826 research outputs found
Transference Principles for Log-Sobolev and Spectral-Gap with Applications to Conservative Spin Systems
We obtain new principles for transferring log-Sobolev and Spectral-Gap
inequalities from a source metric-measure space to a target one, when the
curvature of the target space is bounded from below. As our main application,
we obtain explicit estimates for the log-Sobolev and Spectral-Gap constants of
various conservative spin system models, consisting of non-interacting and
weakly-interacting particles, constrained to conserve the mean-spin. When the
self-interaction is a perturbation of a strongly convex potential, this
partially recovers and partially extends previous results of Caputo,
Chafa\"{\i}, Grunewald, Landim, Lu, Menz, Otto, Panizo, Villani, Westdickenberg
and Yau. When the self-interaction is only assumed to be (non-strongly) convex,
as in the case of the two-sided exponential measure, we obtain sharp estimates
on the system's spectral-gap as a function of the mean-spin, independently of
the size of the system.Comment: 57 page
Autonomous frequency domain identification: Theory and experiment
The analysis, design, and on-orbit tuning of robust controllers require more information about the plant than simply a nominal estimate of the plant transfer function. Information is also required concerning the uncertainty in the nominal estimate, or more generally, the identification of a model set within which the true plant is known to lie. The identification methodology that was developed and experimentally demonstrated makes use of a simple but useful characterization of the model uncertainty based on the output error. This is a characterization of the additive uncertainty in the plant model, which has found considerable use in many robust control analysis and synthesis techniques. The identification process is initiated by a stochastic input u which is applied to the plant p giving rise to the output. Spectral estimation (h = P sub uy/P sub uu) is used as an estimate of p and the model order is estimated using the produce moment matrix (PMM) method. A parametric model unit direction vector p is then determined by curve fitting the spectral estimate to a rational transfer function. The additive uncertainty delta sub m = p - unit direction vector p is then estimated by the cross spectral estimate delta = P sub ue/P sub uu where e = y - unit direction vectory y is the output error, and unit direction vector y = unit direction vector pu is the computed output of the parametric model subjected to the actual input u. The experimental results demonstrate the curve fitting algorithm produces the reduced-order plant model which minimizes the additive uncertainty. The nominal transfer function estimate unit direction vector p and the estimate delta of the additive uncertainty delta sub m are subsequently available to be used for optimization of robust controller performance and stability
Towards a unified theory of Sobolev inequalities
We discuss our work on pointwise inequalities for the gradient which are
connected with the isoperimetric profile associated to a given geometry. We
show how they can be used to unify certain aspects of the theory of Sobolev
inequalities. In particular, we discuss our recent papers on fractional order
inequalities, Coulhon type inequalities, transference and dimensionless
inequalities and our forthcoming work on sharp higher order Sobolev
inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1
Hypercontractive measures, Talagrand's inequality, and influences
We survey several Talagrand type inequalities and their application to
influences with the tool of hypercontractivity for both discrete and
continuous, and product and non-product models. The approach covers similarly
by a simple interpolation the framework of geometric influences recently
developed by N. Keller, E. Mossel and A. Sen. Geometric Brascamp-Lieb
decompositions are also considered in this context
Estimates for measures of sections of convex bodies
A estimate in the hyperplane problem with arbitrary measures has
recently been proved in \cite{K3}. In this note we present analogs of this
result for sections of lower dimensions and in the complex case. We deduce
these inequalities from stability in comparison problems for different
generalizations of intersection bodies
Statistical properties of random matrix product states
We study the set of random matrix product states (RMPS) introduced in
arXiv:0908.3877 as a tool to explore foundational aspects of quantum
statistical mechanics. In the present work, we provide an accurate numerical
and analytical investigation of the properties of RMPS. We calculate the
average state of the ensemble in the non-homogeneous case, and numerically
check the validity of this result. We also suggest using RMPS as a tool to
approximate properties of general quantum random states. The numerical
simulations presented here support the accuracy and efficiency of this
approximation. These results suggest that any generalized canonical state can
be approximated with high probability by the reduced density matrix of a random
MPS, if the average MPS coincide with the associated microcanonical ensemble.Comment: 12 pages, 17 figures; published versio
Moments of unconditional logarithmically concave vectors
We derive two-sided bounds for moments of linear combinations of coordinates
od unconditional log-concave vectors. We also investigate how well moments of
such combinations may be approximated by moments of Gaussian random variables.Comment: 14 page
- …