A formula with some applications to the theory of Lyapunov exponents

We prove an elementary formula about the average expansion of certain products of 2 by 2 matrices. This permits us to quickly re-obtain an inequality by M. Herman and a theorem by Dedieu and Shub, both concerning Lyapunov exponents. Indeed, we show that equality holds in Herman’s result. Finally, we give a result about the growth of the spectral radius of products. Financial support from Pronex-Dynamical Systems, CNP_q 001/2000 and from Faperj is gratefully acknowledged.     

Banach envelopes of vector valued H spaces

In the paper we describe the Banach envelopes of Hardy spaces of analytic functions of several variables on polydiscs taking values in quasi-Banach spaces.     

Guest‐Induced Transformation of a Porphyrin‐Edged FeII4L6 Capsule into a CuIFeII2L4 Fullerene Receptor

The combination of a bent diamino(nickel(II) porphyrin) with 2‐formylpyridine and Fe^II yielded an Fe^II₄L₆ cage. Upon treatment with the fullerenes C₆₀ or C₇₀, this cage was found to transform into a new host–guest complex incorporating three Fe^II centers and four porphyrin ligands, in an arrangement that is hypothesized to maximize π interactions between the porphyrin units of the host and the fullerene guest bound within its central cavity. The new complex shows coordinative unsaturation at one of the Fe^II centers as the result of the incommensurate metal‐to‐ligand ratio, which enabled the preparation of a heterometallic cone‐shaped Cu^IFe^II₂L₄ adduct of C₆₀ or C₇₀.     

Controlled Light‐Mediated Preparation of Gold Nanoparticles by a Norrish Type I Reaction of Photoactive Polymers

Gold nanoparticles (AuNPs) are subjects of broad interest in scientific community due to their promising physicochemical properties. Herein we report the facile and controlled light‐mediated preparation of gold nanoparticles through a Norrish type I reaction of photoactive polymers. These carefully designed polymers act as reagents for the photochemical reduction of gold ions, as well as stabilizers for the in situ generated AuNPs. Manipulating the length and composition of the photoactive polymers allows for control of AuNP size. Nanoparticle diameter can be controlled from 1.5 nm to 9.6 nm.     

Surface Trapping and STM Observation of Conformational Isomers of a Bis(Terpyridine) Ligand from Metallosupramolecular Grids

Tetranuclear Co‐grid complexes incorporating bis‐tridentate ligands, namely 4,6‐bis(2,2′‐bipyrid‐6‐yl)‐2‐phenylpyrimidine, were electrosprayed onto a Au(111) substrate under an ultrahigh vacuum. Fragmentation occurs leaving the ligands in four different conformations. Most ligands are found to form H‐bonded dimers. The most abundant conformer of the ligand on the Au surface is an asymmetric form, which has not been observed before. The present results indicate that the fragmentation of coordination compounds during the deposition process, in principle, allows for surface trapping, identification, and investigation of high energy, out‐of‐equilibrium conformations of the ligand molecules at low temperatures, which otherwise would not be observable.     

Enhanced Photocatalytic Activity of MIL‐125 by Post‐Synthetic Modification with CrIII and Ag Nanoparticles

NH₂‐MIL‐125, [Ti₈O₈(OH)₄(bdc‐NH₂)₆] (bdc^2?=1,4‐benzene dicarboxylate) is a highly porous metal–organic framework (MOF) that has a band gap lying within the ultraviolet region at about 2.6 eV. The band gap may be reduced by a suitable post‐synthetic modification of the nanochannels using conventional organic chemistry methods. Here, it is shown that the photocatalytic activity of NH₂‐MIL‐125 in the degradation of methylene blue under visible light is remarkably augmented by post‐synthetic modification with acetylacetone followed by Cr^III complexation. The latter metal ion extends the absorption from the ultraviolet to the visible light region (band gap 2.21 eV). The photogenerated holes migrate from the MOF’s valence band to the Cr^III valence band, promoting the separation of holes and electrons and increasing the recombination time. Moreover, it is shown that the MOF’s photocatalytic activity is also much improved by doping with Ag nanoparticles, formed in situ by the reduction of Ag⁺ with the acetylacetonate pendant groups (the resulting MOF band gap is 2.09 eV). Presumably, the Ag nanoparticles are able to accept the MOF’s photogenerated electrons, thus avoiding electron–hole recombination. Both, the Cr‐ and Ag‐bearing materials are stable under photocatalytic conditions. These findings open new avenues for improving the photocatalytic activity of MOFs.     

Aza-substitution,benzo-annulation effects and catalytic activity of <Emphasis Type="Italic">β</Emphasis>-octaphenyl-substituted tetrapyrrolic macroheterocyclic cobalt complexes. I. heterogeneous catalysis

The influence of the size of conjugated π-system on catalytic activity of cobalt complex with β-octaphenylporphyrin and its tetraaza-, tetrabenzo and tetrabenzotetraaza derivatives was studied in present work. It is found that catalytic activity for oxidation of sulfur-containing compounds increases under extension of conjugated macrocycle system according to the following series CoP < CoBP ≤ CoPz < CoPPz ? CoPc.