On the placement of an obstacle so as to optimize the Dirichlet heat trace

We prove that among all doubly connected domains of $\R^n$ bounded by two spheres of given radii, $Z(t)$, the trace of the heat kernel with Dirichlet boundary conditions, achieves its minimum when the spheres are concentric (i.e., for the spherical shell). The supremum is attained when the interior sphere is in contact with the outer sphere.This is shown to be a special case of a more general theorem characterizing the optimal placement of a spherical obstacle inside a convex domain so as to maximize or minimize the trace of the Dirichlet heat kernel. In this case the minimizing position of the center of the obstacle belongs to the "heart" of the domain, while the maximizing situation occurs either in the interior of the heart or at a point where the obstacle is in contact with the outer boundary. Similar statements hold for the optimal positions of the obstaclefor any spectral property that can be obtained as a positivity-preserving or positivity-reversing transform of $Z(t)$,including the spectral zeta function and, through it, the regularized determinant.Comment: in SIAM Journal on Mathematical Analysis, Society for Industrial and Applied Mathematics, 201

Universal inequalities for the eigenvalues of Schrodinger operators on submanifolds

We establish inequalities for the eigenvalues of Schr\"odinger operators on compact submanifolds (possibly with nonempty boundary) of Euclidean spaces, of spheres, and of real, complex and quaternionic projective spaces, which are related to inequalities for the Laplacian on Euclidean domains due to Payne, P\'olya, and Weinberger and to Yang, but which depend in an explicit way on the mean curvature. In later sections, we prove similar results for Schr\"odinger operators on homogeneous Riemannian spaces and, more generally, on any Riemannian manifold that admits an eigenmap into a sphere, as well as for the Kohn Laplacian on subdomains of the Heisenberg group. Among the consequences of this analysis are an extension of Reilly's inequality, bounding any eigenvalue of the Laplacian in terms of the mean curvature, and spectral criteria for the immersibility of manifolds in homogeneous spaces.Comment: A paraitre dans Transactions of the AM

On sums of eigenvalues of elliptic operators on manifolds

We use the averaged variational principle introduced in a recent article on graph spectra [7] to obtain upper bounds for sums of eigenvalues of several partial differential operators of interest in geometric analysis, which are analogues of Kr{\"o}ger 's bound for Neumann spectra of Laplacians on Euclidean domains [12]. Among the operators we consider are the Laplace-Beltrami operator on compact subdomains of manifolds. These estimates become more explicit and asymptotically sharp when the manifold is conformal to homogeneous spaces (here extending a result of Strichartz [21] with a simplified proof). In addition we obtain results for the Witten Laplacian on the same sorts of domains and for Schr{\"o}dinger operators with confining potentials on infinite Euclidean domains. Our bounds have the sharp asymptotic form expected from the Weyl law or classical phase-space analysis. Similarly sharp bounds for the trace of the heat kernel follow as corollaries.Comment: in Journal of Spectral Theory, 201

An investigation into the pathogenesis of vulvo-vaginal candidosis

OBJECTIVE: To monitor yeasts isolated from women during and between episodes of recurrent vulvo-vaginal candidosis (VVC) to determine whether vaginal relapse or re-infection occurred. METHODS:Women presenting at the genitourinary medicine clinic with signs and symptoms of VVC were recruited to the study (n = 121). A vaginal washing, high vaginal swab (HVS) and rectal swab were taken and the women treated with a single 500 mg clotrimazole pessary.Women were asked to re-attend after 1, 4, and 12 weeks, or when the VVC recurred, when vaginal washings and HVS were repeated. Candida isolates recovered were strain typed using the Ca3 probe and their similarity assessed. Antifungal susceptibility to fluconazole and clotrimazole were determined. RESULTS: Of the women recruited, 47 completed the study, either returning for four visits or suffering a recurrence during the study period. Of the 22 women who experienced recurrence, the same strain was responsible for the initial and recurrent episode in 17 women. For the remaining five women, four had strain replacement and one had a change of species. None of the isolates recovered from the women demonstrated resistance to either clotrimazole or fluconazole. CONCLUSIONS: Our findings support the theory of vaginal relapse and thus may support the use of more prolonged courses of antifungal therapy initially to increase the chances of eradication of the yeast

Editorial:Metformin: Beyond Diabetes

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Interparticle interaction and structure of deposits for competitive model in (2+1)- dimensions

A competitive (2+1)-dimensional model of deposit formation, based on the combination of random sequential absorption deposition (RSAD), ballistic deposition (BD) and random deposition (RD) models, is proposed. This model was named as RSAD$_{1-s}$(RD$_f$BD$_{1-f}$)$_s$. It allows to consider different cases of interparticle interactions from complete repulsion between near-neighbors in the RSAD model ($s=0$) to sticking interactions in the BD model ($s=1, f=0$) or absence of interactions in the RD model ($s=1$, $f=0$). The ideal checkerboard ordered structure was observed for the pure RSAD model ($s=0$) in the limit of $h \to \infty$. Defects in the ordered structure were observed at small $h$. The density of deposit $p$ versus system size $L$ dependencies were investigated and the scaling parameters and values of $p_\infty=p(L=\infty)$ were determined. Dependencies of $p$ versus parameters of the competitive model $s$ and $f$ were studied. We observed the anomalous behaviour of the eposit density $p_\infty$ with change of the inter-particle repulsion, which goes through minimum on change of the parameter $s$. For pure RSAD model, the concentration of defects decreases with $h$ increase in accordance with the critical law $\rho\propto h^{-\chi_{RSAD}}$, where $\chi_{RSAD} \approx 0.119 \pm 0.04$.Comment: 10 pages,4 figures, Latex, uses iopart.cl

Impact of school lunch type on nutritional quality of English children's diets.

OBJECTIVE: Nutrient and food standards exist for school lunches in English primary schools although packed lunches brought from home are not regulated. The aim of the present study was to determine nutritional and dietary differences by lunch type. DESIGN: A cross-sectional survey was carried out in 2007 assessing diet using the Child and Diet Evaluation Tool (CADET), a validated 24 h estimated food diary. The data were analysed to determine nutritional and dietary intakes over the whole day by school meal type: school meals and packed lunches. SETTING: Fifty-four primary schools across England. SUBJECTS: Children (n 2709) aged 6-8 years. RESULTS: Children having a packed lunch consumed on average 11·0 g more total sugars (95 % CI 6·6, 15·3 g) and 101 mg more Na (95 % CI 29, 173 mg) over the whole day. Conversely, children having a school meal consumed, on average, 4·0 g more protein (95 % CI 2·3, 5·7 g), 0·9 g more fibre (NSP; 95 % CI 0·5, 1·3 g) and 0·4 mg more Zn (95 % CI 0·1, 0·6 mg). There was no difference in daily energy intake by lunch type. Children having a packed lunch were more likely to consume snacks and sweetened drinks; while children having a school meal were more likely to consume different types of vegetables and drink water over the whole day. CONCLUSIONS: Compared with children having a school meal, children taking a packed lunch to school consumed a lower-quality diet over the whole day, including higher levels of sugar and Na and fewer vegetables. These findings support the introduction of policies that increase school meal uptake