A novel approach to the determination of the crosslink density in rubber materials with the dipolar correlation effect in low magnetic fields

We present a novel NMR approach to the determination of crosslink densities in rubber materials. The method is based on the dipolar correlation effect (DCE) on the stimulated echo examined in a series of rubber samples and linear polyisoprene. The parameter evaluated from the echo attenuation curves is the mean‐squared dipolar fluctuation associated with anisotropic reorientations of macromolecular backbones. The contributions to the DCE of the constraints due to excluded volume effects and chemical crosslinks are estimated. A strong dependence of the mean‐squared dipolar fluctuation on the crosslink density of rubber combined with the simplicity of performing the measurements with inexpensive low‐field instruments suggests that the DCE is a useful tool for routine applications. The potential and problems of performing DCE measurements in low‐magnetic‐field conditions are discussed in detail. © 2001 John Wiley & Sons, Inc. J Polym Sci Part B: Polym Phys 39: 2207–2216, 2001     

Stereocontrolled synthesis of enantiopure anticancer cyclopentenone prostaglandin analogues: (−)- and (+)-TEI-9826

The synthesis of both enantiomers of TEI-9826 has been accomplished in seven steps with an overall yield of 44% starting from diastereomeric camphor protected 3-[(dimethoxyphosphoryl)methyl]-4,5-dihydroxycyclopent-2-enones. The key steps include a fully diastereoselective hydrogenation of the endocyclic carbon–carbon double bond in the cyclopentenone ring controlled with a chiral diol moiety and the conversion of the latter into a new transposed olefinic bond.     

Integers without divisors in a given progression

An asymptotic formula is given for the number of integers n ≤ x which do not have divisors in a fixed arithmetical progression. This extends a previous result of Banks et al. (Forum Math 20:1005–1037, 2008) who considered the case of progressions with prime difference.     

Optimal partitioning of a measurable space into countably many sets

Summary A notion of an optimal partition of a measurable space into countably many sets according to given nonatomic probability measures is defined. It is shown that the set of optimal partitions is nonempty. Bounds for the optimal value are given and the set of optimal partitions is characterized. Finally, an example related to statistical decision theory is presented.     

Scattering matrix in conformal geometry

Inventiones mathematicae -     

Harmonic functions of polynomial growth on certain complete manifolds

Given a complete manifold with a particular structure at infinity, we give the dimension of the space of harmonic functions with prescribed polynomial growth.