Optimization problems involving the first Dirichlet eigenvalue and the torsional rigidity

We present some open problems and obtain some partial results for spectral optimization problems involving measure, torsional rigidity and first Dirichlet eigenvalue.Comment: 18 pages, 4 figure

On the torsion function with Robin or Dirichlet boundary conditions

For $p\in (1,+\infty)$ and $b \in (0, +\infty]$ the $p$-torsion function with Robin boundary conditions associated to an arbitrary open set \Om \subset \R^m satisfies formally the equation $-\Delta_p =1$ in \Om and $|\nabla u|^{p-2} \frac{\partial u}{\partial n} + b|u|^{p-2} u =0$ on \partial \Om. We obtain bounds of the $L^\infty$ norm of $u$ {\it only} in terms of the bottom of the spectrum (of the Robin $p$-Laplacian), $b$ and the dimension of the space in the following two extremal cases: the linear framework (corresponding to $p=2$) and arbitrary $b>0$, and the non-linear framework (corresponding to arbitrary $p>1$) and Dirichlet boundary conditions ($b=+\infty$). In the general case, $p\not=2, p \in (1, +\infty)$ and $b>0$ our bounds involve also the Lebesgue measure of \Om.Comment: 19 page

Out of Equilibrium Solutions in the XYXY-Hamiltonian Mean Field model

Out of equilibrium magnetised solutions of the $XY$-Hamiltonian Mean Field ($XY$-HMF) model are build using an ensemble of integrable uncoupled pendula. Using these solutions we display an out-of equilibrium phase transition using a specific reduced set of the magnetised solutions

Large deviations for ideal quantum systems

We consider a general d-dimensional quantum system of non-interacting particles, with suitable statistics, in a very large (formally infinite) container. We prove that, in equilibrium, the fluctuations in the density of particles in a subdomain of the container are described by a large deviation function related to the pressure of the system. That is, untypical densities occur with a probability exponentially small in the volume of the subdomain, with the coefficient in the exponent given by the appropriate thermodynamic potential. Furthermore, small fluctuations satisfy the central limit theorem.Comment: 28 pages, LaTeX 2

Pollen spreads inquiry-based science education throughout Europe

This case study reviews the activity carried out by Pollen, a European network for promoting Inquiry-Based Science Education. The Pollen pilot programme involved working with communities to develop a hands-on approach to science education in primary schools. During the project, ICT played a crucial role in supporting teachers, classes and the project as a whole. The result was a promising model for bringing ICT and innovative pedagogy together in ways that enhance pupils’ experience at school. Pollen was a European research and development project supported by the European Commission Directorate-General for Research under the Science and Society part of the Sixth Framework Programme for research, technological development and demonstration activities. The project began in January 2006 and lasted three-and-a-half years. The research team was made up of a consortium of pedagogical and scientific organisations from 12 European countries and was launched in 12 European cities. Representing all areas of Europe (north, south, east and west), the network addressed their research questions in a diverse set of local educational contexts

Dose and number of applications that maximise fungicide effective life exemplified by Zymoseptoria tritici on wheat - a model analysis

Two key decisions that need to be taken about a fungicide treatment programme are (i) the number of applications that should be used per crop growing season, and (ii) the dosage that should be used in each application. There are two opposing considerations, with control efficacy improved by a higher number of applications and higher dose, and resistance management improved by a lower number of applications and lower dose. Resistance management aims to prolong the effective life of the fungicide, defined as the time between its introduction onto the market for use on the target pathogen, and the moment when effective control is lost due to a build-up of fungicide resistance. Thus, the question is whether there are optimal combinations of dose rate and number of applications that both provide effective control and lead to a longer effective life. In this paper, it is shown how a range of spray programmes can be compared and optimal programmes selected. This is explored with Zymoseptoria tritici on wheat and a quinone outside inhibitor (QoI) fungicide. For this pathogen-fungicide combination, a single treatment provided effective control under the simulated disease pressure, but only if the application timing was optimal and the dose was close to the maximum permitted. Programmes with three applications were generally not optimal as they exerted too much selection for resistance. Two-application fungicide programmes balanced effective control with reasonable flexibility of dose and application timing, and low resistance selection, leading to long effective lives of the fungicide