Characterizations of Lojasiewicz inequalities and applications

The classical Lojasiewicz inequality and its extensions for partial differential equation problems (Simon) and to o-minimal structures (Kurdyka) have a considerable impact on the analysis of gradient-like methods and related problems: minimization methods, complexity theory, asymptotic analysis of dissipative partial differential equations, tame geometry. This paper provides alternative characterizations of this type of inequalities for nonsmooth lower semicontinuous functions defined on a metric or a real Hilbert space. In a metric context, we show that a generalized form of the Lojasiewicz inequality (hereby called the Kurdyka- Lojasiewicz inequality) relates to metric regularity and to the Lipschitz continuity of the sublevel mapping, yielding applications to discrete methods (strong convergence of the proximal algorithm). In a Hilbert setting we further establish that asymptotic properties of the semiflow generated by -∂f are strongly linked to this inequality. This is done by introducing the notion of a piecewise subgradient curve: such curves have uniformly bounded lengths if and only if the Kurdyka- Lojasiewicz inequality is satisfied. Further characterizations in terms of talweg lines -a concept linked to the location of the less steepest points at the level sets of f- and integrability conditions are given. In the convex case these results are significantly reinforced, allowing in particular to establish the asymptotic equivalence of discrete gradient methods and continuous gradient curves. On the other hand, a counterexample of a convex C2 function in R2 is constructed to illustrate the fact that, contrary to our intuition, and unless a specific growth condition is satisfied, convex functions may fail to fulfill the Kurdyka- Lojasiewicz inequality.

Mineralization of the recalcitrant oxalic and oxamic acids by electrochemical advanced oxidation processes using a boron-doped diamond anode

Oxalic and oxamic acids are the ultimate and more persistent by-products of the degradation of N-aromatics by electrochemical advanced oxidation processes (EAOPs). In this paper, the kinetics and oxidative paths of these acids have been studied for several EAOPs using a boron-doped diamond (BDD) anode and a stainless steel or an air-diffusion cathode. Anodic oxidation (AO-BDD) in the presence of Fe2+ (AO-BDD-Fe2+) and under UVA irradiation (AO-BDD-Fe2+-UVA), along with electro-Fenton (EF-BDD), was tested. The oxidation of both acids and their iron complexes on BDD was clarified by cyclic voltammetry. AO-BDD allowed the overall mineralization of oxalic acid, but oxamic acid was removed much more slowly. Each acid underwent a similar decay in AO-BDD-Fe2+ and EFBDD, as expected if its iron complexes were not attacked by hydroxyl radicals in the bulk. The faster and total mineralization of both acids was achieved in AO-BDD-Fe2+-UVA due to the high photoactivity of their Fe(III) complexes that were continuously regenerated by oxidation of their Fe(II) complexes. Oxamic acid always released a larger proportion of NH4 + than NO3- ion, as well as volatile NOx species. Both acids were independently oxidized at the anode in AO-BDD, but in AO-BDD-Fe2+-UVA oxamic acid was more slowlydegraded as its content decreased, without significant effect on oxalic acid decay. The increase in current density enhanced the oxidation power of the latter method, with loss of efficiency. High Fe2+ contents inhibited the oxidation of Fe(II) complexes by the competitive oxidation of Fe2+ to Fe3+. Low current densities and Fe2+ contents are preferable to remove more efficiently these acids by the most potent AO-BDD-Fe2+-UVA method.

Electrochemical synthesis of mesoporous CoPt nanowires for methanol oxidation

A new electrochemical method to synthesize mesoporous nanowires of alloys has been developed. Electrochemical deposition in ionic liquid-in-water (IL/W) microemulsion has been successful to grow mesoporous CoPt nanowires in the interior of polycarbonate membranes. The viscosity of the medium was high, but it did not avoid the entrance of the microemulsion in the interior of the membrane"s channels. The structure of the IL/W microemulsions, with droplets of ionic liquid (4 nm average diameter) dispersed in CoPt aqueous solution, defined the structure of the nanowires, with pores of a few nanometers, because CoPt alloy deposited only from the aqueous component of the microemulsion. The electrodeposition in IL/W microemulsion allows obtaining mesoporous structures in which the small pores must correspond to the size of the droplets of the electrolytic aqueous component of the microemulsion. The IL main phase is like a template for the confined electrodeposition. The comparison of the electrocatalytic behaviours towards methanol oxidation of mesoporous and compact CoPt nanowires of the same composition, demonstrated the porosity of the material. For the same material mass, the CoPt mesoporous nanowires present a surface area 16 times greater than compact ones, and comparable to that observed for commercial carbon-supported platinum nanoparticles.

Electrochemical synthesis of mesoporous CoPt nanowires for methanol oxidation

A new electrochemical method to synthesize mesoporous nanowires of alloys has been developed. Electrochemical deposition in ionic liquid-in-water (IL/W) microemulsion has been successful to grow mesoporous CoPt nanowires in the interior of polycarbonate membranes. The viscosity of the medium was high, but it did not avoid the entrance of the microemulsion in the interior of the membrane"s channels. The structure of the IL/W microemulsions, with droplets of ionic liquid (4 nm average diameter) dispersed in CoPt aqueous solution, defined the structure of the nanowires, with pores of a few nanometers, because CoPt alloy deposited only from the aqueous component of the microemulsion. The electrodeposition in IL/W microemulsion allows obtaining mesoporous structures in which the small pores must correspond to the size of the droplets of the electrolytic aqueous component of the microemulsion. The IL main phase is like a template for the confined electrodeposition. The comparison of the electrocatalytic behaviours towards methanol oxidation of mesoporous and compact CoPt nanowires of the same composition, demonstrated the porosity of the material. For the same material mass, the CoPt mesoporous nanowires present a surface area 16 times greater than compact ones, and comparable to that observed for commercial carbon-supported platinum nanoparticles.

Parallel characterizations of a generalized shapley value and a generalized banzhaf value for cooperative games with level structure of cooperation

We present parallel characterizations of two different values in the framework of restricted cooperation games. The restrictions are introduced as a finite sequence of partitions defined on the player set, each of them being coarser than the previous one, hence forming a structure of different levels of a priori unions. On the one hand, we consider a value first introduced in Ref. [18], which extends the Shapley value to games with different levels of a priori unions. On the other hand, we introduce another solution for the same type of games, which extends the Banzhaf value in the same manner. We characterize these two values using logically comparable properties.