A general wavelet-based profile decomposition in the critical embedding of function spaces

We characterize the lack of compactness in the critical embedding of functions spaces $X\subset Y$ having similar scaling properties in the following terms : a sequence $(u_n)_{n\geq 0}$ bounded in $X$ has a subsequence that can be expressed as a finite sum of translations and dilations of functions $(\phi_l)_{l>0}$ such that the remainder converges to zero in $Y$ as the number of functions in the sum and $n$ tend to $+\infty$. Such a decomposition was established by G\'erard for the embedding of the homogeneous Sobolev space $X=\dot H^s$ into the $Y=L^p$ in $d$ dimensions with $0<s=d/2-d/p$, and then generalized by Jaffard to the case where $X$ is a Riesz potential space, using wavelet expansions. In this paper, we revisit the wavelet-based profile decomposition, in order to treat a larger range of examples of critical embedding in a hopefully simplified way. In particular we identify two generic properties on the spaces $X$ and $Y$ that are of key use in building the profile decomposition. These properties may then easily be checked for typical choices of $X$ and $Y$ satisfying critical embedding properties. These includes Sobolev, Besov, Triebel-Lizorkin, Lorentz, H\"older and BMO spaces.Comment: 24 page

Concentration analysis and cocompactness

Loss of compactness that occurs in may significant PDE settings can be expressed in a well-structured form of profile decomposition for sequences. Profile decompositions are formulated in relation to a triplet $(X,Y,D)$, where $X$ and $Y$ are Banach spaces, $X\hookrightarrow Y$, and $D$ is, typically, a set of surjective isometries on both $X$ and $Y$. A profile decomposition is a representation of a bounded sequence in $X$ as a sum of elementary concentrations of the form $g_kw$, $g_k\in D$, $w\in X$, and a remainder that vanishes in $Y$. A necessary requirement for $Y$ is, therefore, that any sequence in $X$ that develops no $D$-concentrations has a subsequence convergent in the norm of $Y$. An imbedding $X\hookrightarrow Y$ with this property is called $D$-cocompact, a property weaker than, but related to, compactness. We survey known cocompact imbeddings and their role in profile decompositions

Polymer multilayer films obtained by electrochemically catalyzed click chemistry.

We report the covalent layer-by-layer construction of polyelectrolyte multilayer (PEM) films by using an efficient electrochemically triggered Sharpless click reaction. The click reaction is catalyzed by Cu(I) which is generated in situ from Cu(II) (originating from the dissolution of CuSO(4)) at the electrode constituting the substrate of the film. The film buildup can be controlled by the application of a mild potential inducing the reduction of Cu(II) to Cu(I) in the absence of any reducing agent or any ligand. The experiments were carried out in an electrochemical quartz crystal microbalance cell which allows both to apply a controlled potential on a gold electrode and to follow the mass deposited on the electrode through the quartz crystal microbalance. Poly(acrylic acid) (PAA) modified with either alkyne (PAA(Alk)) or azide (PAA(Az)) functions grafted onto the PAA backbone through ethylene glycol arms were used to build the PEM films. Construction takes place on gold electrodes whose potentials are more negative than a critical value, which lies between -70 and -150 mV vs Ag/AgCl (KCl sat.) reference electrode. The film thickness increment per bilayer appears independent of the applied voltage as long as it is more negative than the critical potential, but it depends upon Cu(II) and polyelectrolyte concentrations in solution and upon the reduction time of Cu(II) during each deposition step. An increase of any of these latter parameters leads to an increase of the mass deposited per layer. For given buildup conditions, the construction levels off after a given number of deposition steps which increases with the Cu(II) concentration and/or the Cu(II) reduction time. A model based on the diffusion of Cu(II) and Cu(I) ions through the film and the dynamics of the polyelectrolyte anchoring on the film, during the reduction period of Cu(II), is proposed to explain the major buildup features.journal articleresearch support, non-u.s. gov't2010 Feb 16importe

An In-Out Approach to Disjunctive Optimization

Abstract. Cutting plane methods are widely used for solving convex optimization problems and are of fundamental importance, e.g., to pro-vide tight bounds for Mixed-Integer Programs (MIPs). This is obtained by embedding a cut-separation module within a search scheme. The importance of a sound search scheme is well known in the Constraint Programming (CP) community. Unfortunately, the “standard ” search scheme typically used for MIP problems, known as the Kelley method, is often quite unsatisfactory because of saturation issues. In this paper we address the so-called Lift-and-Project closure for 0-1 MIPs associated with all disjunctive cuts generated from a given set of elementary disjunction. We focus on the search scheme embedding the generated cuts. In particular, we analyze a general meta-scheme for cutting plane algorithms, called in-out search, that was recently proposed by Ben-Ameur and Neto [1]. Computational results on test instances from the literature are presented, showing that using a more clever meta-scheme on top of a black-box cut generator may lead to a significant improvement

Ensemble approach for generalized network dismantling

Finding a set of nodes in a network, whose removal fragments the network below some target size at minimal cost is called network dismantling problem and it belongs to the NP-hard computational class. In this paper, we explore the (generalized) network dismantling problem by exploring the spectral approximation with the variant of the power-iteration method. In particular, we explore the network dismantling solution landscape by creating the ensemble of possible solutions from different initial conditions and a different number of iterations of the spectral approximation.Comment: 11 Pages, 4 Figures, 4 Table