S 1 actions and elliptic genera

A proof is given of Witten's conjectures for the rigidity of the index of the Dirac-Ramond operator on the loop space of a spin manifold which admits anS ¹ symmetry.Research supported in part by the National Science Foundation     

Self-organized ZnAl2O4 nanostructures grown on -sapphire

Self-organized ZnAl₂O₄ nanostructures with the appearance (in SEM) of high aspect ratio horizontal nanowires are grown on uncatalysed c-sapphire by vapour phase transport. The nanostructures grow as three equivalent crystallographic variants on c-sapphire. Raman and cathodoluminescence spectroscopy confirm that the nanostructures are not ZnO and TEM shows that they are the cubic spinel, zinc aluminate, ZnAl₂O₄, formed by the reaction of Zn and O with the sapphire substrate.     

Periodic and quasi-periodic solutions of nonlinear wave equations via KAM theory

In this paper the nonlinear wave equation $$u_u - u_{xx} + v(x)u(x,t) + \varepsilon u^3 (x,t) = 0$$ is studied. It is shown that for a large class of potentials,v(x), one can use KAM methods to construct periodic and quasi-periodic solutions (in time) for this equation.     

A Tensor product generalized ADI method for the method of planes

We consider solving separable, second order, linear elliptic prtial differential equations in three independent variables. If the partial differential opertor separates into two terms, one depending on x and y, and one depending on z, then we use the method of planes to obtain a discrete problem, which we write in tensor product from as We apply a new interative method, the tensor product generalized alternating direction implicit method, to solve the discrete problem. We study a specific implementation that uses Hermite bicubic collocation in the xy direction and symmetric finite differences in the z direction. We demostrate that this method is a fast and accurate way to solve the large linear systems arising from three-dimensional elliptic problems.     

Diels-Alder chemistry on alkene functionalized films

A substrate-independent method has been devised for ring formation at solid surfaces. This entails the aminolysis reaction of allylamine with maleic anhydride pulsed plasma polymer films to yield terminal alkene groups at the surface. Subsequent exposure to 1,3-cyclohexadiene leads to a Diels-Alder type (4 + 2) cycloaddition reaction to give a mixture of endo- and exo-bicyclo[2.2.2]oct-2-ene rings.     

Comparison of localization and delocalization indices obtained with Hartree-Fock and conventional correlated methods: effect of Coulomb correlation

Atomic populations and localization [lambda(A)] and delocalization [delta(A,B)] indices (LIs and DIs) are calculated for a large set of molecules at the Hartree-Fock (HF), MP2, MP4(SDQ), CISD, and QCISD levels with the 6-311++G(2d,2p) basis set. The HF method and the conventional correlation methods [MP2, MP4(SDQ), CISD, and QCISD] yield distinct sets of LIs and DIs. Yet, within the four conventional correlation methods the differences in atomic populations and LIs and DIs are small. Relative to HF, the conventional correlation methods [MP2, MP4(SDQ), CISD, QCISD] yield virtually the same LIs and DIs for molecules with large charge separations while LIs and DIs that differ significantly from the HF values--the LIs are increased and DIs decreased--are obtained for bonds with no or small charge separations. Such is the case in the archetypal homopolar molecules HC(triple bond)CH, H2C=CH2, CH3-CH3, and "protonated cyclopropane" C(3)H(7) (+), in which case the bonding may be atypical. Relative to HF, the typical effect of the conventional correlation methods is to decrease the DI between atoms.