On the minimization of Dirichlet eigenvalues of the Laplace operator

We study the variational problem \inf \{\lambda_k(\Omega): \Omega\ \textup{open in}\ \R^m,\ |\Omega| < \infty, \ \h(\partial \Omega) \le 1 \}, where $\lambda_k(\Omega)$ is the $k$'th eigenvalue of the Dirichlet Laplacian acting in $L^2(\Omega)$, \h(\partial \Omega) is the $(m-1)$- dimensional Hausdorff measure of the boundary of $\Omega$, and $|\Omega|$ is the Lebesgue measure of $\Omega$. If $m=2$, and $k=2,3, \cdots$, then there exists a convex minimiser $\Omega_{2,k}$. If $m \ge 2$, and if $\Omega_{m,k}$ is a minimiser, then $\Omega_{m,k}^*:= \textup{int}(\overline{\Omega_{m,k}})$ is also a minimiser, and $\R^m\setminus \Omega_{m,k}^*$ is connected. Upper bounds are obtained for the number of components of $\Omega_{m,k}$. It is shown that if $m\ge 3$, and $k\le m+1$ then $\Omega_{m,k}$ has at most $4$ components. Furthermore $\Omega_{m,k}$ is connected in the following cases : (i) $m\ge 2, k=2,$ (ii) $m=3,4,5,$ and $k=3,4,$ (iii) $m=4,5,$ and $k=5,$ (iv) $m=5$ and $k=6$. Finally, upper bounds on the number of components are obtained for minimisers for other constraints such as the Lebesgue measure and the torsional rigidity.Comment: 16 page

Invisibility in billiards

The question of invisibility for bodies with mirror surface is studied in the framework of geometrical optics. We construct bodies that are invisible/have zero resistance in two mutually orthogonal directions, and prove that there do not exist bodies which are invisible/have zero resistance in all possible directions of incidence

On the boundary of the attainable set of the Dirichlet spectrum

Denoting by $\mathcal{E}\subseteq \R^2$ the set of the pairs $(\lambda_1(\Omega),\lambda_2(\Omega))$ for all the open sets $\Omega\subseteq\R^N$ with unit measure, and by $\Theta\subseteq\R^N$ the union of two disjoint balls of half measure, we give an elementary proof of the fact that \partial\E has horizontal tangent at its lowest point $(\lambda_1(\Theta),\lambda_2(\Theta))$.Comment: 7 pages, 3 figure

Apresentando alguns aspectos históricos do desenvolvimento da lógica clássica, ciência das idéias e dos processos da mente

Lógica é a ciência que tem por objeto determinar, entre as operações intelectuais orientadas para o conhecimento da verdade, as que são válidas e as que não são. Estuda os processos e as condições de verdade de todo e qualquer raciocínio. O conhecimento só é científico quando, além de universal, é metódico e sistemático, ou seja, lógico. Assim, a lógica se entende como método, ou caminho que as ciências trilham para determinar e conhecer seu objeto, e como característica geral do conhecimento científico

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co

Overview of the SMART-BEAR Technical Infrastructure

This paper describes a cloud-based platform that offers evidence-based, personalised interventions powered by Artificial Intelligence to help support efficient remote monitoring and clinician-driven guidance to people over 65 who suffer or are at risk of hearing loss, cardiovascular diseases, cognitive impairments, balance disorders, and mental health issues. This platform has been developed within the SMART-BEAR integrated project to power its large-scale clinical pilots and comprises a standards-based data harmonisation and management layer, a security component, a Big Data Analytics system, a Clinical Decision Support tool, and a dashboard component for efficient data collection across the pilot sites

Characterisation and categorisation of the diversity in viscoelastic vibrational properties between 98 wood types

International audienceContext Increased knowledge on diversity in wood properties would have implications both for fundamental research and for promoting a diversification of uses as material. *Aims The objective is to contribute to overcoming the critical lack of data on the diversity of wood dynamic mechanical/viscoelastic vibrational properties, by testing lesser-known species and categorizing sources of variability. *Methods Air-dry axial specific dynamic modulus of elasticity (E'/γ) and damping coefficient (tanδ) were measured on a wide sampling (1792 specimens) of 98 wood types from 79 species. An experimental device and protocol was designed for conducting systematic (i.e. rapid and reproducible) characterizations. *Results Diversity at the specimens' level corroborates the "standard" relationship between tanδ and E'/γ, which is discussed in terms of orientation of wood elements and of chemical composition. Diversity at the species level is expressed on the basis of results for normal heartwood, with specific gravity (γ) ranging from 0.2 to 1.3. Axial E'/γ ranges from 9 to 32 GPa and tanδ from 4×10-3 to 19×10-3. Properties distribution follows a continuum, but with group characteristics. The lowest values of tanδ are only found in certain tropical hardwoods. Results can also suggest alternative species for musical instruments making

Compact Polyelectrolyte Complexes: “Saloplastic” Candidates for Biomaterials

Precipitates of polyelectrolyte complexes were transformed into rugged shapes suitable for bioimplants by ultracentrifugation in the presence of high salt concentration. Salt ions dope the complex, creating a softer material with viscous fluid-like properties. Complexes that were compacted under the centrifugal field (CoPECs) were made from poly(diallyldimethyl ammonium), PDADMA, as polycation, and poly(styrene sulfonate), PSS, or poly(methacrylic acid), PMAA, as polyanion. Dynamic mechanical testing revealed a rubbery plateau at lower frequencies for PSS/PDADMA with moduli that decreased with increasing salt concentration, as internal ion pair cross-links were broken. CoPECs had significantly lower modulii compared to similar polyelectrolyte complexes prepared by the “multilayering ” method. The difference in mechanical properties was ascribed to higher water content (located in micropores) for the former and, more importantly, to their nonstoichiometric polymer composition. The modulus of PMAA/PDADMA CoPECs, under physiological conditions, demonstrated dynamic mechanical properties that were close to those of the nucleus pulposus in an intervertebral disk

Metatypical basal cell carcinoma: a clinical review

Background. Metatypical cell carcinoma can be considered as a new entity of skin cancer, being an intermediate typology between basal cell carcinomas and squamous cell carcinomas. The behaviour of the metatypical cell carcinoma lies between these two varieties of skin cancer. It is difficult to perform a differential diagnosis based on morphological and clinical features - therefore it is only possible by accurate histology. Methods. The authors have retrospectively analysed clinical records of 240 patients who were affected by metatypical skin cancer and who were treated by surgery, radiotherapy and chemotherapy. Results. MTC affected more males than females (62.5% vs 37.5%) than males. The most affected site was the cervicofacial area, 71.7%; then the trunk, 10%; the limbs, 9.6%; the scalp 3.7%; and other regions 5%. A recurrence occurred in 24 cases (10%), mainly in head and neck area. Conclusion. In this manuscript, the authors have emphasised the importance of conducting a differential diagnosis, and the importance of the specific treatment for metatypical skin cancer, even though more clinical studies and long-term follow-ups are required before establishing specific guidelines. © 2008 Tarallo et al; licensee BioMed Central Ltd