Proof of the Log-Convex Density Conjecture

We completely characterize isoperimetric regions in R^n with density e^h, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.Comment: 40 pages, 7 figure

Calibrated Weighting for Small Area Estimation

Calibrated weighting methods for estimation of survey population characteristics are widely used. At the same time, model-based prediction methods for estimation of small area or domain characteristics are becoming increasingly popular. This paper explores weighting methods based on the mixed models that underpin small area estimates to see whether they can deliver equivalent small area estimation performance when compared with standard prediction methods and superior population level estimation performance when compared with standard calibrated weighting methods. A simple MSE estimator for weighted small area estimation is also developed

Parental problems, case plan requirements, and service targeting in child welfare reunification

Only about half of parents attempting to reunify with their children in foster care succeed in their efforts. Parents are ordered by the court to use treatment services in order to resolve their problems. These treatment services thus play a critical role in reunification, and in fact the use of services appropriately matched to parents\u27 problems has been found to be associated with a greater likelihood of reunification. However, there is little in the literature regarding the specific requirements of reunification case plans, and whether they are accurately targeted at reunifying parents\u27 problems. This mostly descriptive study uses case file data to examine the relationship between parental problems and case plan requirements for a sample of parents reunifying with their children in one large urban California county. Findings show that most reunifying parents had multiple problems, and were required to attend approximately 8 service events per week. There was a positive correlation between the total number of concerns (treatment problems and life challenges) and required weekly service events. While 85% of parents were ordered treatment services for all their identified problems, over 30% were ordered services targeting problems they were not known to have. Overall, 58% of parents were ordered both all appropriate and only appropriate services. Implications for policy and practice are discussed, including the need for models of service delivery that limit the burden of accessing multiple service locations for reunifying parents

Constructing monotone homotopies and sweepouts

This article investigates when homotopies can be converted to monotone homotopies without increasing the lengths of curves. A monotone homotopy is one which consists of curves which are simple or constant, and in which curves are pairwise disjoint. We show that, if the boundary of a Riemannian disc can be contracted through curves of length less than $L$, then it can also be contracted monotonously through curves of length less than $L$. This proves a conjecture of Chambers and Rotman. Additionally, any sweepout of a Riemannian $2$-sphere through curves of length less than $L$ can be replaced with a monotone sweepout through curves of length less than $L$. Applications of these results are also discussed.Comment: 16 pages, 6 figure

Fluid or fuel? The context of consuming a beverage is important for satiety

Energy-containing beverages have a weak effect on satiety, limited by their fluid characteristics and perhaps because they are not considered ‘food’. This study investigated whether the context of consuming a beverage can influence the satiating power of its nutrients. Eighty participants consumed a lower- (LE, 75 kcal) and higher-energy (HE, 272 kcal) version of a beverage (covertly manipulated within-groups) on two test days, in one of four beverage contexts (between-groups): thin versions of the test-drinks were consumed as a thirst-quenching drink (n = 20), a filling snack (n = 20), or without additional information (n = 20). A fourth group consumed subtly thicker versions of the beverages without additional information (n = 20). Lunch intake 60 minutes later depended on the beverage context and energy content (p = 0.030): participants who consumed the thin beverages without additional information ate a similar amount of lunch after the LE and HE versions (LE = 475 kcal, HE = 464 kcal; p = 0.690) as did those participants who believed the beverages were designed to quench-thirst (LE = 442 kcal, HE = 402 kcal; p = 0.213), despite consuming an additional 197 kcal in the HE beverage. Consuming the beverage as a filling snack led participants to consume less at lunch after the HE beverage compared to the LE version (LE = 506 kcal, HE = 437 kcal; p = 0.025). This effect was also seen when the beverages were subtly thicker, with participants in this group displaying the largest response to the beverage’s energy content, consuming less at lunch after the HE version (LE = 552 kcal, HE = 415 kcal; p<0.001). These data indicate that beliefs about the consequences of consuming a beverage can affect the impact of its nutrients on appetite regulation and provide further evidence that a beverage’s sensory characteristics can limit its satiating power

Optimising foods for satiety

Foods that generate strong satiety sensations have obvious ben- efits for weight management. This review builds on the under- standing that a food’s satiating power is dependent on the amount of protein, carbohydrate, fat and fibre it contains by examining evidence that the consumer’s sensory and cognitive appraisal of the food is also important. It is concluded that numerous features of a food product can be manipulated to enhance the consumer’s experience of satiety but the combi- nation of these features will ultimately determine its effect on appetite control. Taking this integrated approach to satiety will optimise the development of high satiety foods