Multiple ergodic theorems

Given measure preserving transformationsT ₁,T ₂,...,T _s of a probability space (X,B, μ) we are interested in the asymptotic behaviour of ergodic averages of the form $$\frac{1}{N}\sum\limits_{n = 0}^{N - 1} {T_1^n f_1 \cdot T_2^n f_2 } \cdot \cdots \cdot T_s^n f_s $$ wheref ₁,f ₂,...,f _s ?L ^∞(X,B,μ). In the general case we study, mainly for commuting transformations, conditions under which the limit of (1) inL ²-norm is ∫ _x f ₁ dμ·∫ _x f ₂ dμ...∫ _x f _s dμ for anyf ₁,f ₂...,f _s ?L ^∞(X,B,μ). If the transformations are commuting epimorphisms of a compact abelian group, then this limit exists almost everywhere. A few results are also obtained for some classes of non-commuting epimorphisms of compact abelian groups, and for commuting epimorphisms of arbitrary compact groups.     

Correlation length bounds for disordered Ising ferromagnets

Thed-dimensional, nearest-neighbor disordered Ising ferromagnet: $$H = - \sum {J_{ij} \sigma _i \sigma _j }$$ is studied as a function of both temperature,T, and a disorder parameter,λ, which measures the size of fluctuations of couplingsJ _ij ≧0. A finite-size scaling correlation length,ζ _f (T, λ), is defined in terms of the magnetic response of finite samples. This correlation length is shown to be equivalent, in the scaling sense, to the quenched average correlation lengthζ(T, λ), defined as the asymptotic decay rate of the quenched average two-point function. Furthermore, the magnetic response criterion which definesζ _f is shown to have a scale-invariant property at the critical point. The above results enable us to prove that the quenched correlation length satisfies: $$C\left| {\log \xi (T)} \right|\xi (T) \geqq \left| {T - T_c } \right|^{ - {2 \mathord{\left/ {\vphantom {2 d}} \right. \kern-\nulldelimiterspace} d}}$$ which implies the boundv≧2/d for the quenched correlation length exponent.     

1,3,6,9,12,14,17,20-octaethynyltetrabenz[a,b,f,j,k,o]-4,5,10,11,15,16,21,22- octadehydro[18]annulene: a carbon-rich hydrocarbon

Dodecaynes 1a-d have been prepared via a convergent strategy that employs Sonogashira couplings as the carbon-carbon bond-forming tool. Due to the steric bulk of the DMTS groups, 1c adopts a nonplanar conformation, the dynamics of which have been probed by VT-NMR. The cobalt-catalyzed isomerization of 1a,b produced the new conjugated phenylenes 2a,b and 3a,b, respectively. [structure: see text]     

Conformation control of bilatrienes by peptides: A survey

Summary A variety of bilipeptides has been synthesized and investigated by UV-VIS and CD spectroscopy to explore the conformational influence of peptide ligands on the covalently bound tetrapyrrol moiety. The results obtained are complemented by those reported previously. From the comprehensive data thus available several features of the peptidic chain regulating interactions with the bilatriene moiety have been elucidated. Accordingly, the presence of three amide linkages at the ligand is prerequiste for induction of pronounced conformational changes consisting either in a complete transition ofM intoP helical bilatriene species by inversion of all torsional angles, or in a stretching of the chromophore. The kind of conformational influence can be controlled by varying the position of the prolyl entity within a given peptide.Dedicated to Prof. Dr. Kurt Schaffner on occasion of his 60th birthday     

Bis(amino acid) derivatives of 1,4-diamino-2-butyne that adopt a C2-symmetric turn conformation

1,4-Diamino-2-butyne was prepared from 1,4-dichloro-2-butyne via 1,4-diazido-2-butyne. Bis(amino acid) derivatives of 1,4-diamino-2-butyne having the general structure (Boc-Xxx-NHCH2C[triple bond])2 (Xxx = Ala, Phe and Met) were prepared and examined by 1H NMR spectroscopy. Using chemical shift, coupling constant and DMSO titration data it is found that these compounds adopt a C2-symmetric turn conformation featuring two intramolecular hydrogen bonds.     

Cover Picture

The cover picture shows in the background the whole cell of a methanotrophic bacterium on which are superimposed components of methane monooxygenase (the structure of the hydroxylase component (top), one of the two four-helix bundles that house the catalytic diiron centers (left)) and a schematic diagram of the catalytic cycle by which the enzyme converts dioxygen and methane into methanol and water. More about this unusual enzyme system is reported by Lippard et al. on p. 2782 ff.     

Enzymatic synthesis of neopentylpolyol esters in organic media

Utilization of lipases for synthesis of esters of hydrophilic polyols has been investigated. The choice of a suitable solvent is crucial in this type of reaction. An interesting case is fatty acid esters from neopentylpolyols, such as trimethylolpropane, which are of great interest as high temperature lubricants. Enzymatic synthesis of trimethylolpropane tricaprylate was studied as an alternative to chemical manufacturing. Triester production occurred only if the water produced by esterification was continuously removed from the medium. In these condition, kinetics of appearance and transformation of mono-, di- and triesters were determined in order to define optimal conditions.     

AN INEXPENSIVE STIRRING DEVICE FOR THE 'MERRY-GO-ROUND' PHOTOREACTOR FOR THE DETERMINATION OF REACTION QUANTUM YIELDS

A simple bar magnet is employed to effect stirring of the contents of reaction cells in a'merry-go-round'photoreactor.