On extensions of wavelet systems to dual pairs of frames

It is an open problem whether any pair of Bessel sequences with wavelet structure can be extended to a pair of dual frames by adding a pair of singly generated wavelet systems. We consider the particular case where the given wavelet systems are generated by the multiscale setup with trigonometric masks and provide a positive answer under extra assumptions. We also identify a number of conditions that are necessary for the extension to dual (multi-) wavelet frames with any number of generators, and show that they imply that an extension with two pairs of wavelet systems is possible. Along the way we provide examples that demonstrate the extra flexibility in the extension to dual pairs of frames compared with the more popular extensions to tight frames.     

Aromatic hydroxylation in a copper bis(imine) complex mediated by a micro-eta2:eta2 peroxo dicopper core: a mechanistic scenario

Detailed mechanistic studies on the ligand hydroxylation reaction mediated by a copper bis(imine) complex are presented. Starting from a structural analysis of the CuI complex and the CuII product with a hydroxylated ligand, the optical absorption and vibrational spectra of starting material and product are analyzed. Kinetic analysis of the ligand hydroxylation reaction shows that O2 binding is the rate-limiting step. The reaction proceeds much faster in methanol than in acetonitrile. Moreover, an inverse kinetic isotope effect (KIE) is evidenced for the reaction in acetonitrile, which is attributed to a sterically congested transition state leading to the peroxo adduct. In methanol, however, no KIE is observed. A DFT analysis of the oxygenation reaction mediated by the micro-eta2:eta2 peroxo core demonstrates that the major barrier after O2 binding corresponds to electrophilic attack on the arene ring. The relevant orbital interaction occurs between the sigma* orbital of the Cu2O2 unit and the HOMO of the ligand. On the basis of the activation energy for the rate-limiting step (18.3 kcal mol(-1)) this reaction is thermally allowed, in agreement with the experimental observation. The calculations also predict the presence of a stable dienone intermediate which, however, escaped experimental detection so far. Reasons for these findings are considered. The implications of the results for the mechanism of tyrosinase are discussed.     

Perturbation of operators and applications to frame theory

A celebrated classical result states that an operator U on a Banach space is invertible if it is close enough to the identity operator I in the sense that ‖I−U‖<1. Here we show that U actually is invertible under a much weaker condition. As an application we prove new theorems concerning stability offrames (and frame-like decompositions) under perturbation in both Hilbert spaces and Banach spaces. The first named author is partially supported by grants from the U.S. National Science Foundation (grant no. NSF DMS-9201357), the Danish Natural Science Research Council (Grant no. 9401598), and grants from the University of Missouri System Research Board, and the MU Research Council. The second named author thanks the University of Missouri for its hospitality during a visit, where the first draft of the paper was written.     

Hydrogen Production Using a Molybdenum Sulfide Catalyst on a Titanium-Protected n(+) p-Silicon Photocathode

A low-cost substitute: A titanium protection layer on silicon made it possible to use silicon under highly oxidizing conditions without oxidation of the silicon. Molybdenum sulfide was electrodeposited on the Ti-protected n(+) p-silicon electrode. This electrode was applied as a photocathode for water splitting and showed a greatly enhanced efficiency.     

On the pathway of photoexcited electrons: probing photon-to-electron and photon-to-phonon conversions in silicon by ATR-IR

Photoexcitation and charge carrier thermalization inside semiconductor photocatalysts are two important steps in solar fuel production. Here, photoexcitation and charge carrier thermalization in a silicon wafer are for the first time probed by a novel, yet simple and user-friendly Attenuated Total Reflectance Infrared spectroscopy (ATR-IR) system.     

Evaporation of water droplets on "lock-and-key" structures with nanoscale features

Highly ordered poly(dimethylsiloxane) microbowl arrays (MBAs) and microcap arrays (MCAs) with "lock-and-key" properties are successfully fabricated by self-assembly and electrochemical deposition. The wetting properties and evaporation dynamics of water droplets for both cases have been investigated. For the MBAs case, the wetting radius of the droplets remains unchanged until the portion of the droplet completely dries out at the end of the evaporation process. The pinning state extends for more than 99.5% of the total evaporation time, and the pinning-shrinking transition is essentially prevented whereas in the case of the MCAs the contact radius exhibits distinct stages during evaporation and the contact line retreats significantly in the middle of the evaporation process. We explain the phenomenon by a qualitative energy balance argument based on the different shrinkage types of the nanoscale-folded contact line.