On the honeycomb conjecture for Robin Laplacian eigenvalues

We prove that the optimal cluster problem for the sum of the first Robin eigenvalue of the Laplacian, in the limit of a large number of convex cells, is asymptotically solved by (the Cheeger sets of) the honeycomb of regular hexagons. The same result is established for the Robin torsional rigidity

Extremal functions for the anisotropic Sobolev inequalities

The existence of multiple nonnegative solutions to the anisotropic critical problem - \sum_{i=1}^{N} \frac{\partial}{\partial x_i} (| \frac{\partial u}{\partial x_i} |^{p_i-2} \frac{\partial u}{\partial x_i}) = |u|^{p^*-2} u {in} \mathbb{R}^N is proved in suitable anisotropic Sobolev spaces. The solutions correspond to extremal functions of a certain best Sobolev constant. The main tool in our study is an adaptation of the well-known concentration-compactness lemma of P.-L. Lions to anisotropic operators. Futhermore, we show that the set of nontrival solutions \calS is included in $L^\infty(\R^N)$ and is located outside of a ball of radius $\tau >0$ in $L^{p^*}(\R^N)$

A novel collaboratively designed robot to assist carers

© Springer International Publishing Switzerland 2014. This paper presents a co-design process and an assisted navigation strategy that enables a novel assistive robot, Smart Hoist, to aid carers transferring non-ambulatory residents. Smart Hoist was codesigned with residents and carers at IRT Woonona residential care facility to ensure that the device can coexist in the facility, while providing assistance to carers with the primary aim of reducing lower back injuries, and improving the safety of carers and patients during transfers.The Smart Hoist is equipped with simple interfaces to capture user intention in order to provide assisted manoeuvring. Using the RGB-D sensor attached to the device, we propose a method of generating a repulsive force that can be combined with the motion controller’s output to allow for intuitive manoeuvring of the Smart Hoist, while negotiating with the environment.Extensive user trials were conducted on the premises of IRTWoonona residential care facility and feedback from end users confirm its intended purpose of intuitive behaviour, improved performance and ease of use

On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results

Given an open bounded subset $\Omega$ of $\mathbb{R}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u = 1$ in $\Omega$, subject to the homogeneous boundary condition $u = 0$ on $\partial \Omega$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C ^1(\Omega)$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $|\nabla u| = a$ on $\partial \Omega$; by using a suitable $P$-function we prove that, if $\Omega$ satisfies the same assumptions as above and in addition contains a ball with touches $\partial \Omega$ at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of $\Omega$ coincide. In turn, in dimension $n=2$, this entails that $\Omega$ must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class $C^2$.Comment: 26 pages, 1 figur

Neck-cooling improves repeated sprint performance in the heat

The present study evaluated the effect of neck-cooling during exercise on repeated sprint ability in a hot environment. Seven team-sport playing males completed two experimental trials involving repeated sprint exercise (5 × 6 s) before and after two 45 min bouts of a football specific intermittent treadmill protocol in the heat (33.0 ± 0.2°C; 53 ± 2% relative humidity). Participants wore a neck-cooling collar in one of the trials (CC). Mean power output and peak power output declined over time in both trials but were higher in CC (540 ± 99 v 507 ± 122 W, d = 0.32; 719 ± 158 v 680 ± 182 W, d = 0.24 respectively). The improved power output was particularly pronounced (d = 0.51–0.88) after the 2nd 45 min bout but the CC had no effect on % fatigue. The collar lowered neck temperature and the thermal sensation of the neck (P 0.05). There were no trial differences but interaction effects were demonstrated for prolactin concentration and rating of perceived exertion (RPE). Prolactin concentration was initially higher in the collar cold trial and then was lower from 45 min onwards (interaction trial × time P = 0.04). RPE was lower during the football intermittent treadmill protocol in the collar cold trial (interaction trial × time P = 0.01). Neck-cooling during exercise improves repeated sprint performance in a hot environment without altering physiological or neuroendocrinological responses. RPE is reduced and may partially explain the performance improvement

