Random Convex Hulls and Extreme Value Statistics

In this paper we study the statistical properties of convex hulls of $N$ random points in a plane chosen according to a given distribution. The points may be chosen independently or they may be correlated. After a non-exhaustive survey of the somewhat sporadic literature and diverse methods used in the random convex hull problem, we present a unifying approach, based on the notion of support function of a closed curve and the associated Cauchy's formulae, that allows us to compute exactly the mean perimeter and the mean area enclosed by the convex polygon both in case of independent as well as correlated points. Our method demonstrates a beautiful link between the random convex hull problem and the subject of extreme value statistics. As an example of correlated points, we study here in detail the case when the points represent the vertices of $n$ independent random walks. In the continuum time limit this reduces to $n$ independent planar Brownian trajectories for which we compute exactly, for all $n$, the mean perimeter and the mean area of their global convex hull. Our results have relevant applications in ecology in estimating the home range of a herd of animals. Some of these results were announced recently in a short communication [Phys. Rev. Lett. {\bf 103}, 140602 (2009)].Comment: 61 pages (pedagogical review); invited contribution to the special issue of J. Stat. Phys. celebrating the 50 years of Yeshiba/Rutgers meeting

Higher-dose DHA supplementation modulates immune responses in pregnancy and is associated with decreased preterm birth

Pregnancy and parturition involve extensive changes in the maternal immune system. In our randomized, multi-site, double-blind superiority trial using a Bayesian adaptive design, we demonstrated that 1000 mg/day of docosahexaenoic acid (DHA) was superior to 200 mg/day in preventing both early preterm birth (less than 34 weeks’ gestation) and preterm birth (less than 37 weeks’ gestation). The goal of this secondary study is to compare the effects of 1000 mg/day versus 200 mg/day on maternal inflammation, a possible mechanism by which DHA may prevent preterm birth. Maternal blood samples were collected at enrollment (12–20 weeks’ gestation) and at delivery. Red blood cell DHA levels were measured by gas chromatography, and plasma concentrations of sRAGE, IL-6, IL-1β, TNFα, and INFγ were measured by ELISA. Data were analyzed for associations with the DHA dose, gestational age at birth, and preterm birth (<37 weeks). Higher baseline and lower delivery levels of maternal sRAGE were associated with a greater probability of longer gestation and delivery at term gestation. Higher-dose DHA supplementation increased the probability of a smaller decrease in delivery sRAGE levels. Higher IL-6 concentrations at delivery were associated with the probability of delivering after 37 weeks, and higher-dose DHA supplementation increased the probability of greater increases in IL-6 concentrations between enrollment and delivery. These data provide a proposed mechanistic explanation of how a higher dose of DHA during pregnancy provides immunomodulatory regulation in the initiation of parturition by influencing sRAGE and IL-6 levels, which may explain its ability to reduce the risk of preterm birth. © 2021 by the authors. Licensee MDPI, Basel, Switzerland.Open access journalThis item from the UA Faculty Publications collection is made available by the University of Arizona with support from the University of Arizona Libraries. If you have questions, please contact us at [email protected]