13,645 research outputs found
Fast rates for empirical vector quantization
We consider the rate of convergence of the expected loss of empirically
optimal vector quantizers. Earlier results show that the mean-squared expected
distortion for any fixed distribution supported on a bounded set and satisfying
some regularity conditions decreases at the rate O(log n/n). We prove that this
rate is actually O(1/n). Although these conditions are hard to check, we show
that well-polarized distributions with continuous densities supported on a
bounded set are included in the scope of this result.Comment: 18 page
When are the invariant submanifolds of symplectic dynamics Lagrangian?
Let L be a D-dimensional submanifold of a 2D-dimensional exact symplectic
manifold (M, w) and let f be a symplectic diffeomorphism onf M. In this
article, we deal with the link between the dynamics of f restricted to L and
the geometry of L (is L Lagrangian, is it smooth, is it a graph...?). We prove
different kinds of results. - for D=3, we prove that if a torus that carries
some characteristic loop, then either L is Lagrangian or the restricted
dynamics g of f to L can not be minimal (i.e. all the orbits are dense) with
(g^k) equilipschitz; - for a Tonelli Hamiltonian of the cotangent bundle M of
the 3-dimenional torus, we give an example of an invariant submanifold L with
no conjugate points that is not Lagrangian and such that for every symplectic
diffeomorphism f of M, if , then is not minimal; - with some
hypothesis for the restricted dynamics, we prove that some invariant Lipschitz
D-dimensional submanifolds of Tonelli Hamiltonian flows are in fact Lagrangian,
C^1 and graphs; -we give similar results for C^1 submanifolds with weaker
dynamical assumptions.Comment: 17 page
Large deviation functional of the weakly asymmetric exclusion process
We obtain the large deviation functional of a density profile for the
asymmetric exclusion process of L sites with open boundary conditions when the
asymmetry scales like 1/L. We recover as limiting cases the expressions derived
recently for the symmetric (SSEP) and the asymmetric (ASEP) cases. In the ASEP
limit, the non linear differential equation one needs to solve can be analysed
by a method which resembles the WKB method
On hyperbolic analogues of some classical theorems in spherical geometry
We provide hyperbolic analogues of some classical theorems in spherical
geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the
spherical results are also made more precise
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