Applications of Bourgain-Brezis inequalities to Fluid Mechanics and Magnetism

We apply the borderline Sobolev inequalities of Bourgain-Brezis to the vorticity equation and Navier-Stokes equation in 2D. We take the initial vorticity to be in the space of functions of Bounded variation(BV). We obtain the subsequent vorticity to be in the space of functions of bounded variation, uniformly for small time, and the velocity vector to be uniformly bounded for small time. Such a conclusion cannot follow for initial vorticity taken to be just a measure or in L^1 from the Lamb-Oseen vortex example. Secondly we apply an improved Strichartz inequality obtained earlier by the first and third authors to the Maxwell equations of Electromagnetism. In particular we estimate the size of the magnetic field vector in terms of the gradient of the current density vector. The main point is that in this inequality only the L^1 norm in space appears for the gradient of the current density vector. Such a result is only possible because of a vanishing divergence inhomogeneity in the wave equation for the Magnetic field vector stemming from the Maxwell equations. A key ingredient in the proof of the improved Strichartz inequality is the Bourgain-Brezis borderline Sobolev inequalities.Comment: References added to the work of M. Ben-Artzi and also Haim Brezis in ARMA. Introduction revise

Bourgain-Brezis Estimates on Symmetric Spaces of Non-compact Type

Let M be a globally Riemannian symmetric space. We prove a duality estimate between pairings of vector fields with divergence zero and and in L^1 with vector fields in a critical Sobolev space on M. As a consequence we get a sharp Calderon-Zygmund estimate for solutions to Poisson's equation on M, where the right side data is manufactured from divergence free vector fields which are in L^1. Such a result was proved earlier by Jean Bourgain and Haim Brezis on Euclidean space.Comment: Final version of the paper to appear in J. Functional Analysi

Limiting Sobolev and Hardy inequalities on stratified homogeneous groups

We give a sufficient condition for limiting Sobolev and Hardy inequalities to hold on stratified homogeneous groups. In the Euclidean case, this condition reduces to the known cancelling necessary and sufficient condition. We obtain in particular endpoint Korn--Sobolev and Korn--Hardy inequalities on stratified homogeneous groups.Comment: 32 page

Going to Lorentz when fractional Sobolev, Gagliardo and Nirenberg estimates fail

In the cases where there is no Sobolev-type or Gagliardo-Nirenberg-type fractional estimate involving $\lvert u\rvert_{W^{s,p}}$, we establish alternative estimates where the strong $L^p$ norms are replaced by Lorentz norms.Comment: 12 pages. arXiv admin note: text overlap with arXiv:2003.0521

Real-World Evolution Adapts Robot Morphology and Control to Hardware Limitations

For robots to handle the numerous factors that can affect them in the real world, they must adapt to changes and unexpected events. Evolutionary robotics tries to solve some of these issues by automatically optimizing a robot for a specific environment. Most of the research in this field, however, uses simplified representations of the robotic system in software simulations. The large gap between performance in simulation and the real world makes it challenging to transfer the resulting robots to the real world. In this paper, we apply real world multi-objective evolutionary optimization to optimize both control and morphology of a four-legged mammal-inspired robot. We change the supply voltage of the system, reducing the available torque and speed of all joints, and study how this affects both the fitness, as well as the morphology and control of the solutions. In addition to demonstrating that this real-world evolutionary scheme for morphology and control is indeed feasible with relatively few evaluations, we show that evolution under the different hardware limitations results in comparable performance for low and moderate speeds, and that the search achieves this by adapting both the control and the morphology of the robot.Comment: Accepted to the 2018 Genetic and Evolutionary Computation Conference (GECCO

L'affaire Khashoggi. Quelles implications régionales et internationales pour la Turquie ?

L’assassinat à Istanbul du journaliste saoudien Jamal Khashoggi révèle la forte tension entre la Turquie et l’Arabie saoudite. Ce sont deux conceptions de l’islam sunnite qui s’affrontent. C’est aussi une question de leadership sur le monde musulman. Ce n’est pas sans lien avec la question iranienne, ni avec la position des États-Unis qui hébergent le prédicateur Fethullah Gülen, ennemi numéro un du président turc

Transcriptomic analysis of Arabidopsis developing stems: a close-up on cell wall genes

<p>Abstract</p> <p>Background</p> <p>Different strategies (genetics, biochemistry, and proteomics) can be used to study proteins involved in cell biogenesis. The availability of the complete sequences of several plant genomes allowed the development of transcriptomic studies. Although the expression patterns of some <it>Arabidopsis thaliana </it>genes involved in cell wall biogenesis were identified at different physiological stages, detailed microarray analysis of plant cell wall genes has not been performed on any plant tissues. Using transcriptomic and bioinformatic tools, we studied the regulation of cell wall genes in <it>Arabidopsis </it>stems, <it>i.e. </it>genes encoding proteins involved in cell wall biogenesis and genes encoding secreted proteins.</p> <p>Results</p> <p>Transcriptomic analyses of stems were performed at three different developmental stages, <it>i.e.</it>, young stems, intermediate stage, and mature stems. Many genes involved in the synthesis of cell wall components such as polysaccharides and monolignols were identified. A total of 345 genes encoding predicted secreted proteins with moderate or high level of transcripts were analyzed in details. The encoded proteins were distributed into 8 classes, based on the presence of predicted functional domains. Proteins acting on carbohydrates and proteins of unknown function constituted the two most abundant classes. Other proteins were proteases, oxido-reductases, proteins with interacting domains, proteins involved in signalling, and structural proteins. Particularly high levels of expression were established for genes encoding pectin methylesterases, germin-like proteins, arabinogalactan proteins, fasciclin-like arabinogalactan proteins, and structural proteins. Finally, the results of this transcriptomic analyses were compared with those obtained through a cell wall proteomic analysis from the same material. Only a small proportion of genes identified by previous proteomic analyses were identified by transcriptomics. Conversely, only a few proteins encoded by genes having moderate or high level of transcripts were identified by proteomics.</p> <p>Conclusion</p> <p>Analysis of the genes predicted to encode cell wall proteins revealed that about 345 genes had moderate or high levels of transcripts. Among them, we identified many new genes possibly involved in cell wall biogenesis. The discrepancies observed between results of this transcriptomic study and a previous proteomic study on the same material revealed post-transcriptional mechanisms of regulation of expression of genes encoding cell wall proteins.</p