Fractional div-curl quantities and applications to nonlocal geometric equations

We investigate a fractional notion of gradient and divergence operator. We generalize the div-curl estimate by Coifman-Lions-Meyer-Semmes to fractional div-curl quantities, obtaining, in particular, a nonlocal version of Wente's lemma. We demonstrate how these quantities appear naturally in nonlocal geometric equations, which can be used to obtain a theory for fractional harmonic maps analogous to the local theory. Firstly, regarding fractional harmonic maps into spheres, we obtain a conservation law analogous to Shatah's conservation law and give a new regularity proof analogous to H\'elein's for harmonic maps into spheres. Secondly, we prove regularity for solutions to critical systems with nonlocal antisymmetric potentials on the right-hand side. Since the half-harmonic map equation into general target manifolds has this form, as a corollary, we obtain a new proof of the regularity of half-harmonic maps into general target manifolds following closely Rivi\`{e}re's celebrated argument in the local case. Lastly, the fractional div-curl quantities provide also a new, simpler, proof for H\"older continuity of $W^{s,n/s}$-harmonic maps into spheres and we extend this to an argument for $W^{s,n/s}$-harmonic maps into homogeneous targets. This is an analogue of Strzelecki's and Toro-Wang's proof for $n$-harmonic maps into spheres and homogeneous target manifolds, respectively

Electronic Dangerous Citizens

Helen Tartar, Editorial Director at Fordham University Press, discusses how the electronic Dangerous Citizens project came to be, and what the inter-institutional partnership enables for the book's author, anthropologist Neni Panourgia

A variational approach to the local character of G-closure: the convex case

This article is devoted to characterize all possible effective behaviors of composite materials by means of periodic homogenization. This is known as a $G$-closure problem. Under convexity and $p$-growth conditions ($p>1$), it is proved that all such possible effective energy densities obtained by a $\Gamma$-convergence analysis, can be locally recovered by the pointwise limit of a sequence of periodic homogenized energy densities with prescribed volume fractions. A weaker locality result is also provided without any kind of convexity assumption and the zero level set of effective energy densities is characterized in terms of Young measures. A similar result is given for cell integrands which enables to propose new counter-examples to the validity of the cell formula in the nonconvex case and to the continuity of the determinant with respect to the two-scale convergence.Comment: 24 pages, 1 figur

Universal bounds on the electrical and elastic response of two-phase bodies and their application to bounding the volume fraction from boundary measurements

Universal bounds on the electrical and elastic response of two-phase (and multiphase) ellipsoidal or parallelopipedic bodies have been obtained by Nemat-Nasser and Hori. Here we show how their bounds can be improved and extended to bodies of arbitrary shape. Although our analysis is for two-phase bodies with isotropic phases it can easily be extended to multiphase bodies with anisotropic constituents. Our two-phase bounds can be used in an inverse fashion to bound the volume fractions occupied by the phases, and for electrical conductivity reduce to those of Capdeboscq and Vogelius when the volume fraction is asymptotically small. Other volume fraction bounds derived here utilize information obtained from thermal, magnetic, dielectric or elastic responses. One bound on the volume fraction can be obtained by simply immersing the body in a water filled cylinder with a piston at one end and measuring the change in water pressure when the piston is displaced by a known small amount. This bound may be particularly effective for estimating the volume of cavities in a body. We also obtain new bounds utilizing just one pair of (voltage, flux) electrical measurements at the boundary of the body.Comment: 5 figures, 27 page

Homogenization in a thin domain with an oscillatory boundary

In this paper we analyze the behavior of the Laplace operator with Neumann boundary conditions in a thin domain of the type $R^\epsilon = \{(x,y) \in \R^2; x \in (0,1), 0 < y < \epsilon G(x, x/\epsilon)\}$ where the function G(x,y) is periodic in y of period L. Observe that the upper boundary of the thin domain presents a highly oscillatory behavior and, moreover, the height of the thin domain, the amplitude and period of the oscillations are all of the same order, given by the small parameter $\epsilon$.Comment: 27 pages, 2 figure

