Multiple radial positive solutions of semilinear elliptic problems with Neumann boundary conditions

Assuming $B_{R}$ is a ball in $\mathbb R^{N}$, we analyze the positive solutions of the problem $$\begin{cases} -\Delta u+u= |u|^{p-2}u, &\text{ in } B_{R},\newline \partial_{\nu}u=0,&\text{ on } \partial B_{R}, \end{cases}$$ that branch out from the constant solution $u=1$ as $p$ grows from $2$ to $+\infty$. The non-zero constant positive solution is the unique positive solution for $p$ close to $2$. We show that there exist arbitrarily many positive solutions as $p\to\infty$ (in particular, for supercritical exponents) or as $R \to \infty$ for any fixed value of $p>2$, answering partially a conjecture in [Bonheure-Noris-Weth]. We give the explicit lower bounds for $p$ and $R$ so that a given number of solutions exist. The geometrical properties of those solutions are studied and illustrated numerically. Our simulations motivate additional conjectures. The structure of the least energy solutions (among all or only among radial solutions) and other related problems are also discussed.Comment: 37 pages, 24 figure

Characteristic Functions Describing the Power Absorption Response of Periodic Structures to Partially Coherent Fields

Many new types of sensing or imaging surfaces are based on periodic thin films. It is explained how the response of those surfaces to partially coherent fields can be fully characterized by a set of functions in the wavenumber spectrum domain. The theory is developed here for the case of 2D absorbers with TE illumination and arbitrary material properties in the plane of the problem, except for the resistivity which is assumed isotropic. Sum and difference coordinates in both spatial and spectral domains are conveniently used to represent the characteristic functions, which are specialized here to the case of periodic structures. Those functions can be either computed or obtained experimentally. Simulations rely on solvers based on periodic-boundary conditions, while experiments correspond to Energy Absorption Interferometry (EAI), already described in the literature. We derive rules for the convergence of the representation versus the number of characteristic functions used, as well as for the sampling to be considered in EAI experiments. Numerical examples are given for the case of absorbing strips printed on a semi-infinite substrate.Comment: Submitted to JOSA

Characterization of Power Absorption Response of Periodic 3D Structures to Partially Coherent Fields

In many applications of absorbing structures it is important to understand their spatial response to incident fields, for example in thermal solar panels, bolometric imaging and controlling radiative heat transfer. In practice, the illuminating field often originates from thermal sources and is only spatially partially coherent when reaching the absorbing device. In this paper, we present a method to fully characterize the way a structure can absorb such partially coherent fields. The method is presented for any 3D material and accounts for the partial coherence and partial polarization of the incident light. This characterization can be achieved numerically using simulation results or experimentally using the Energy Absorption Interferometry (EAI) that has been described previously in the literature. The absorbing structure is characterized through a set of absorbing functions, onto which any partially coherent field can be projected. This set is compact for any structure of finite extent and the absorbing function discrete for periodic structures

Atmospheric Stellar Parameters from Cross-Correlation Functions

The increasing number of spectra gathered by spectroscopic sky surveys and transiting exoplanet follow-up has pushed the community to develop automated tools for atmospheric stellar parameters determination. Here we present a novel approach that allows the measurement of temperature ($T_{\rm eff}$), metallicity ($[{\rm Fe}/{\rm H}]$) and gravity ($\log g$) within a few seconds and in a completely automated fashion. Rather than performing comparisons with spectral libraries, our technique is based on the determination of several cross-correlation functions (CCFs) obtained by including spectral features with different sensitivity to the photospheric parameters. We use literature stellar parameters of high signal-to-noise ($\textrm{SNR}$), high-resolution HARPS spectra of FGK Main Sequence stars to calibrate $T_{\rm eff}$, $[{\rm Fe}/{\rm H}]$ and $\log g$ as a function of CCFs parameters. Our technique is validated using low $\textrm{SNR}$ spectra obtained with the same instrument. For FGK stars we achieve a precision of $\sigma_{T_{\rm eff}} = 50$ K, $\sigma_{\log g} = 0.09~ \textrm{dex}$ and $\sigma_{\textrm{Fe}/\textrm{H}]} =0.035~ \textrm{dex}$ at $\textrm{SNR}=50$, while the precision for observation with $\textrm{SNR} \gtrsim 100$ and the overall accuracy are constrained by the literature values used to calibrate the CCFs. Our approach can be easily extended to other instruments with similar spectral range and resolution, or to other spectral range and stars other than FGK dwarfs if a large sample of reference stars is available for the calibration. Additionally, we provide the mathematical formulation to convert synthetic equivalent widths to CCF parameters as an alternative to direct calibration. We have made our tool publicly available.Comment: Accepted by MNRAS. 12 pages, 12 figures. The code to retrieve the atmospheric stellar parameters from HARPS and HARPS-N spectra is available "at this url, https://github.com/LucaMalavolta/CCFpams

Réutilisation d'un corpus pour une nouvelle analyse des réseaux sociaux grâce à l'adaptation de l'outil KSV

In the field of interaction analysis in computer supported collaborative learning, this paper revisits a previous analysis by applying a new tool on a corpus now available. The goal is to analyze cliques (Social Network Analysis) by traveling along the dimensions of intensity and time, thanks to an adapted version of KSV: Knowledge Space Visualizer. The theory invoked is SNA. This new exploration alows us to suggest new forms of indicators for cohesion and espescially to compare cohesion of different groups. This work also made possible some improvements for KSV that gives it new perspectives.Dans le champ de l'analyse des interactions en situation d'apprentissage collaboratif en ligne, nous revisitons un corpus connu et accessible pour en reprendre l'analyse avec un outil qui offre de nouvelles dimensions d'exploration : l'intensité et le temps. Le cadre d'analyse est ici celui des réseaux sociaux. Cette nouvelle exploration des données nous permet de proposer de nouveaux indicateurs pour comparer la cohésion des groupes au regard de l'intensité de communication. Cette rencontre entre un corpus et un outil a aussi permis d'étendre les capacités de traitement et de visualisation de l'outil KSV, qui lui offrent de nouvelles perspectives

Destabilization of a flow focused suspension of magnetotactic bacteria

Active matter is a new class of material, intrinsically out-of equilibrium with intriguing properties. So far, the recent upsurge of studies has mostly focused on the spontaneous behavior of these systems --in the absence of external constraints or driving--. Yet, many real life systems evolve under constraints, being both submitted to flow and various taxis. In the present work, we demonstrate a new experimental system which opens up the way for quantitative investigations, and discriminating examinations, of the challenging theoretical description of such systems. We explore the behavior of magnetotactic bacteria as a particularly rich and versatile class of driven matter, which behavior can be studied under contrasting and contradicting stimuli. In particular we demonstrate that the competing driving of an orienting magnetic field and hydrodynamic flow lead not only to jetting, but also unveils a new pearling instability. This illustrates new structuring capabilities of driven active matter