Total syntheses of (±)‐cis‐ and trans‐11‐methoxydeethyleburnamonine

The total syntheses of 11‐methoxydeethyleburnamonines ( 4 ) and ( 13 ) were carried out with use of 6‐methoxytryptophyl bromide ( 5 ) as starting material. Compound 5 was converted in three steps to trans‐ester 8 . Acid‐catalysed epimerization of 8 , lithium aluminum hydride reduction of the ester group, tosylation and substitution with cyanide gave the cis‐nitrile 12 . Acid‐induced cyclization of 12 yielded mainly (±)‐trans‐11‐methoxydeethyleburnamonine ( 13 ), whereas base‐induced cyclization gave (±)‐cis‐11‐methoxydeethyleburnamonine ( 4 ).     

Algebraic aspects of renormalization

This paper uses the formulation of renormalization theory in a pure algebraic way providing the notion of Hopf algebras as introduced by A. Connes and D. Kreimer [Commun. Math. Phys. 199 (1998) 203]. First an introduction to the Hopf algebra of rooted trees will be given. The second section explores how renormalization is achieved using the Hopf algebra of rooted trees. So far the paper reviews the results of D. Kreimer as published in the paper cited above. The review will end with a sketch of a category-theoretical interpretation which is under thorough investigation at the time of writing.     

Sumset phenomenon in countable amenable groups

Jin proved that whenever A and B are sets of positive upper density in Z, A+B is piecewise syndetic. Jin's theorem was subsequently generalized by Jin and Keisler to a certain family of abelian groups, which in particular contains Zd. Answering a question of Jin and Keisler, we show that this result can be extended to countable amenable groups. Moreover we establish that such sumsets (or — depending on the notation — “product sets”) are piecewise Bohr, a result which for G=Z was proved by Bergelson, Furstenberg and Weiss. In the case of an abelian group G, we show that a set is piecewise Bohr if and only if it contains a sumset of two sets of positive upper Banach density.     

Bipower-type estimation in a noisy diffusion setting

We consider a new class of estimators for volatility functionals in the setting of frequently observed Itō diffusions which are disturbed by i.i.d. noise. These statistics extend the approach of pre-averaging as a general method for the estimation of the integrated volatility in the presence of microstructure noise and are closely related to the original concept of bipower variation in the no-noise case. We show that this approach provides efficient estimators for a large class of integrated powers of volatility and prove the associated (stable) central limit theorems. In a more general Itō semimartingale framework this method can be used to define both estimators for the entire quadratic variation of the underlying process and jump-robust estimators which are consistent for various functionals of volatility. As a by-product we obtain a simple test for the presence of jumps in the underlying semimartingale.     

Percycloalkylated cyclohexanes: attempted synthesis of a trispropellane

The acid catalyzed rearrangement of two cyclohexanols of spiroannelated four-membered rings has been studied. In accordance with molecular mechanics calculations, far-reaching reorganizations with formation of unsaturated hexacyclic systems, including a fully cycloalkylated cyclohexene with a bispropellane partial structure, were observed. Attempts to convert this bispropellane to a trispropellane failed.