30 research outputs found
Functional central limit theorems for vicious walkers
We consider the diffusion scaling limit of the vicious walker model that is a
system of nonintersecting random walks. We prove a functional central limit
theorem for the model and derive two types of nonintersecting Brownian motions,
in which the nonintersecting condition is imposed in a finite time interval
for the first type and in an infinite time interval for
the second type, respectively. The limit process of the first type is a
temporally inhomogeneous diffusion, and that of the second type is a temporally
homogeneous diffusion that is identified with a Dyson's model of Brownian
motions studied in the random matrix theory. We show that these two types of
processes are related to each other by a multi-dimensional generalization of
Imhof's relation, whose original form relates the Brownian meander and the
three-dimensional Bessel process. We also study the vicious walkers with wall
restriction and prove a functional central limit theorem in the diffusion
scaling limit.Comment: AMS-LaTeX, 20 pages, 2 figures, v6: minor corrections made for
publicatio
On Poincare and logarithmic Sobolev inequalities for a class of singular Gibbs measures
This note, mostly expository, is devoted to Poincar{\'e} and log-Sobolev
inequalities for a class of Boltzmann-Gibbs measures with singular interaction.
Such measures allow to model one-dimensional particles with confinement and
singular pair interaction. The functional inequalities come from convexity. We
prove and characterize optimality in the case of quadratic confinement via a
factorization of the measure. This optimality phenomenon holds for all beta
Hermite ensembles including the Gaussian unitary ensemble, a famous exactly
solvable model of random matrix theory. We further explore exact solvability by
reviewing the relation to Dyson-Ornstein-Uhlenbeck diffusion dynamics admitting
the Hermite-Lassalle orthogonal polynomials as a complete set of
eigenfunctions. We also discuss the consequence of the log-Sobolev inequality
in terms of concentration of measure for Lipschitz functions such as maxima and
linear statistics.Comment: Minor improvements. To appear in Geometric Aspects of Functional
Analysis -- Israel Seminar (GAFA) 2017-2019", Lecture Notes in Mathematics
225
Chronic non-specific low back pain - sub-groups or a single mechanism?
Copyright 2008 Wand and O'Connell; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0),
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.Background: Low back pain is a substantial health problem and has subsequently attracted a
considerable amount of research. Clinical trials evaluating the efficacy of a variety of interventions
for chronic non-specific low back pain indicate limited effectiveness for most commonly applied
interventions and approaches.
Discussion: Many clinicians challenge the results of clinical trials as they feel that this lack of
effectiveness is at odds with their clinical experience of managing patients with back pain. A
common explanation for this discrepancy is the perceived heterogeneity of patients with chronic
non-specific low back pain. It is felt that the effects of treatment may be diluted by the application
of a single intervention to a complex, heterogeneous group with diverse treatment needs. This
argument presupposes that current treatment is effective when applied to the correct patient.
An alternative perspective is that the clinical trials are correct and current treatments have limited
efficacy. Preoccupation with sub-grouping may stifle engagement with this view and it is important
that the sub-grouping paradigm is closely examined. This paper argues that there are numerous
problems with the sub-grouping approach and that it may not be an important reason for the
disappointing results of clinical trials. We propose instead that current treatment may be ineffective
because it has been misdirected. Recent evidence that demonstrates changes within the brain in
chronic low back pain sufferers raises the possibility that persistent back pain may be a problem of
cortical reorganisation and degeneration. This perspective offers interesting insights into the
chronic low back pain experience and suggests alternative models of intervention.
Summary: The disappointing results of clinical research are commonly explained by the failure of
researchers to adequately attend to sub-grouping of the chronic non-specific low back pain
population. Alternatively, current approaches may be ineffective and clinicians and researchers may
need to radically rethink the nature of the problem and how it should best be managed