Variance asymptotics and scaling limits for Gaussian Polytopes

Let $K_n$ be the convex hull of i.i.d. random variables distributed according to the standard normal distribution on $\R^d$. We establish variance asymptotics as $n \to \infty$ for the re-scaled intrinsic volumes and $k$-face functionals of $K_n$, $k \in \{0,1,...,d-1\}$, resolving an open problem. Variance asymptotics are given in terms of functionals of germ-grain models having parabolic grains with apices at a Poisson point process on $\R^{d-1} \times \R$ with intensity $e^h dh dv$. The scaling limit of the boundary of $K_n$ as $n \to \infty$ converges to a festoon of parabolic surfaces, coinciding with that featuring in the geometric construction of the zero viscosity solution to Burgers' equation with random input

Variance Asymptotics and Scaling Limits for Random Polytopes

Let K be a convex set in R d and let K $\lambda$ be the convex hull of a homogeneous Poisson point process P $\lambda$ of intensity $\lambda$ on K. When K is a simple polytope, we establish scaling limits as $\lambda$ $\rightarrow$ $\infty$ for the boundary of K $\lambda$ in a vicinity of a vertex of K and we give variance asymptotics for the volume and k-face functional of K $\lambda$, k $\in$ {0, 1, ..., d -- 1}, resolving an open question posed in [18]. The scaling limit of the boundary of K $\lambda$ and the variance asymptotics are described in terms of a germ-grain model consisting of cone-like grains pinned to the extreme points of a Poisson point process on R d--1 $\times$ R having intensity \sqrt de dh dhdv

Ambient Home Server: Location Aware Home Automation

Currently in the realm of home automation, there are two primary approaches. One approach creates independent devices which can be controlled by a smart phone or send notifications when its task is complete, such as door locks and washing machines. The other approach has devices with sensors on them which attempts to determine if people are around before performing an action, such as the Nest thermostat and the WeMo motion sensors. The goal of this project is to merge both of these approaches together by using a smart phone to create a location aware device which can inform the home of where people are located. This project requires a Java server and MariaDB to handle interactions between various sensors and controllers, sensors built with off the shelf parts to monitor the home and an Android application to inform the server of its location, as well as view and control sensors inside of the home

Some Classical Problems in Random Geometry

International audienceThis chapter is intended as a first introduction to selected topics in random geometry. It aims at showing how classical questions from recreational mathematics can lead to the modern theory of a mathematical domain at the interface of probability and geometry. Indeed, in each of the four sections, the starting point is a historical practical problem from geometric probability. We show that the solution of the problem, if any, and the underlying discussion are the gateway to the very rich and active domain of integral and stochastic geometry, which we describe at a basic level. In particular, we explain how to connect Buffon’s needle problem to integral geometry, Bertrand’s paradox to random tessellations, Sylvester’s four-point problem to random polytopes and Jeffrey’s bicycle wheel problem to random coverings. The results and proofs selected here have been especially chosen for non-specialist readers. They do not require much prerequisite knowledge on stochastic geometry but nevertheless comprise many of the main results on these models

Increased expression of CART, nNOS, VIP, PACAP, SP and GAL in enteric neurons of the porcine stomach prepyloric region following hydrochloric acid infusion

Introduction. Stomach hyperacidity leads to damage of the mucus/bicarbonate barrier, ulcerations and the development of stomach cancer. Key regulators of the mucosal barrier/luminal acid balance are neurotransmitters secreted by intramural neurons. The aim of the current study was to determine the expression of gastric neuropeptides and nNOS in the porcine stomach following hydrochloric acid instillation. We report on increased expression of enteric neurotransmitters involved in adaptive reaction to an experimentally-induced hyperacidity state. Material and methods. The investigation was conducted on eight 12–18 kg pigs. The influence of intragastric infusion of hydrochloric acid on the expression of cocaine- and amphetamine-regulated transcript peptide (CART), neuronal nitric oxide synthase (nNOS), vasoactive intestinal polypeptide (VIP), pituitary adenylate cyclase-activating peptide (PACAP), substance P (SP) and galanin (GAL) in the submucous and myenteric gastric neurons of the pig has been studied with double immunofluorescence. Results. A mimicked hyperacidity state significantly increased the proportion of enteric neurons immunoreactive to CART, nNOS, VIP, PACAP, SP and GAL in the submucous gastric neurons. In the myenteric plexus, a significant increase of the number of VIP-, CART- and GAL-immunoreactive (IR) neurons was found. Similarly, the percentage of myenteric nNOS-IR and PACAP-IR neurons tended to increase, while the fraction of SP-IR cells did not change. Conclusions. Stomach hyperacidity modifies the expression of the studied neurotransmitters in a specific way depending on the location of the neurons in particular plexuses of the stomach. Increased numbers of neurons expressing CART, nNOS, VIP, PACAP, SP and GAL clearly indicate their regulatory engagement in the restoration of the physiological gastric balance following hyperacidity

Why Do Female Faculty Members Still Earn Less? Gender Pay Gap in Higher Education in Science, Health and Engineering Fields

This study investigates faculty gender pay gap in higher education, while controlling for academic, demographic and family factors. By using data from NSF and drawing on various economic theories such as human capital theory, comparable worth theory, and structural theory, this study will build a framework for examining and comparing the differences in wages for full time-faculty in Science, Health, and Engineering Fields. Rank, discipline, hours worked per week, race/ethnicity, employer size, year of award of the highest degree and number of weeks worked per year and family were all variables used to consider the extent of a pay gap. Multiple hierarchical regression results showed that even when controlling for academic, demographic and family factors female faculty members earned 3.0% less than their male counterparts. These findings are consistent with other studies that prove that the gender pay gap still exists in academia

Variance asymptotics for random polytopes in smooth convex bodies

Let $K \subset \R^d$ be a smooth convex set and let \P_\la be a Poisson point process on $\R^d$ of intensity \la. The convex hull of \P_\la \cap K is a random convex polytope K_\la. As \la \to \infty, we show that the variance of the number of $k$-dimensional faces of K_\la, when properly scaled, converges to a scalar multiple of the affine surface area of $K$. Similar asymptotics hold for the variance of the number of $k$-dimensional faces for the convex hull of a binomial process in $K$