Homogeneity and local symmetry of complex (<Emphasis Type="Italic">κ</Emphasis>, µ)-spaces

We prove that a complex (κ, μ)-space with κ < 1 is a locally homogeneous complex contact metric manifold. Also, a complex (κ, μ)-space has either κ = 1 or is GH-locally symmetric.     

New lithium-zincate approaches for the selective functionalisation of pyrazine: direct dideprotozincation vs. nucleophilic alkylation

Comparing the reactivity of the related lithium zincates [(THF)LiZn(TMP)(t)Bu(2)] (1) and [(PMDETA)LiZn(t)Bu(3)] (2) towards pyrazine discloses two new bimetallic approaches for the selective 2,5-dideprotonation and room temperature C-H alkylation of this sensitive heterocycle.     

The total synthesis of (−)-aplysin via a lithiation–borylation–propenylation sequence

A concise, highly enantioselective synthesis of sesquiterpene natural products (?)-debromoaplysin and (?)-aplysin has been completed. The key steps included lithiation–borylation of a secondary benzylic carbamate to give a tertiary boronic ester followed by propenylation which installed the quaternary stereocenter with complete enantioselectivity. Subsequent RCM followed by deprotection and in situ cyclization led to debromoaplysin with good diastereoselectivity from which the target compound was prepared in just eight overall steps.     

Facile separation of chiral 1,3-dihydrobenzo[c]furan derivatives using a -xylose moiety as a protecting group

1,2-O-Isopropylidene-α- -xylofuranose has been used to protect one aldehyde group of o-phthalaldehyde. This chiral protecting group acts as a resolving agent and this leads to separable diastereoisomers when a new stereogenic centre is created by the conversion of the second aldehyde group to a benzyloxyhydroxyethyl chain. These separated diastereoisomers were cyclised to 1,3-dihydrobenzo[c]furans with retention of chiral integrity at the C3 site thus allowing further elaboration to enantiomerically pure nucleoside analogues.     

Values of the Euler phi function not divisible by a given odd prime

An asymptotic formula is given for the number of integers n≤x for which φ(n) is not divisible by a given odd prime.     

The computational complexity of multi-level linear programs

We show that (L+1)-level linear programs are as difficult as levelL of the polynomial-time hierarchy, even if one only considers problems with unique optimal solutions.Dedicated to Robert Jeroslow (1942–1988)