The Mean Number of Extreme Lines in a Convex Hull of Lines

Abstract. The dual problem of the mean number of extreme lines among n chosen at random from a convex polygon of lines is defined, evaluations made for n=4 , 6, 20 and the asymptotic mean are shown to be the same as in the classic case of n points.     

Variant superfield representations

We discuss some variant superfield representations which can arise by the replacement of some of the usual fields in a multiplet with p-form gauge fields.     

NMR identification of C11 to C40 branched hydrocarbons derived from the decomposition of polyisobutylene

The composition of nonvolatile fluids obtained from thermally cracking and hydrogenating polyisobutylene was determined by using a combination of gas chromatography and nuclear magnetic resonance spectroscopy (NMR). This work involved the separation and characterization of a homologous series, C₁₁–C₄₀, of sixteen branched hydrocarbon species consisting of repeating isobutylene structures. As a result of this investigation, useful correlations between NMR spectra and molecular structure for highly branched hydrocarbons were developed. The data demonstrate that these hydrocarbons are unique species characterized by “crowded” and sterically hindered geminal methyl and isolated methylene groups. NMR solvent shift studies in benzene solutions indicate that it is possible to differentiate between maximally crowded geminal methyl groups and between maximally crowded methylene groups in these structures. Results of the benzene-induced solvent effects are discussed with respect to the stereochemistry of these molecules and related to existing solvent shift data. These results suggest that these hydrocarbons are polar or nearly polar materials. Successive losses of isobutylene units from stabilized tertiary radicals can account for the formation of the major species identified in these fluids. Higher carbon numbered species have lower refractive indices and densities and higher molal volumes than predicted by calculations.     

Ferromagnetic spin alignment in head-to-tail coupled oligo(1, 4-phenyleneethynylene)s and Oligo(1,4-phenylenevinylene)s bearing pendant p-phenylenediamine radical cations

The intramolecular and long-range ferromagnetic coupling between p-phenylenediamine radical cations in head-to-tail coupled oligo(1, 4-phenyleneethynylene)s and oligo(1,4-phenylenvinylene)s between neighbors and next-nearest neighbors is described. UV/vis/near-IR experiments show that the radical cations are localized in the pendant p-phenylenediamine units of the conjugated oligomers. The ESR spectra of these oligo(1,4-phenyleneethynylene) and oligo(1, 4-phenylenvinylene) di(radical cation)s are consistent with those of a triplet state. A linear behavior is observed for the doubly integrated ESR intensity of the DeltaM(s) = +/-1 and DeltaM(s) = +/-2 signals with the inverse temperature (I approximately 1/T), consistent with Curie's law. This behavior indicates a triplet ground-state diradical with a large triplet-singlet energy gap or possibly a degeneracy of singlet and triplet states.     

Catalysis by supported gold: correlation between catalytic activity for CO oxidation and oxidation states of gold

X-ray absorption near-edge spectra and temperature-programmed oxidation and reduction data demonstrate that Au(I) and Au(0) are both present in working MgO-supported gold catalysts for CO oxidation. EXAFS data indicate gold clusters with essentially the same average diameter (about 30 A) in each catalyst sample. Thus, the results provide no evidence of an effect of gold cluster size on the catalytic activity, but both the catalytic activity and the surface concentration of Au(I) were found to decrease with increasing CO partial pressure (as Au(0) was increasingly formed), demonstrating that the catalytic sites incorporate Au(I).