Systematic derivation of amplitude equations and normal forms for dynamical systems

We present a systematic approach to deriving normal forms and related amplitude equations for flows and discrete dynamics on the center manifold of a dynamical system at local bifurcations and unfoldings of these. We derive a general, explicit recurrence relation that completely determines the amplitude equation and the associated transformation from amplitudes to physical space. At any order, the relation provides explicit expressions for all the nonvanishing coefficients of the amplitude equation together with straightforward linear equations for the coefficients of the transformation. The recurrence relation therefore provides all the machinery needed to solve a given physical problem in physical terms through an amplitude equation. The new result applies to any local bifurcation of a flow or map for which all the critical eigenvalues are semisimple (i.e., have Riesz index unity). The method is an efficient and rigorous alternative to more intuitive approaches in terms of multiple time scales. We illustrate the use of the method by deriving amplitude equations and associated transformations for the most common simple bifurcations in flows and iterated maps. The results are expressed in tables in a form that can be immediately applied to specific problems. (c) 1998 American Institute of Physics.     

Finite element error bounds for a curve shrinking with prescribed normal contact to a fixed boundary

We approximate the evolution of a curve subject to motion bycurvature by linear finite elements. The curve evolves insidea given domain and meets orthogonally. We derive optimal boundsfor the error with respect to the L²- and H¹-norms and presentsome computed examples.     

An enantioselective synthesis of FR182877 provides a chemical rationalization of its structure and affords multigram quantities of its direct precursor

The evolution of a strategy culminating in an efficient, enantioselective synthesis of the potent microtubule-stabilizing agent FR182877 is described. Guided by a proposed biogenesis of this complex natural product, a solution emerged that involved the first reported example of a double transannular Diels-Alder reaction to fashion the key elements of its hexacyclic structure. This pivotal transformation creates a complex pentacycle from a 19-membered macrocyclic pentaene, forming seven new stereogenic centers in a fully diastereocontrolled fashion. The efficiency of the approach ultimately enabled the preparation of multigram quantities of the direct precursor of FR182877 for conversion to the relatively unstable natural product when required. The reactivity of the strained, bridgehead olefin of this secondary metabolite with biologically relevant nucleophiles is also described.     

Mixed monolayers of poly-alanine and lipids at the air-water interface

Summary The mixed monolayers of poly-alanine + stearyl alcohol and poly-alanine + cholesterol were studied at the air-water interface. In the mixed monolayers the surface pressure-area isotherms showed three collapse states. The first and the third collapse pressures were identical in magnitude with the collapse pressures of pure components. The intermediate collapse pressure in the poly-alanine + stearyl alcohol was found to be ca. 5 dyne/cm higher than that was observed in the poly-alanine + cholesterol system. Further, the mixed films in both systems were found to show no deviation from the ideality rule. The magnitude o f the intermediate collapse state is shown to be related to the van der Waals forces present in the lipid films.With 6 figures