Best Local Approximation in Orlicz Spaces

We get results in Orlicz spaces L ^φ about best local approximation on non-balanced neighborhoods when φ satisfies a certain asymptotic condition. This fact generalizes known previous results in L ^p spaces.     

Nonlinear Chebyshev approximation to set-valued functions

In this paper, we give a characterization of best Chebyshev approximation to set-valued functions from a family of continuous functions with the weak betweeness property. As a consequence, we obtain a characterization of Kolmogorov type for best simultaneous approximation to an infinity set of functions. We introduce the concept of a set-sun and give a characterization of it. In addition, we prove a property of Amir–Ziegler type for a family of real functions and we get a characterization of best simultaneous approximation to two functions     

Gateaux differentiability in Orlicz–Lorentz spaces and applications

Let Λ_w,? be the Orlicz–Lorentz space. We study Gateaux differentiability of the functional ψ_w,? (f) = ? (f *)w and of the Luxemburg norm. More precisely, we obtain the one‐sided Gateaux derivatives in both cases and we characterize those points where the Gateaux derivative of the norm exists. We give a characterization of best ψ_w,? ‐approximants from convex closed subsets and we establish a relation between best ψ_w,? ‐approximants and best approximants from a convex set. A characterization of best constant ψ_w,? ‐approximants and the algorithm to construct the best constant for maximum and minimum ψ_w,? ‐pproximants are given. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Coarsening phenomena of metal nanoparticles and the influence of the support pre-treatment: Pt/TiO2(110)

One of the technologically most important requirements for the application of oxide-supported metal nanoparticles (NPs) in the fields of molecular electronics, plasmonics, and catalysis is the achievement of thermally stable systems. For this purpose, a thorough understanding of the different pathways underlying thermally-driven coarsening phenomena, and the effect of the nanoparticle synthesis method, support morphology, and degree of support reduction on NP sintering is needed. In this study, the sintering of supported metal NPs has been monitored via scanning tunneling microscopy combined with simulations following the Ostwald ripening and diffusion-coalescence models. Modifications were introduced to the diffusion-coalescence model to incorporate the correct temperature dependence and energetics. Such methods were applied to describe coarsening phenomena of physical-vapor deposited (PVD) and micellar Pt NPs supported on TiO₂(110). The TiO₂(110) substrates were exposed to different pre-treatments, leading to reduced, oxidized and polymer-modified TiO₂ surfaces. Such pre-treatments were found to affect the coarsening behavior of the NPs.No coarsening was observed for the micellar Pt NPs, maintaining their as-prepared size of ~ 3 nm after annealing in UHV at 1060 °C. Regardless of the initial substrate pre-treatment, the average size of the PVD-grown NPs was found to increase after identical thermal cycles, namely, from 0.5 ± 0.2 nm to 1.0 ± 0.3 nm for pristine TiO₂, and from 0.8 ± 0.3 nm to 1.3 ± 0.6 nm for polymer-coated TiO₂ after identical thermal treatments. Although no direct real-time in situ microscopic evidence is available to determine the dominant coarsening mechanism of the PVD NPs unequivocally, our simulations following the diffusion-coalescence coarsening route were in significantly better agreement with the experimental data as compared to those based on the Ostwald-ripening model. The enhanced thermal stability of the micellar NPs as compared to the PVD clusters might be related to their initial larger NP size, narrower size distribution, and larger interparticle distances.     

Dynamic Changes in the Structure,Chemical State and Catalytic Selectivity of Cu Nanocubes during CO2 Electroreduction: Size and Support Effects

In situ and operando spectroscopic and microscopic methods were used to gain insight into the correlation between the structure, chemical state, and reactivity of size‐ and shape‐controlled ligand‐free Cu nanocubes during CO₂ electroreduction (CO₂RR). Dynamic changes in the morphology and composition of Cu cubes supported on carbon were monitored under potential control through electrochemical atomic force microscopy, X‐ray absorption fine‐structure spectroscopy and X‐ray photoelectron spectroscopy. Under reaction conditions, the roughening of the nanocube surface, disappearance of the (100) facets, formation of pores, loss of Cu and reduction of CuO_x species observed were found to lead to a suppression of the selectivity for multi‐carbon products (i.e. C₂H₄ and ethanol) versus CH₄. A comparison with Cu cubes supported on Cu foils revealed an enhanced morphological stability and persistence of Cu^I species under CO₂RR in the former samples. Both factors are held responsible for the higher C₂/C₁ product ratio observed for the Cu cubes/Cu as compared to Cu cubes/C. Our findings highlight the importance of the structure of the active nanocatalyst but also its interaction with the underlying substrate in CO₂RR selectivity.     

Cp Condition and the Best Local Approximation

In this paper, we introduce a condition weaker than the Lpdifferentiability,which we call Cpcondition. We prove that if a function satisfies this condition at a point, then there exists the best local approximation at that point. We also give a necessary and sufficient condition for that a function be Lpdifferentiable. In addition, we study the convexity of the set of cluster points of the net of best appoximations of f,{Pe( f)} as e → 0.