The Ups and Downs of Percent (and Some Interesting Connections)

Percent is a standardized ratio comparison that is often used to describe relative amounts of increase and decrease. In this paper, the nonsymmetric relationship between percent of increase and percent of decrease is explored in several ways. Beginning with a simple graphical representation of the relationship, the functional relationship is developed, with relative decrease expressed as a function of relative increase. Once expressed in the form f(x) = 100x/(100 + x), an immediate connection is made to two other topics commonly found within the study of rates—the work problem and the harmonic mean. Finally, successive percents of increase or decrease are shown to be everyday applications of the geometric mean. This deeper look at a common mathematical concept has uncovered rich structural relationships between topics that are often isolated in the curriculum.     

QCD sum rules: charmonium

The dimensiond=4 gluon condensateφ ₁ is determined from an analysis of charmonium, taking model dependent estimates ofd=6.8 condensates into account. ForS-waves, there is a compatibility region, where both these and model dependent higher state corrections to ground state dominance are small, enablingφ ₁ to be determined from a fit to the data. ForP-waves the estimated higher dimension contributions are large in the region of ground state dominance, and noφ ₁ determination is possible. Theφ ₁ value obtained from the fit to theS-wave ratios is 3 5 times bigger than that obtained by RRY using the plateau method. The methods are compared for the vector current. We show that the higher state correction used by RRY does not fit the data; and that the plateau method is much more sensitive to thed=6.8 condensates than the ratio used here. When reasonable estimates of both these and the higher state corrections are taken into account, there is compatibility between the two methods.     

A convenient synthesis of sulfonylureas from carboxylic acids and sulfonamides via an in situ Curtius rearrangement

A one-pot reaction of carboxylic acids with sulfonamides to afford sulfonylureas is described.