Searcharxiv⌕ Search

arXiv subjects

Jean-Jacques Loeb

Publications and source records attributed to Jean-Jacques Loeb.

4 recordsLinked to original sources
arXiv2023

Geometric approach for non pharmaceutical interventions in epidemiology

Various non pharmaceutical interventions have been settled to minimise the burden of the COVID-19 outbreak. We build a framework to analyse the dynamics of non pharmaceutical interventions, to distinguish between mitigations measures leading to objective scientific improvements and mitigations based on both political and scientific considerations. We analyse two possible strategies within this framework. Namely, we consider mitigations driven by the limited resources of the health system and mitigations where a constant set of measures is applied at different moments. We describe the optimal interventions for these scenarios. Our approach is mathematical and involves sir differential systems, it is qualitative and geometrical rather than computational. Along with the analysis of these scenarios, we collect several results that may be useful on their own, in particular on the ground when the variables are not known in real time.

q-bio.PE↗
arXiv2013

Strong Logarithmic Sobolev Inequalities for Log-Subharmonic Functions

We prove an intrinsic equivalence between strong hypercontractivity and a strong logarithmic Sobolev inequality for the cone of logarithmically subharmonic functions. We introduce a new large class of measures, Euclidean regular and exponential type, in addition to all compactly-supported measures, for which this equivalence holds. We prove a Sobolev density theorem through log-subharmonic functions, and use it to prove the equivalence of strong hypercontractivity and the strong log Sobolev inequality for such log-subharmonic functions.

math.FA↗
arXiv2008

Hypercontractivity for log-subharmonic functions

We prove strong hypercontractivity (SHC) inequalities for logarithmically subharmonic functions on $\RR^n$ and different classes of measures: Gaussian measures on $\RR^n$, symmetric Bernoulli and symmetric uniform probability measures on $\RR$, as well as their convolutions. Surprisingly, a slightly weaker strong hypercontractivity property holds for {\em any} symmetric measure on $\RR$. For all measures on $\R$ for which we know the (SHC) holds, we prove that a log--Sobolev inequality holds in the log-subharmonic category with a constant {\em smaller} than the one for Gaussian measure in the classical context. This result is extended to all dimensions for compactly-supported measures.

math.FA↗