New formulas for decreasing rearrangements and a class of Orlicz-Lorentz spaces

Using a nonlinear version of the well known Hardy-Littlewood inequalities, we derive new formulas for decreasing rearrangements of functions and sequences in the context of convex functions. We use these formulas for deducing several properties of the modular functionals defining the function and sequence spaces $M_{\varphi,w}$ and $m_{\varphi,w}$ respectively, introduced earlier in \cite{HKM} for describing the K\"othe dual of ordinary Orlicz-Lorentz spaces in a large variety of cases ($\varphi$ is an Orlicz function and $w$ a {\it decreasing} weight). We study these $M_{\varphi,w}$ classes in the most general setting, where they may even not be linear, and identify their K\"othe duals with ordinary (Banach) Orlicz-Lorentz spaces. We introduce a new class of rearrangement invariant Banach spaces $\mathcal{M}_{\varphi,w}$ which proves to be the K\"othe biduals of the $M_{\varphi,w}$ classes. In the case when the class $M_{\varphi,w}$ is a separable quasi-Banach space, $\mathcal{M}_{\varphi,w}$ is its Banach envelope.Comment: 25 page

Asymptotically Hilbertian Modular Banach Spaces: Examples of Uncountable Categoricity

We give a criterion ensuring that the elementary class of a modular Banach space E (that is, the class of Banach spaces, some ultrapower of which is linearly isometric to an ultrapower of E) consists of all direct sums E\oplus_m H, where H is an arbitrary Hilbert space and \oplus_m denotes the modular direct sum. Also, we give several families of examples in the class of Nakano direct sums of finite dimensional normed spaces that satisfy this criterion. This yields many new examples of uncountably categorical Banach spaces, in the model theory of Banach space structures.Comment: 20 page

Ultrapowers of Köthe function spaces

The ultrapowers, relative to a fixed ultrafilter, of all the Köthe function spaces with non trivial concavity over the same measure space can be represented as Köthe function spaces over the same (enlarged) measure space. The existence of a uniform homeomorphism between the unit spheres of two such Köthe function spaces is reproved

Une démarche pour l'enseignement des réseaux et de la communication

Cet article se propose de faire le point sur l'enseignement de l'informatique dans le domaine des réseaux. Il s'appuie sur nos expériences pédagogiques post-baccalauréat dans les filières informatiques ( BTS, IUT, MIAG, MAITRISE, DEA, DESS... ).Nous décrivons dans une première partie le « concept réseau » et son rôle prépondérant dans l'informatique d'aujourd'hui. Face aux problèmes posés par ce domaine complexe, nous énonçons quelques « règles d'or » pour une approche progressive et applicative conduisant à expérimenter des systèmes de communications locaux.Nous exposons notre démarche didactique pour l'une des règles énoncées : « Apprendre la communication ». Nous l'illustrons à travers l'utilisation d'un logiciel d'enseignement assisté par ordinateur. Ce produit réalisé par notre équipe concerne le R.N.I.S. (Réseau Numérique à Intégration de Service). Il permet à un étudiant de se familiariser avec les concepts, les services, l'architecture et la mise en oeuvre d'un réseau R.N.I.S

On complemented subspaces of rearrangement invariant function spaces

A necessary and sufficient condition is given for a r.i. function space to contain a complemented isomorphic copy of $\ell_1(\ell_2)$

2-positive contractive projections on noncommutative Lp\mathrm{L}^p-spaces

We prove the first theorem on projections on general noncommutative $\mathrm{L}^p$-spaces associated with non-type I von Neumann algebras where $1 \leq p < \infty$. This is the first progress on this topic since the seminal work of Arazy and Friedman [Memoirs AMS 1992] where the problem of the description of contractively complemented subspaces of noncommutative $\mathrm{L}^p$-spaces is explicitly raised. We show that the range of a 2-positive contractive projection on an arbitrary noncommutative $\mathrm{L}^p$-space is completely order and completely isometrically isomorphic to some noncommutative $\mathrm{L}^p$-space. This result is sharp and is even new for Schatten spaces $S^p$. Our approach relies on non tracial Haagerup's noncommutative $\mathrm{L}^p$-spaces in an essential way, even in the case of a projection acting on a Schatten space and is unrelated to the methods of Arazy and Friedman.Comment: 28 pages. arXiv admin note: text overlap with arXiv:1909.00391, arXiv:1910.1389

Modeling the effect of soil meso- and macropores topology on the biodegradation of a soluble carbon substrate

Soil structure and interactions between biotic and abiotic processes are increasingly recognized as important for explaining the large uncertainties in the outputs of macroscopic SOM decomposition models. We present a numerical analysis to assess the role of meso- and macropore topology on the biodegradation of a soluble carbon substrate in variably water saturated and pure diffusion conditions . Our analysis was built as a complete factorial design and used a new 3D pore-scale model, LBioS, that couples a diffusion Lattice-Boltzmann model and a compartmental biodegradation model. The scenarios combined contrasted modalities of four factors: meso- and macropore space geometry, water saturation, bacterial distribution and physiology. A global sensitivity analysis of these factors highlighted the role of physical factors in the biodegradation kinetics of our scenarios. Bacteria location explained 28% of the total variance in substrate concentration in all scenarios, while the interactions among location, saturation and geometry explained up to 51% of it