Approximation in higher-order Sobolev spaces and Hodge systems

Let $d\ge 2$ be an integer, $1\le l\le d?1$ and φ be a differential l-form on ${\mathbb{R}}^d$ with ${\dot{W}}^{1,d}$ coefficients. It was proved by Bourgain and Brezis ([5, Theorem 5]) that there exists a differential l-form ψ on ${\mathbb{R}}^d$ with coefficients in $L^{\infty }\cap {\dot{W}}^{1,d}$ such that $d\phi =d\psi$. In the same work, Bourgain and Brezis also left as an open problem the extension of this result to the case of differential forms with coefficients in the higher order space ${\dot{W}}^{2,d/2}$ or more generally in the fractional Sobolev spaces ${\dot{W}}^{s,p}$ with $sp=d$. We give a positive answer to this question, provided that $d?\kappa \le l\le d?1$, where κ is the largest positive integer such that $\kappa <\min ?(p,d)$. The proof relies on an approximation result (interesting in its own right) for functions in ${\dot{W}}^{s,p}$ by functions in ${\dot{W}}^{s,p}\cap L^{\infty }$, even though ${\dot{W}}^{s,p}$ does not embed into $L^{\infty }$ in this critical case. The proofs rely on some techniques due to Bourgain and Brezis but the context of higher order and/or fractional Sobolev spaces creates various difficulties and requires new ideas and methods.     

Capture of Gaseous Iodine in Isoreticular Zirconium-Based UiO-n Metal-Organic Frameworks: Influence of Amino Functionalization,DFT Calculations,Raman and EPR Spectroscopic Investigation

A series of Zr-based UiO-n MOF materials (n=66, 67, 68) have been studied for iodine capture. Gaseous iodine adsorption was collected kinetically from a home-made set-up allowing the continuous measurement of iodine content trapped within UiO-n compounds, with organic functionalities (−H, −CH₃, −Cl, −Br, −(OH)₂, −NO₂, −NH₂, (−NH₂)₂, −CH₂ NH₂) by in-situ UV-Vis spectroscopy. This study emphasizes the role of the amino groups attached to the aromatic rings of the ligands connecting the {Zr₆O₄(OH)₄} brick. In particular, the preferential interaction of iodine with lone-pair groups, such as amino functions, has been experimentally observed and is also based on DFT calculations. Indeed, higher iodine contents were systematically measured for amino-functionalized UiO-66 or UiO-67, compared to the pristine material (up to 1211 mg/g for UiO-67-(NH₂)₂). However, DFT calculations revealed the highest computed interaction energies for alkylamine groups (−CH₂NH₂) in UiO-67 (−128.5 kJ/mol for the octahedral cavity), and pointed out the influence of this specific functionality compared with that of an aromatic amine. The encapsulation of iodine within the pore system of UiO-n materials and their amino-derivatives has been analyzed by UV-Vis and Raman spectroscopy. We showed that a systematic conversion of molecular iodine (I₂) species into anionic I⁻ ones, stabilized as I⁻⋅⋅⋅I₂ or I₃⁻ complexes within the MOF cavities, occurs when I₂@UiO-n samples are left in ambient light.     

Environmentally responsive particles: from superhydrophobic particle films to water-dispersible microspheres

We describe the preparation, by precipitation copolymerization, of multifunctional divinylbenzene-co-pentafluorostyrene microspheres able to produce superhydrophobic surfaces or disperse in aqueous media upon annealing either in air or water, respectively. For that purpose, an amphiphilic block copolymer, polystyrene-b-poly(acrylic acid), was introduced in the initial feed composed of divinylbenzene and 2,3,4,5,6-pentafluorostyrene. As a result, fluorinated particles were obtained in which the diblock copolymer was encapsulated during the polymerization step. Upon annealing in dry air, the particles are completely hydrophobic and form superhydrophobic surfaces. On the contrary, annealing in water induces the reorientation of the PAA groups toward the particle interface, thus the particles can be dispersed in aqueous media. In addition, the presence of carboxylic acid groups at the particle interface permits us to switch the surface charge between negative and neutral depending on the environmental pH.     

Fabrication and superhydrophobic behavior of fluorinated microspheres

We describe the preparation of fluorinated microspheres by precipitation polymerization and their use to fabricate superhydrophobic surfaces. For that purpose, two different approaches have been employed. In the first approach, a fluorinated monomer (either 4-fluorostyrene or 2,3,4,5,6-pentafluorostyrene) was added to the initial mixture of monomers constituted by styrene (S) and divinylbenzene (DVB). The second approach is based on the encapsulation of a block copolymer, polystyrene-b-poly(2,3,4,5,6-pentafluorostyrene), during the polymerization of the monomers (S and DVB), thus enabling the formation of particles with perfluorinated chains instead of single functional groups at the interface. Both approaches led to narrow polydisperse particles with fluoro-functional groups at the interface as demonstrated by scanning electron microscopy (SEM), infrared (IR) spectroscopy, and X-ray photoelectron spectroscopy (XPS). Surface array of particles obtained by simple solvent casting presented superhydrophobic behavior with contact angles of water droplets of ca. 160-165°.     

Weak approximation by bounded Sobolev maps with values into complete manifolds

We have recently introduced the trimming property for a complete Riemannian manifold $N^n$ as a necessary and sufficient condition for bounded maps to be strongly dense in $W^{1,p}(B^m;N^n)$ when $p\in \{1,\dots ,m\}$. We prove in this note that, even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.     

On the Lipschitz character of orthotropic <Emphasis Type="Italic">p</Emphasis>-harmonic functions

We prove that local weak solutions of the orthotropic p-harmonic equation are locally Lipschitz, for every \(p\ge 2\) and in every dimension. More generally, the result holds true for more degenerate equations with orthotropic structure, with right-hand sides in suitable Sobolev spaces.     

Integration of satellite and terrestrial systems in future multimedia communications

In this article we examine the role of satellite communications in future telecommunication networks and service provision. Lessons from the past indicate that satellites are successful as a result of their wide area coverage or speed to market for new services. Niche areas such as coverage of air and sea will persist, but for land masses convergence of fixed, mobile, and broadcasting will dictate that the only way forward for satellites is in an integrated format with terrestrial systems. We outline future ways forward for satellites, and discuss the research challenges and technology advances needed to facilitate this integrated approach.     

New approaches to statistical learning theory

We present new tools from probability theory that can be applied to the analysis of learning algorithms. These tools allow to derive new bounds on the generalization performance of learning algorithms and to propose alternative measures of the complexity of the learning task, which in turn can be used to derive new learning algorithms.