Periodicity and Circle Packings of the Hyperbolic Plane

We prove that given a fixed radius r, the set of isometry-invariant probability measures supported on 'periodic' radius r-circle packings of the hyperbolic plane is dense in the space of all isometry-invariant probability measures on the space of radius r-circle packings. By a periodic packing, we mean one with cofinite symmetry group. As a corollary, we prove the maximum density achieved by isometry-invariant probability measures on a space of radius r-packings of the hyperbolic plane is the supremum of densities of periodic packings. We also show that the maximum density function varies continuously with radius.     

Guaiacol transprotection: replacement of the phenoxy isopropyl protecting function by acetyl

A one-step method for the conversion of isopropyl protected guaiacols to the corresponding acetates is reported. Treating 6-substituted isopropyl protected guaiacols with trimethylsilyl trifluoromethanesulfonate in a mixture of acetic anhydride and acetonitrile affords 6-substituted guaiacol acetates in yields ranging from 35% to 99%.     

Muon spin rotation (μSR) in a metal-cluster carbonyl compound

μSR spectra from a metal carbonyl compound are reproted for the first time. The metal cluster compound Cs₂[Os₁₀C(CO)₂₄] shows a temperature dependent signal for implanted muons with the onset of motional narrowing at ca. 200 K. Also observed is a probable change in the width within the temperature range 40–140 K. A tentative assignment is provided for the muon sites.     

Contours of Liapunov functions

As is well known, the stability of a dynamical system in two dimensions may be demonstrated in a very intuitive fashion from the existence of a suitable positive-definite Liapunov function, providing the contours of this function in a neighborhood of the stable point are Jordan curves. It is shown that the Liapunov function will certainly have this property if the stable point is an isolated stationary point in the sense of the Clarke calculus, but a counterexample is given if this assumption is weakened to the stable point being an isolated local extremum.This work was carried out with the support of the Natural Sciences and Engineering Council of Canada, which is gratefully acknowledged.     

The level of a witt ring

We show that the level of the Witt ring W(F) of symmetric bilinear forms over a field F can only take the values 1, 3, 4, or x and that the level of F determines the level of W(F).     

An analysis tool for piezoelectric gages

The Piezo-electric Gage Analysis System U.S. (pegasus) couples a two-dimensional dynamic structural finite element code to a two-dimensional electrostatics code for analysis of piezoelectric gages. The method has a sound theoretical basis and is built around two powerful finite element anlysis codes. The analysis codes provide the solution of the time dependent stress state in the gage and the solution of the electrostatic equation for each time step. pegasus provides the link between the two codes and the steps required to carry the analysis through to prediction of gage currents. Post-processing of the results allows visual interpretation of the the electric fields within the gage. Here we briefly describe the code and show that it can be a valuable tool for understanding the nature of piezoelectric gages. Received 6 May 1996 / Accepted 31 October 1996