The nilpotence height of

A 20-year-old conjecture about the mod 2 Steenrod algebra The method of Walker and Wood is used to completely determine the nilpotence height of the elements in the Steenrod algebra at the prime 2. In particular, it is shown that for all , . In addition, several interesting relations in are developed in order to carry out the proof.

     

Application of the shape group method to conformational processes: Shape and conjugation changes in the conformers of 2-phenyl pyrimidine

The shape group method (SGM) and the associated (a,b)-parameter maps provide a detailed shape characterization of molecular charge distributions. This method is applied to the study of the variations of shape and conjugation of conformers of 2-phenyl pyrimidine in their electronic ground state. Within the SGM framework, the method of (a,b)-parameter maps provides a concise, nonvisual, algorithmic technique for shape characterization of molecules with fixed nuclear geometries. Moreover, shape codes derived from the (a,b)-parameter maps afford a practical means for efficiently storing the shape properties of molecules in an electronic database. The shape codes of two or more charge distributions can be compared directly, and numerical measures of molecular shape similarity can be computed using a technique that is simple, fast, and inexpensive, especially in relation to direct, pairwise comparisons of electronic charge densities. The quantitative and automated nature of the method suggests applications in the field of computer-aided molecular design. In this study, the method is used for the first time to determine detailed numerical shape codes and shape similarity measures for a nontrivial conformational problem involving changes in energy and in conjugation. Numerical shape similarity measures of eight conformers of 2-phenyl pyrimidine are determined and correlated with variations in conformational energy and conjugation. The competing effects of steric repulsion and conjugation lead to important correlations between conformational energy and shape. © 1995 John Wiley & Sons, Inc.