Centralizers in symmetric inverse semigroups: Structure and order

The representation [5] of the centralizerC(x) of a permutationx in (a symmetric inverse semigroup)C _n involves direct products of wreath products. Indeed, this semigroup case extends its group theory counterpart. Here, the last case (forx nilpotent) is addressed: A quotient of a wreath product is introduced and used to obtain a representation of the correspondingC(x). It follows that, for anyx∈C _n ,C(x) can be imbedded in a direct product of wreath products with a quotient of a wreath product. A formula for calculating the order ofC(x) is given. The independent parameters in the formula are precisely those that define the path structure ofx∈C _n . Part of this research was supported by a Mary Washington College faculty development grant.     

Ultrafast dynamics of polybutadiene probed by optically heterodyne-detected optical-Kerr-effect spectroscopy

The ultrafast dynamics of polybutadiene have been studied using ultrafast optical-Kerr-effect spectroscopy. The data are compared with measurements on 1,3- and 1,4-pentadiene. The two diene derivatives have quite distinct subpicosecond dynamics, with an important contribution from an intramolecular torsional mode in the 1,4-derivative. The main part of the polymer spectral density can be assigned, by analogy with the data for 1,4-pentadiene, to intramolecular torsional motion about carbon–carbon single bonds. The picosecond diffusive orientational relaxation times of the dienes are not well described by simple hydrodynamics.     

Is there a universal scale of time?

The assumption that the same universal scale of time applies for different physical laws depends uponthe hypothesis that there is a unique universal scale of time in the universe. Intrinsic to that hypothesis is the assumption that if extraneous factors that affect physical clocks and processes that run on different physical principles could be removed, all clocks are synchronizable with one another. The consequences of that hypothesis not being true are considered. It was found thatclock asynchrony implies that: (1) a mass variation theory is probable; (2) there is a need for multiple clocks; and (3) t in the equations of physics is referring to clocks rather than time such that the properties of time are reduced to its direction (as characterized by thermodynamics and cosmology).     

Solid phase synthesis of 5-aminopyrazoles and derivatives

The development of a novel solid phase synthesis of some 5-aminopyrazoles and derivatives is described. Reaction of hydrazines with solid supported β-keto-nitrile (1) affords 5-aminopyrazoles (2) the amino group of which is readily acylated or sulphonylated. Generation of the solid supported β-keto-nitrile (1) is non trivial and represents a key step in the overall synthesis.