The area measure of log-concave functions and related inequalities

On the class of log-concave functions on $\R^n$, endowed with a suitable algebraic structure, we study the first variation of the total mass functional, which corresponds to the volume of convex bodies when restricted to the subclass of characteristic functions. We prove some integral representation formulae for such first variation, which lead to define in a natural way the notion of area measure for a log-concave function. In the same framework, we obtain a functional counterpart of Minkowski first inequality for convex bodies; as corollaries, we derive a functional form of the isoperimetric inequality, and a family of logarithmic-type Sobolev inequalities with respect to log-concave probability measures. Finally, we propose a suitable functional version of the classical Minkowski problem for convex bodies, and prove some partial results towards its solution.Comment: 36 page

An Evidence-Based Comparison of Operational Criteria for the Presence of Sarcopenia

Background. Several consensus groups have previously published operational criteria for sarcopenia, incorporating lean mass with strength and/or physical performance. The purpose of this manuscript is to describe the prevalence, agreement, and discrepancies between the Foundation for the National Institutes of Health (FNIH) criteria with other operational definitions for sarcopenia. Methods. The FNIH Sarcopenia Project used data from nine studies including: Age, Gene and Environment Susceptibility-Reykjavik Study; Boston Puerto Rican Health Study; a series of six clinical trials from the University of Connecticut; Framingham Heart Study; Health, Aging, and Body Composition Study; Invecchiare in Chianti; Osteoporotic Fractures in Men Study; Rancho Bernardo Study; and Study of Osteoporotic Fractures. Participants included in these analyses were aged 65 and older and had measures of body mass index, appendicular lean mass, grip strength, and gait speed. Results. The prevalence of sarcopenia and agreement proportions was higher in women than men. The lowest prevalence was observed with the FNIH criteria (1.3% men and 2.3% women) compared with the International Working Group and the European Working Group for Sarcopenia in Older Persons (5.1% and 5.3% in men and 11.8% and 13.3% in women, respectively). The positive percent agreements between the FNIH criteria and other criteria were low, ranging from 7% to 32% in men and 5% to 19% in women. However, the negative percent agreement were high (all >95%). Conclusions. The FNIH criteria result in a more conservative operational definition of sarcopenia, and the prevalence was lower compared with other proposed criteria. Agreement for diagnosing sarcopenia was low, but agreement for ruling out sarcopenia was very high. Consensus on the operational criteria for the diagnosis of sarcopenia is much needed to characterize populations for study and to identify adults for treatment

Cutpoints for Low Appendicular Lean Mass That Identify Older Adults With Clinically Significant Weakness

Background. Low lean mass is potentially clinically important in older persons, but criteria have not been empirically validated. As part of the FNIH (Foundation for the National Institutes of Health) Sarcopenia Project, this analysis sought to identify cutpoints in lean mass by dual-energy x-ray absorptiometry that discriminate the presence or absence of weakness (defined in a previous report in the series as grip strength <26kg in men and <16kg in women). Methods. In pooled cross-sectional data stratified by sex (7,582 men and 3,688 women), classification and regression tree (CART) analysis was used to derive cutpoints for appendicular lean body mass (ALM) that best discriminated the presence or absence of weakness. Mixed-effects logistic regression was used to quantify the strength of the association between lean mass category and weakness. Results. In primary analyses, CART models identified cutpoints for low lean mass (ALM <19.75kg in men and <15.02kg in women). Sensitivity analyses using ALM divided by body mass index (BMI: ALMBMI) identified a secondary definition (ALMBMI <0.789 in men and ALMBMI <0.512 in women). As expected, after accounting for study and age, low lean mass (compared with higher lean mass) was associated with weakness by both the primary (men, odds ratio [OR]: 6.9 [95% CI: 5.4, 8.9]; women, OR: 3.6 [95% CI: 2.9, 4.3]) and secondary definitions (men, OR: 4.3 [95% CI: 3.4, 5.5]; women, OR: 2.2 [95% CI: 1.8, 2.8]). Conclusions. ALM cutpoints derived from a large, diverse sample of older adults identified lean mass thresholds below which older adults had a higher likelihood of weakness