Magnetic Laplacian in Polyhedral Bodies
The asymptotic behavior of the first eigenvalues of magnetic Laplacian
operators with large magnetic fields and Neumann realization in polyhedral
domains is characterized by a hierarchy of model problems. We investigate
properties of the model problems (continuity, semi-continuity, existence of
generalized eigenfunctions). We prove estimates for the remainders of our
asymptotic formula. Lower bounds are obtained with the help of a classical IMS
partition based on adequate coverings of the polyhedral domain, whereas upper
bounds are established by a novel construction of quasimodes, qualified as
sitting or sliding according to spectral properties of local model problems.Comment: 59 page
Geometry of Higher-Order Markov Chains
We determine an explicit Gr\"obner basis, consisting of linear forms and
determinantal quadrics, for the prime ideal of Raftery's mixture transition
distribution model for Markov chains. When the states are binary, the
corresponding projective variety is a linear space, the model itself consists
of two simplices in a cross-polytope, and the likelihood function typically has
two local maxima. In the general non-binary case, the model corresponds to a
cone over a Segre variety.Comment: 9 page
Local entropy averages and projections of fractal measures
We show that for families of measures on Euclidean space which satisfy an
ergodic-theoretic form of "self-similarity" under the operation of re-scaling,
the dimension of linear images of the measure behaves in a semi-continuous way.
We apply this to prove the following conjecture of Furstenberg: Let m,n be
integers which are not powers of the same integer, and let X,Y be closed
subsets of the unit interval which are invariant, respectively, under times-m
mod 1 and times-n mod 1. Then, for any non-zero t:
dim(X+tY)=min{1,dim(X)+dim(Y)}. A similar result holds for invariant measures,
and gives a simple proof of the Rudolph-Johnson theorem. Our methods also apply
to many other classes of conformal fractals and measures. As another
application, we extend and unify Results of Peres, Shmerkin and Nazarov, and of
Moreira, concerning projections of products self-similar measures and Gibbs
measures on regular Cantor sets. We show that under natural irreducibility
assumptions on the maps in the IFS, the image measure has the maximal possible
dimension under any linear projection other than the coordinate projections. We
also present applications to Bernoulli convolutions and to the images of
fractal measures under differentiable maps.Comment: 55 pages. Version 2: Corrected an error in proof Thm. 4.3; some new
references; various small correction
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