Tandem cross metathesis and intramolecular aza-Michael reaction to synthesize bicyclic piperidines and indolizidine 167E

We have successfully transformed the terminal alkenes of dihydropyridones to the α,β-unsaturated esters by cross metathesis (CM). After detosylation the secondary amides can undergo the intramolecular aza-Michael reaction to give the bicyclic piperidine structures. The stereoselectivity of the aza-Michael reaction is determined by the size of the newly formed ring. With simple transformations we have also achieved the synthesis of indolizidine 167E.     

Lewis acid-catalyzed tandem acylation–Nazarov cyclization for the syntheses of fused cyclopentenones

Treatment of N-tosylpyrroles or N-tosylindoles with α-unsubstituted α,β-unsaturated carboxylic acids or unsaturated carboxylic acids having an α electron-withdrawing substituent in the presence of TFAA and a Lewis acid catalyst resulted in the formation of fused cyclopentenones via a tandem acylation–Nazarov cyclization sequence, while either the acylation product obtained or no reaction occurred in the absence of the Lewis acid catalyst.     

Higher-order Sobolev embeddings and isoperimetric inequalities

Optimal higher-order Sobolev type embeddings are shown to follow via isoperimetric inequalities. This establishes a higher-order analogue of a well-known link between first-order Sobolev embeddings and isoperimetric inequalities. Sobolev type inequalities of any order, involving arbitrary rearrangement-invariant norms, on open sets in Rⁿ${\mathbb{R}}^n$, possibly endowed with a measure density, are reduced to much simpler one-dimensional inequalities for suitable integral operators depending on the isoperimetric function of the relevant sets. As a consequence, the optimal target space in the relevant Sobolev embeddings can be determined both in standard and in non-standard classes of function spaces and underlying measure spaces. In particular, our results are applied to any-order Sobolev embeddings in regular (John) domains of the Euclidean space, in Maz'ya classes of (possibly irregular) Euclidean domains described in terms of their isoperimetric function, and in families of product probability spaces, of which the Gauss space is a classical instance.     

Gravitational interpretation of the Hitchin equations

By referring to theorems of Donaldson and Hitchin, we exhibit a rigorous AdS/CFT-type correspondence between classical 2+1-dimensional vacuum general relativity theory on Σ×R$\Sigma \times \mathbb{R}$ and SO(3) Hitchin theory (regarded as a classical conformal field theory) on the spacelike past boundary Σ$\Sigma$, a compact, oriented Riemann surface of genus greater than 1. Within this framework we can interpret the 2+1-dimensional vacuum Einstein equation as a decoupled “dual” version of the two-dimensional SO(3) Hitchin equations.     

Twisted Poincaré Duality for some Quadratic Poisson Algebras

We exhibit a Poisson module restoring a twisted Poincaré duality between Poisson homology and cohomology for the polynomial algebra endowed with Poisson bracket arising from a uniparametrised quantum affine space. This Poisson module is obtained as the semiclassical limit of the dualising bimodule for Hochschild homology of the corresponding quantum affine space. As a corollary we compute the Poisson cohomology of R, and so retrieve a result obtained by direct methods (so completely different from ours) by Monnier. This research of the first author was supported by a Marie Curie Intra-European Fellowship within the 6th European Community Framework Programme. The second author was supported by EPSRC Grant EP/D034167/1.     

High pressure inactivation of Clostridium botulinum type E endospores in model emulsion systems

Clostridium botulinum type E is a cold-tolerant, neurotoxigenic, endospore-forming organism, primarily associated with aquatic environments. High pressure thermal (HPT) processing presents a promising tool to enhance food safety and stability. The effect of fat on HPT inactivation of C. botulinum type E spores was investigated using an emulsion model system. The distribution of spores in oil-in-water (O/W) emulsions and their HPT (300–750?MPa, 45–75?°C, 10?min) inactivation was determined as a function of emulsion fat content (30–70% (v/v) soybean oil in buffer). Approximately 26% and 74% of the spores were located at the oil–buffer interface and the continuous phase, respectively. Spore inactivation in emulsion systems decreased with increasing oil contents, which suggests that the fat content of food plays an important role in the protection of C. botulinum type E endospores against HPT treatments. These results can be helpful for future safety considerations.     

On Shellsort and the frobenius problem

A boundO(N ^1+1/k ) for the running time of Shellsort, withO(logN) passes, is proved very simply by application of a Frobenius basis withk elements.