The Nights of Our Lives: Why We Sleep & Dream

Explore the most recent theories and evidence of why we sleep, and what happens in the brain during sleep and the consequences of sleep loss and sleep disorders

The non-photosynthetic, pathogenic green alga Helicosporidium sp. has retained a modified, functional plastid genome

A fragment of the Helicosporidium sp. (Chlorophyta: Trebouxiophyceae) plastid genome has been sequenced. The genome architecture was compared to that of both a non-photosynthetic relative (Prototheca wickerhamii) and a photosynthetic relative (Chlorella vulgaris). Comparative genomic analysis indicated that Helicosporidium and Prototheca are closely related genera. The analyses also revealed that the Helicosporidium sp. plastid genome has been rearranged. In particular, two ribosomal protein-encoding genes (rpl19 and rps23) appeared to have been transposed, or lost from the Helicosporidium sp. plastid genome. RT-PCR reactions demonstrated that the retained plastid genes were transcribed, suggesting that, despite rearrangement(s), the Helicosporidium sp. plastid genome has remained functional. The modified plastid genome architecture is a novel apomorphy that indicates that the Helicosporidia are highly derived green algae, more so than Prototheca spp. As such, they represent a promising model to study organellar genome reorganizations in parasitic protists

A-quasiconvexity : relaxation and homogenization

Integral representation of relaxed energies and of Γ-limits of functionals (u, v)↦→�Ω f(x, u(x), v(x))dx are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell’s equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W 1,p are recovered

On a differential inclusion related to the Born-Infeld equations

We study a partial differential relation that arises in the context of the Born-Infeld equations (an extension of the Maxwell's equations) by using Gromov's method of convex integration in the setting of divergence free fields

Interactions langagières et communicatives au collège : postures discursives et gestion de la tension

International audienceQue fait-on de la relation interpersonnelle entre enseignant et élève, entre professionnel et adolescent au collège ? Quels effets peut avoir la gestion de l'interaction didactique conflictuelle par l'enseignant sur la construction de la relation interpersonnelle enseignant / élève(s) ? Nous répondrons à ces questions en rendant compte d'une recherche conduite dans deux collèges inscrits en Z.E.P. de Vitrolles et Perpignan et dans un collège dit favorisé de Marseille. A partir d'un corpus constitué d'une dizaines d'heures d'enregistrements audio, la gestion par les enseignants des énoncés et interactions potentiellement conflictuels survenant dans le cadre de l'interaction pédago-didactique (réprimandes, questionnements, ...) et leur impact sur la conduite de l'interaction didactique sont interrogés. L'analyse a pour objet les variables structuration de l'échange didactique et élaboration de la relation interpersonnelle pour l'étude desquelles des indicateurs sont retenus (séquences de travail et parasitaires, atteintes portées à la face et (non) réaction, clarté et audibilité des énoncés, niveau sonore, posture élève/adolescent, posture enseignant/interlocuteur lambda, distance relationnelle). Leur mise en correspondance vise à permettre de comprendre les stratégies discursives et de voir dans quelles mesures la relation interpersonnelle enseignant/élèves altère ou non la construction de l'échange didactique. Les effets de la gestion de la relation interpersonnelle par l'enseignant sur l'interaction pédago-didactique seront donc questionnés. L'analyse s'inspire de travaux en pragmatique et en linguistique interactionnelle et a pour objet principal de décrire les relations qui se construisent, par le biais de l'échange verbal entre les interactants. En tant que linguiste et psychologue, et conformément à nos travaux antérieurs (Romain, 2005 ; Tartar-Goddet 2006), nous croiserons nos regards sur l'analyse des interactions interpersonnelles de type « conflictuel » et sur leurs effets sur l'apprentissage. Afin d'analyser ces pratiques discursives, les travaux de Kerbrat-Orecchioni (1996) et de Filliettaz (2008) seront convoqués