Borel fibrations and G-spaces

Considering discrete groups G only, we present an elementary proof of the familiar equivalence of the category of G-spaces (with maps equivariant up to homotopy) and the category of Borel fibrations over BG.     

Dynamical systems with relaxation time

We define mathematically a class of dynamical systems that exhibit relaxation corresponding to that observed in physical systems, and then show that this class is identical with the class ofK-mixing dynamical systems.     

Symbolic dynamics for the renormalization map of a quasiperiodic Schrödinger equation

A rigorous analysis is given of the dynamics of the renormalization map associated to a discrete Schrödinger operatorH onl ²(), defined byH(n)=(n+1)+(n–1)+Vf(n)(n), whereV is a real parameter,f is a certain discontinuous period-1 function, and is the golden mean. The renormalization map forH is a diffeomorphism,T, of ³, preserving a cubic surfaceS _V . ForV8 we prove that the non-wandering set of the restriction ofT toS _v is a hyperbolic set, on whichT is conjugate to a subshift on six symbols. It follows from results in dynamical systems theory that the optimally approximating periodic operators toH have spectra which obey a global scaling law. We also define a set which we call the pseudospectrum of the operatorH. We prove it to be a Cantor set of measure zero, and obtain bounds on its Hausdorff dimension. It is an open question whether the pseudospectrum coincides with the spectrum ofH.     

Effect of time on the interfacial and foaming properties of beta-lactoglobulin/acacia gum electrostatic complexes and coacervates at pH 4.2

The electrostatic complexation between beta-lactoglobulin and acacia gum was investigated at pH 4.2 and 25 degrees C. The binding isotherm revealed a spontaneous exothermic reaction, leading to a DeltaHobs = -2108 kJ mol(-1) and a saturation protein to polysaccharide weight mixing ratio of 2:1. Soluble electrostatic complexes formed in these conditions were characterized by a hydrodynamic diameter of 119 +/- 0.6 nm and a polydispersity index of 0.097. The effect of time on the interfacial and foaming properties of these soluble complexes was investigated at a concentration of 0.1 wt % at two different times after mixing (4 min, referred as t approximately 0 h and t = 24 h). At t approximately 0 h, the mixture is mainly made of aggregating soluble electrostatic complexes, whereas after 24 h these complexes have already insolubilize to form liquid coacervates. The surface elasticity, viscosity and phase angle obtained at low frequency (0.01 Hz) using oscillating bubble tensiometry revealed higher fluidity and less rigidity in the film formed at t approximately 0 h. This observation was confirmed by diminishing bubble experiments coupled with microscopy of the thin film. It was thicker, more homogeneous and contained more water at t approximately 0 h as compared to t = 24 h (thinner film, less water). This led to very different gas permeability's of Kt approximately 0 h = 0.021 cm s(-1) and Kt=24 h) = 0.449 cm s(-1), respectively. Aqueous foams produced with the beta-lactoglobulin/acacia gum electrostatic complexes or coacervates exhibited very different stability. The former (t approximately 0 h) had a stable volume, combining low drainage rate and mainly air bubble disproportionation as the destabilization mechanism. By contrast, using coacervates aged for 24 h, the foam was significantly less stable, combining fast liquid drainage and air bubble destabilization though fast gas diffusion followed by film rupture and bubble coalescence. The strong effect of time on the air/water interfacial properties of the beta-lactoglobulin/acacia gum electrostatic complexes can be understood by their reorganization at the interface to form a coacervate phase that is more fluid/viscous at t approximately 0 h vs rigid/elastic at t = 24 h.     

Donor—acceptor interactions of substituted benzenes with molecular chlorine and carbon disulfide

CNDO molecular orbital calculations have been performed to analyze donor—acceptor interactions between molecular chlorine and benzene, toluene, mesitylene and hexamethylbenzene and the, as yet, unreported chlorine—hexafluorobenzene and carbon disulfide—benzene pairs. The stabilization energy and the dipole moment and its derivative (?p/?R_CICI) calculated for the benzene—chlorine complex are in good agreement with the estimated experimental values. The trends in the experimental stabilization energies and the Cl-Cl vibrational frequencies with increasing methyl substitution appear to be well reproduced by the calculations. The charge transferred from the benzene donor is polarized toward the outer chlorine atom or sulfur atom. For hexafluorobenzene-chlorine the direction of electronic charge polarization is reversed from that of the benzene and methylbenzene complexes. The calculated results are discussed within the framework of Muliiken's simplified resonance theory for complexes.     

Alternativ-Liganden. XXIII. Rhodium(I)-Komplexe mit Donor/Akzeptor-Liganden des Typs (Me2PCH2CH2SiX2 und (2-Me2PC6H4)SiXMe2 (X = F,Cl)

Alternative Ligands. XXIII Rhodium(I) Complexes with Donor/Acceptor Ligands of the Type (Me₂PCH₂CH₂)₂SiX₂ and (2-Me₂PC₆H₄)SiXMe₂ (X = F, Cl) Donor/acceptor ligands of the type (Me₂PCH₂CH₂)₂SiX₂ and (2-Me₂PC₆H₄)SiXMe₂ (X = F, Cl) react with [Rh(CO)₂Cl]₂ (₁) to give the mononuclear complexes RhCl(CO)(Me₂PCH₂CH₂)₂SiX₂ [X = F( 4 ), Cl ( 5 )] and RhCl(CO)[2-Me₂PC₆H₄)SixMe₂]₂ [X = F ( 8 ), Cl ( 9 )], respectively. In case of the ligands (Me₂PCH₂CH₂)₂SiCl₂ ( 3 ) and (2-Me₂PC₆H₆)SiClMe₂ ( 7 ) the Rh(I) complexes formed in the first step partly undergo oxidative addition reactions of SiCl bonds yielding rhodium(III) compounds of low solubility. Only for 8 the coordination shifts Δδ = δ(complex)?δ(ligand) and coupling constants give some indication to possible Rh→Si interactions. However, the molecular structure of 8 determined by X-ray diffraction does not show RhSi or RhF bonding contacts. The new compounds were characterized by analytical (C, H) and spectroscopic investigations (MS, IR,-NMR).     

Pincer-porphyrin hybrids

[reaction: see text] Starting from tetrakis(3,5-bis(bromomethyl)phenyl)porphyrin, pincer-porphyrin hybrid molecules (tetrakis(ECE-pincer)porphyrin; E = N, P, S) based on a tetraphenylporphyrin skeleton have been prepared in high yields. These multi-ligand site compounds could be selectively metalated at their peripheries, which was shown by X-ray crystallography.