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

Effects of histamine H1-, H2- and H3-receptor selective drugs on the mechanical activity of guinea-pig small and large intestine

1. In this study we have evaluated the possible contribution of acetylcholine release in histamine-induced contractions of guinea-pig large and small intestinal smooth muscle. Moreover, the presence of the histamine receptor types involved in smooth muscle relaxations and inhibition of electrically-induced twitches was studied by use of several selective agents. 2. Histamine-induced contractions appeared to be a pure H1-receptor-mediated effect. Responses were not attenuated by the presence of 0.1 microM atropine and were competitively and stereoselectively inhibited by the two enantiomers of chlorpheniramine with pA2 values of 6.73 +/- 0.08, 7.30 +/- 0.06, 6.93 +/- 0.03 and 7.19 +/- 0.04 for the L-isomer and 8.63 +/- 0.09, 8.85 +/- 0.09, 9.01 +/- 0.16 and 0.11 for the D-isomer in the duodenum, jejunum, ileum and colon, respectively. 3. There appeared to be a marked regional difference in sensitivity to histamine. In ileal and jejunal preparations pD2 values of 6.24 +/- 0.06 (n = 22) and 6.37 +/- 0.07 (n = 22) were found, whereas the pD2 values in the duodenum and colon were 5.55 +/- 0.05 (n = 36) and 5.68 +/- 0.06 (n = 31) respectively. 4. This regional difference in sensitivity to histamine was not due to variations in receptor affinity since pA2 values for the two enantiomers of chlorpheniramine did not differ markedly among the four tested preparations. Since a similar variation in sensitivity was found for methacholine, it is likely that the signal transfer mechanism in guinea-pig ileum and jejunum is more efficient than in the duodenum and colon. 5. The H2-agonists dimaprit and impromidine relaxed methacholine-precontracted (+/- 70% of maximum contraction) intestine at high concentrations (pD2 values of 3.79 + 0.03 and 4.44 + 0.09 for the jejunum). These relaxations could not be antagonized by 0.1 microM tiotidine, famotidine or mifentidine and were observed in all parts of the intestine investigated. 6. The dimaprit analogues nordimaprit and homodimaprit (inactive at H2-receptors) were equipotent in relaxing the methacholine-precontracted smooth muscle. Since several H2-antagonists were also able to produce relaxations, we do not consider these relaxations to be mediated by a H2-receptor subtype, but to be due to some nonspecific effects at the high concentrations used. 7. The histamine receptor involved in the inhibition of electrically-induced contractions in the presence of atropine could be classified as an H3-receptor effect. In all parts of the intestine the H3-agonist R-alpha-amethylhistamine inhibited the twitches with pD2 values ranging from 8.10 + 0.06 (ileum) to 8.27 + 0.03(colon). This effect was competitively antagonized with the selective H3-antagonist thioperamine (pA2 values are 8.09 + 0.07, 8.13 + 0.05, 8.15 + 0.04 and 8.36 + 0.04 in duodenum, jejunum, ileum and colon, respectively. 8. The guinea-pig intestine is a suitable preparation for the evaluation of either H1- or H3-receptor effects. H2-receptors, causing smooth muscle relaxation appear not to be present in our preparations. At high concentrations of H2-receptor agents (agonists and antagonists) relaxations might be observed, due to unknown nonspecific effects