Spectral bounds for the Neumann-Poincaré operator on planar domains with corners

The boundary double layer potential, or the Neumann-Poincaré operator, is studied on the Sobolev space of order 1/2 along the boundary, coinciding with the space of charges giving rise to double layer potentials with finite energy in the whole space. Poincaré’s program of studying the spectrum of the boundary double layer potential is developed in complete generality on closed Lipschitz hypersurfaces in euclidean space. Furthermore, the Neumann-Poincaré operator is realized as a singular integral transform bearing similarities to the Beurling-Ahlfors transform in 2 dimensions. As an application, in the case of planar curves with corners, bounds for the spectrum of the Neumann-Poincaré operator are derived from recent results in quasi-conformal mapping theory

Las vitaminas y su utilización clínica

Data do prólogo: 193

On a modification of the Poisson integral operator

Given a quasisymmetric automorphism $\gamma$ of the unit circle $\mathbb{T}$ we define and study a modification $P_{\gamma}$ of the classical Poisson integral operator in the case of the unit disk $\mathbb{D}$. The modification is done by means of the generalized Fourier coefficients of $\gamma$. For a Lebesgue’s integrable complexvalued function $f$ on $\mathbb{T}$, $P_{\gamma}[f]$ is a complex-valued harmonic function in $\mathbb{D}$ and it coincides with the classical Poisson integral of $f$ provided $\gamma$ is the identity mapping on $\mathbb{T}$. Our considerations are motivated by the problem of spectral values and eigenvalues of a Jordan curve. As an application we establish a relationship between the operator $P_{\gamma}$, the maximal dilatation of a regular quasiconformal Teichmuller extension of $\gamma$ to $\mathbb{D}$ and the smallest positive eigenvalue of $\gamma$

Usual dietary anthocyanin intake, sources and their association with blood pressure in a representative sample of Australian adults

Background: Anthocyanins represent an important subgroup of non-nutritive components of food as evidence continues to build related to their beneficial bioactive effects. Using a recently developed Australian anthocyanin database, the present study aimed to estimate the intake of both total anthocyanins and their subclasses, identify food sources of anthocyanins, and determine associations between anthocyanin intake and measured blood pressure (BP). Methods: The present study comprised was a secondary analysis of the 2011-12 National Nutrition and Physical Activity component of the Australian Health Survey. Anthocyanin intake was estimated using an Australian anthocyanin database. Usual anthocyanin intake, as estimated from 24-h diet recall data, was computed using multiple source methods, whereas food sources were determined by calculating contribution of food groups to total anthocyanin intake. Regression analysis, adjusted for covariates (age, gender, body mass index, high BP) diagnosis, smoking status and physical activity) assessed the relationship between anthocyanin intake and BP in adults aged ≥50 years. Results: Mean anthocyanin intake was 24.17 ± 0.32 mg day −1 . Across age groups, berries were the top sources: blackberry (5-65%), cherry (2-24%), blueberry (2-13%) and raspberry (3-12%). There was a significant inverse association between anthocyanin intake and systolic BP (β = −0.04, F = 16.8, d.f. = 6, r2 = 0.05, P \u3c 0.01) and diastolic BP (β = 0.01, F = 5.35, d.f. = 6, R2 = 0.013, P \u3c 0.01), in models that adjusted for covariates. Conclusions: In comparison with the world composite database, anthocyanin intake in the Australian population was above average [mean (SD): 24.17 (0.32) mg day−1 versus 18.05 (21.14) mg day−1]. Berries comprised up the primary source of anthocyanins. Anthocyanin intake in older adults aged aged ≥50 years was inversely associated with BP