Synthèse d'analogues azotés d'esters chrysanthémiques et pyréthriques

3,3-Dimethyl-2-vinyl-l-ethoxycarbonylaziridines were prepared from 2-acylaziridines and phosphoranes, phosphonate or silyl anions. A 3,3-di-methyl-2-ethoxycarbonyl N-vinylaziridine was also prepared. These compounds constitute some analogues of chrysanthemic and pyrethric acid esters.     

Internal Nucleophilic Termination in Acid-Mediated Polyene Cyclisations: Synthetic access to methyle homologs of (±)-Ambrox® and its diastereoisomers

Treatment of ten monocyclic dienols 8–11 with an excess of fluorosulfonic acid in 2-nitropropane at ?90° afforded diastereoisomeric mixtures of racemic tricyclic ethers 12–14 in 81–91% yield (see tables 1 and 2). These transformations represent further examples of biomimetic acid-mediated cyclisations in which an OH group serves as the internal nucleophilic terminator. A non-synchronous process is postulated, and the examples described strongly re-inforce our working mechanistic hypothesis, whereby the stereochemical course of cyclisation is directed by the orientation of the side chain vicinal to the intermediate cyclohexyl cation (see Schemes 4 and 5). It is also demonstrated that the efficiency of this process is independent of the nature of the OH group, which may be primary, secondary, or tertiary. In addition, the organoleptic properties of 12–14 , Me homologs of known odorants such as Ambrox® ((?)- 3a ) and its diastereoisomer, are briefly discussed.     

A reproducing kernel condition for indeterminacy in the multidimensional moment problem

Using the smallest eigenvalues of Hankel forms associated with a multidimensional moment problem, we establish a condition equivalent to the existence of a reproducing kernel. This result is a multivariate analogue of Berg, Chen, and Ismail's 2002 result. We also present a class of measures for which the existence of a reproducing kernel implies indeterminacy.

     

On Poincaré Type Inequalities

Using estimates of the heat kernel we prove a Poincaré inequality for star-shape domains on a complete manifold. The method also gives a lower bound for the gap of the first two Neumann eigenvalues of a Schrödinger operator.