4,333 research outputs found
Relativistic Comparison Theorems
Comparison theorems are established for the Dirac and Klein--Gordon
equations. We suppose that V^{(1)}(r) and V^{(2)}(r) are two real attractive
central potentials in d dimensions that support discrete Dirac eigenvalues
E^{(1)}_{k_d\nu} and E^{(2)}_{k_d\nu}. We prove that if V^{(1)}(r) \leq
V^{(2)}(r), then each of the corresponding discrete eigenvalue pairs is ordered
E^{(1)}_{k_d\nu} \leq E^{(2)}_{k_d\nu}. This result generalizes an earlier more
restrictive theorem that required the wave functions to be node free. For the
the Klein--Gordon equation, similar reasoning also leads to a comparison
theorem provided in this case that the potentials are negative and the
eigenvalues are positive.Comment: 6 page
Sub-Laplacian comparison theorems on totally geodesic Riemannian foliations
We develop a variational theory of geodesics for the canonical variation of
the metric of a totally geodesic foliation. As a consequence, we obtain
comparison theorems for the horizontal and vertical Laplacians. In the case of
Sasakian foliations, we show that sharp horizontal and vertical comparison
theorems for the sub-Riemannian distance may be obtained as a limit of
horizontal and vertical comparison theorems for the Riemannian distances
approximations.Comment: Typos corrected, some improved bound
Comparison Theorems for Gibbs Measures
The Dobrushin comparison theorem is a powerful tool to bound the difference
between the marginals of high-dimensional probability distributions in terms of
their local specifications. Originally introduced to prove uniqueness and decay
of correlations of Gibbs measures, it has been widely used in statistical
mechanics as well as in the analysis of algorithms on random fields and
interacting Markov chains. However, the classical comparison theorem requires
validity of the Dobrushin uniqueness criterion, essentially restricting its
applicability in most models to a small subset of the natural parameter space.
In this paper we develop generalized Dobrushin comparison theorems in terms of
influences between blocks of sites, in the spirit of Dobrushin-Shlosman and
Weitz, that substantially extend the range of applicability of the classical
comparison theorem. Our proofs are based on the analysis of an associated
family of Markov chains. We develop in detail an application of our main
results to the analysis of sequential Monte Carlo algorithms for filtering in
high dimension.Comment: 55 page
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