Courant-sharp Robin eigenvalues for the square and other planar domains

This paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, Bérard-Helffer, Helffer-Persson Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter $h$, we focus on the case when $h$ is large. We hope to come back to the analysis when $h$ is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with $C^{2,\alpha}$ boundary ($\alpha >0$) as $h$ large varies

Courant-sharp Robin eigenvalues for the square: the case with small Robin parameter

This article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter $h$). For the square, our first paper focused on the case where $h$ is large and extended results that were obtained by Pleijel, B\'erard--Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of $h$, where $h$ is positive and not close to $0$. In this second paper, we extend results that were obtained by Helffer--Persson-Sundqvist for the Neumann problem to the case where $h>0$ is small

Courant-sharp Robin eigenvalues for the square -the case with small Robin parameter

International audienceThis article is the continuation of our first work on the determination of the cases where there is equality in Courant's Nodal Domain theorem in the case of a Robin boundary condition (with Robin parameter h). For the square, our first paper focused on the case where h is large and extended results that were obtained by Pleijel, Bérard-Helffer, for the problem with a Dirichlet boundary condition. There, we also obtained some general results about the behaviour of the nodal structure (for planar domains) under a small deformation of h, where h is positive and not close to 0. In this second paper, we extend results that were obtained by Helffer-Persson-Sundqvist for the Neumann problem to the case where h > 0 is small. MSC classification (2010): 35P99, 58J50, 58J37

Courant-sharp Robin eigenvalues for the square and other planar domains

International audienceThis paper is devoted to the determination of the cases where there is equality in Courant's nodal domain theorem in the case of a Robin boundary condition. For the square, we partially extend the results that were obtained by Pleijel, Bérard-Helffer, Helffer-Persson-Sundqvist for the Dirichlet and Neumann problems. After proving some general results that hold for any value of the Robin parameter h, we focus on the case when h is large. We hope to come back to the analysis when h is small in a second paper. We also obtain some semi-stability results for the number of nodal domains of a Robin eigenfunction of a domain with C ^(2,α) boundary (α > 0) as h large varies. MSC classification (2010): 35P99, 58J50, 58J37

Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: We prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is Steklov isospectral to the disk

Some spectral applications of McMullen's Hausdorff dimension algorithm

Using McMullen's Hausdorff dimension algorithm, we study numerically the dimension of the limit set of groups generated by reflections along three geodesics on the hyperbolic plane. Varying these geodesics, we found four minima in the two-dimensional parameter space, leading to a rigorous result why this must be so. Extending the algorithm to compute the limit measure and its moments, we study orthogonal polynomials on the unit circle associated with this measure. Several numerical observations on certain coefficients related to these moments and on the zeros of the polynomials are discussed. - See more at: http://www.ams.org/journals/ecgd/2012-16-10/S1088-4173-2012-00244-5/home.html#sthash.MXrRFUVZ.dpu

Membranous nephropathy in the UK Biobank

Background Despite MN being one of the most common causes of nephrotic syndrome worldwide, its biological and environmental determinants are poorly understood in large-part due to it being a rare disease. Making use of the UK Biobank, a unique resource holding a clinical dataset and stored DNA, serum and urine for ~500,000 participants, this study aims to address this gap in understanding. Methods The primary outcome was putative MN as defined by ICD-10 codes occurring in the UK Biobank. Univariate relative risk regression modelling was used to assess the associations between the incidence of MN and related phenotypes with sociodemographic, environmental exposures, and previously described increased-risk SNPs. Results 502,507 patients were included in the study of whom 100 were found to have a putative diagnosis of MN; 36 at baseline and 64 during the follow-up. Prevalence at baseline and last follow-up were 72 and 199 cases/million respectively. At baseline, as expected, the majority of those previously diagnosed with MN had proteinuria, and there was already evidence of proteinuria in patients diagnosed within the first 5 years of follow-up. The highest incidence rate for MN in patients was seen in those homozygous for the high-risk alleles (9.9/100,000 person-years). Conclusion It is feasible to putatively identify patients with MN in the UK Biobank and cases are still accumulating. This study shows the chronicity of disease with proteinuria present years before diagnosis. Genetics plays an important role in disease pathogenesis, with the at-risk group providing a potential population for recall

Polyelectrolyte Multilayering on a Charged Planar Surface

The adsorption of highly \textit{oppositely} charged flexible polyelectrolytes (PEs) on a charged planar substrate is investigated by means of Monte Carlo (MC) simulations. We study in detail the equilibrium structure of the first few PE layers. The influence of the chain length and of a (extra) non-electrostatic short range attraction between the polycations and the negatively charged substrate is considered. We show that the stability as well as the microstructure of the PE layers are especially sensitive to the strength of this latter interaction. Qualitative agreement is reached with some recent experiments.Comment: 28 pages; 11 (main) Figs - Revtex4 - Higher resolution Figs can be obtained upon request. To appear in Macromolecule

The promoter from SlREO, a highly-expressed, root-specific Solanum lycopersicum gene, directs expression to cortex of mature roots

Root-specific promoters are valuable tools for targeting transgene expression, but many of those already described have limitations to their general applicability. We present the expression characteristics of SlREO, a novel gene isolated from tomato (Solanum lycopersicum L.). This gene was highly expressed in roots but had a very low level of expression in aerial plant organs. A 2.4-kb region representing the SlREO promoter sequence was cloned upstream of the uidA GUS reporter gene and shown to direct expression in the root cortex. In mature, glasshouse-grown plants this strict root specificity was maintained. Furthermore, promoter activity was unaffected by dehydration or wounding stress but was somewhat suppressed by exposure to NaCl, salicylic acid and jasmonic acid. The predicted protein sequence of SlREO contains a domain found in enzymes of the 2-oxoglutarate and Fe(II)-dependent dioxygenase superfamily. The novel SlREO promoter has properties ideal for applications requiring strong and specific gene expression in the bulk of tomato root tissue growing in soil, and is also likely to be useful in other Solanaceous crop