Convexity and gradient estimates for fully nonlinear curvature flows

We study deformations of hypersurfaces with normal velocity given by a smooth symmetric increasing function of the principal curvatures. Specifically we study flows where the speed is a nonlinear concave function, so that at the coordinate level the evolution is governed by a fully nonlinear parabolic PDE. For each $k \geq 3$ we construct the first flows of this kind which smoothly deform any compact $k$-convex hypersurface of Euclidean space through a family of hypersurfaces which are also $k$-convex, before forming finite-time singularities which are necessarily convex (by $k$-convexity we mean that the sum of the smallest $k$ principal curvatures is everywhere positive). That is, we show that $k$-convexity is preserved and establish an analogue of the Huisken-Sinestrari convexity estimate, which implies convexity of singularities for mean-convex mean curvature flow. In contrast to the mean curvature flow, the fully nonlinear flows constructed here also preserve $k$-convexity in a Riemannian background, and we show that the convexity estimate carries over to this setting as long as the ambient curvature is suitably pinched. We then employ our convexity estimate to prove Harnack and derivative estimates for the second fundamental form of solutions which are embedded. These results imply for example that sequences of rescalings about a singularity satisfy universal bounds for the second fundamental form and all of its higher derivatives on compact subsets of spacetime. The estimates are obtained by generalising an induction on scales technique introduced by Brendle-Huisken for two-convex flows to the $k$-convex setting. Our arguments apply to a general class of flows including mean-convex mean curvature flow, and in this case we recover the influential global Harnack inequality of Haslhofer-Kleiner, but without using Huisken's monotonicity formula

Programming in the Mathematics Curriculum at Manchester Metropolitan University

An increasing number of schools are teaching programming to their pupils and there is also an increase in programming in Higher Education with recent reports recommending this approach. At Manchester Metropolitan University (MMU) we wanted to attract and retain mathematics students and prepare them for careers upon graduation. By integrating Mathematics/Statistics/Operational Research packages across the curriculum and by solving real world problems we have managed to make the course highly desirable and loved by our students. In this case study, we show how it is possible to integrate programming and mathematical/computational modelling across the curriculum

Uniqueness of convex ancient solutions to hypersurface flows

We show that every convex ancient solution of mean curvature flow with Type I curvature growth is either spherical, cylindrical, or planar. We then prove the corresponding statement for flows by a natural class of curvature functions which are convex or concave in the second fundamental form. Neither of these results assumes interior noncollapsing

Variation in Insurance Coverage Across Congressional Districts: New Estimates From 2008

Examines trends in rates of private health insurance coverage, public coverage, and uninsurance by congressional district and poverty rate. Identifies districts that have the most to gain from health reforms designed to increase coverage

Patient participation, encounter, and methadone-reinforcement in the treatment of heroin addicts

Tho present thesis represents a summary or research done by the author (and others) that was conducted with heroin addicts and drug abusers undergoing behavioral and pharmacological therapy at Stockton State Hospital, Stockton, California. From June 1970 to December 1970 the Research Department of Stockton State Hospital, in conjunction with the Drug Abuse Program at Stockton State Hospital, conducted research investigating a number of difference facets relating to inpatient programs for heroin addicts undergoing methadone maintenance and drug abusers. These facets included the investigation and evaluation of (a) motivational factors; affecting the voluntary participation of inpatient heroin addicts and drug abusers in behavioral and pharmacological therapy, (b) the effectiveness of the synthetic narcotic methadone hydrocloride as a primary reinforcing technique for appropriate behavior, (c) the effectiveness of various therapeutic approaches used in conjunction with behavioral modification techniques, and (d) the effect of methadone on perceptual and motor functioning in the heroin addict under-going methadone maintenance. The present thesis is a compilation cf these research projects