Normal functional and spaces of weights

It is shown that the cone of positive normal functionals on a hypergraphA may be substantially larger than the cone of positive weights onA. For a semiclassical hypergraphA the two cones coincide if and only if the number of edges ofA of cardinality 2 is finite. This disproves an earlier statement of T. A. Cook.     

Bi-invariant Integrals on GL(n) with Applications

In this paper measures and functions on GL(n) are called bi-invariant if they are invariant under left and right multiplication of their arguments. If v is any bi-invariant Borel measure on GL(n), then there exists a unique Borel measure v^* on D _{+ ≥}(n), the set of all diagonal matrices of rank n with positive non-increasing diagonal entries, such that holds for each v-integrable bi-invariant function f:GL(n) → IR. An explicit formula for v^* will be derived in case v equals the Lebesgue measure on GL(n) and the above integral formula will be applied to concrete integration problems. In particular, if v is a probability measure, then v^* can be interpreted as the distribution of the singular value vector. This fact will be used to derive a stochastic version of a theorem from perturbation theory concerning the numerical computation of the polar decomposition.     

Synthesis and characterization of branched polystyrene. Part II. Fractionation

Fractionations have been carried out by a column-elution method on mixtures of linear and branched polystyrene. These had been prepared by anionic polymerization followed by a coupling reaction to form star-shaped branched polymers. Therefore, the components of the mixture were each of narrow distribution, and their molecular weights differed from each other as multiples of the original single chain length. For single chains in the 25,000–30,000 molecular weight range, separations were accomplished in which the multiples (number of branches) were 1, 2, 4, and 6. By this means, virtually monodisperse samples of the branched polymers could be isolated.