On the Stability of Numerical Integration Routines for Ordinary Differential Equations

Numerical integration methods for the solution of initial valueproblems for ordinary vector differential equations may be modelledas discrete time feedback systems. The stability criteria discoveredin modern control theory are applied to these systems and criteriainvolving the routine, the step size and the differential equationare derived. Linear multistep, Runge-Kutta, and predictor-correctormethods are all investigated.     

Thermotropic and lyotropic liquid crystalline behaviour of 4-alkoxyphenyl beta-D-glucopyranosides

The liquid crystalline properties of a series of 4-alkoxyphenyl beta -D-glucopyranosides (methoxy to decyloxy and dodecyloxy) were studied using polarized light microscopy and differential scanning calorimetry. The compounds with the shortest alkoxy substituents are not liquid crystalline. The butoxy derivative displays a monotropic smectic A phase and the higher homologues display enantiotropic smectic A phases. The lyotropic behaviour was studied as a function of concentration and temperature. Hexagonal, cubic and lamellar phases were observed for compounds with alkoxy chains longer than butoxy. The nonyloxy derivative forms long ribbons in dilute solution as revealed by electron microscopy.