Ring Contraction Through Epoxide Rearrangement: A Formal Synthesis of Capsorubin

An eight-step synthesis of the cyclopentane keto-alcohol 2, which has previously been converted in one step into the carotenoid pigment capsorubin (1), is described. The key step in our synthesis is a stereospecific epoxide rearrangement with ring contraction, thus producing the cyclopentane ring from an epoxide of a cyclohexene.     

Synthesis,characterization and antibacterial activity of 3‐(2‐methoxyphenyl)‐2‐sulfanylpropenoic acid and di‐isopropylammonium [3‐(2‐methoxyphenyl)‐2‐sulfanylpropenoato] triphenylstannate(IV). The crystal structure of [HQ][SnPh3(o‐mpspa)]

Reaction of 3‐(2‐methoxyphenyl)‐2‐sulfanylpropenoic acid [H₂(o‐mpspa)] with SnPh₃OH in the presence of di‐isopropylamine resulted in the formation of the complex [HQ][SnPh₃(o‐mpspa)] (where HQ = di‐isopropylammonium cation and o‐mpspa = 3‐(2‐methoxyphenyl)‐2‐sulfanylpropenoato), which was characterized by mass spectrometry and vibrational spectroscopy, as well as by ¹H, ¹³C and ¹¹⁹Sn NMR spectroscopy. The single‐crystal X‐ray structural analysis of the new complex shows a trigonal‐bipyramidal coordination geometry around the Sn atom where o‐mpspa behaves as a bidentate chelating ligand. Dimeric units arise from the existence of N? H…O hydrogen bonds between the NH₂ group of the di‐isopropylammonium cation and the oxygen atoms of the two neighbouring carboxylato groups. The bacteriostatic activity of the complex is also reported. Copyright © 2007 John Wiley & Sons, Ltd.     

Herman Müntz: A Mathematician’s Odyssey

This column is a forum for discussion of mathematical communities throughout the world, and through all time. Our definition of ‘mathematical community’ is the broadest. We include ‘schools’ of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.     

Positive definite completions of partial Hermitian matrices

The question of which partial Hermitian matrices (some entries specified, some free) may be completed to positive definite matrices is addressed. It is shown that if the diagonal entries are specified and principal minors, composed of specified entries, are positive, then, if the undirected graph of the specified entries is chordal, a positive definite completion necessarily exists. Furthermore, if this graph is not chordal, then examples exist without positive definite completions. In case a positive definite completion exists, there is a unique matrix, in the class of all positive definite completions, whose determinant is maximal, and this matrix is the unique one whose inverse has zeros in those positions corresponding to unspecified entries in the original partial Hermitian matrix. Additional observations regarding positive definite completions are made.     

Linear programming approaches to the measurement and analysis of productive efficiency

Invited by J. T. Pastor     

Higher dimensional classicalW-algebras

ClassicalW-algebras in higher dimensions are constructed. This is achieved by generalizing the classical Gel'fand-Dickey brackets to the commutative limit of the ring of classical pseudodifferential operators in arbitrary dimension. TheseW-algebras are the Poisson structures associated with a higher dimensional version of the Khokhlov-Zabolotskaya hierarchy (dispersionless KP-hierarchy). The two dimensional case is worked out explicitly and it is shown that the role of DiffS(1) is taken by the algebra of generators of local diffeomorphisms in two dimensions.     

Maximin location: Discretization not always works

In this note we show by means of a simple example that, if the maximin problem with (nonlinear) concave increasing utility functions is solved by inspecting the extreme points of the (generalized) Voronoi diagram (as usually proposed), one may have to inspect an infinite number of candidate points. The research of the second and third authors is partially supported by Grant PB96-1416-C02-02 of Ministerio de Educación y Cultura, Spain