The Lie Bracket of Adapted Vector Fields on Wiener Spaces

Let W(M) be the based (at o∈ M) path space of a compact Riemannian manifold M equipped with Wiener measure ν . This paper is devoted to considering vector fields on W(M) of the form X _s ^h ( σ )=P _s ( σ )h _s ( σ ) where P _s ( σ ) denotes stochastic parallel translation up to time s along a Wiener path σ ∈ W(M) and {h _s } _{s∈ [0,1]} is an adapted T _o M -valued process on W(M). It is shown that there is a large class of processes h (called adapted vector fields) for which we may view X ^h as first-order differential operators acting on functions on W(M) . Moreover, if h and k are two such processes, then the commutator of X ^h with X ^k is again a vector field on W(M) of the same form. Accepted 5 May 1997     

Exploring multivariate Padé approximants for multiple hypergeometric series

We investigate the approximation of some hypergeometric functions of two variables, namely the Appell functions F _i , i = 1,...,4, by multivariate Padé approximants. Section 1 reviews the results that exist for the projection of the F _i onto ϰ=0 or y=0, namely, the Gauss function ₂ F ₁(a, b; c; z), since a great deal is known about Padé approximants for this hypergeometric series. Section 2 summarizes the definitions of both homogeneous and general multivariate Padé approximants. In section 3 we prove that the table of homogeneous multivariate Padé approximants is normal under similar conditions to those that hold in the univariate case. In contrast, in section 4, theorems are given which indicate that, already for the special case F ₁(a, b, b′; c; x; y) with a = b = b′ = 1 and c = 2, there is a high degree of degeneracy in the table of general multivariate Padé approximants. Section 5 presents some concluding remarks, highlighting the difference between the two types of multivariate Padé approximants in this context and discussing directions for future work. This revised version was published online in June 2006 with corrections to the Cover Date.     

Structural phases formed by NO(2)∕CO co-adsorption on Au{111} surfaces

Exposing a Au{111} surface to NO(2) and then to CO at temperatures around 120 K in ultra-high vacuum gives rise to molecular overlayers in which the two species are co-adsorbed, which we have investigated using low-temperature scanning tunnelling microscopy. Under NO(2)-rich conditions, a (√7 × √7)R19.1° phase with 3:1 NO(2):CO stoichiometry forms. Under CO-rich conditions, this phase co-exists with other phases having 2:1 and 1:1 NO(2):CO stoichiometries and different symmetries, and with bare Au surface. Structural models for these phases are discussed. Individual domains of the (√7 × √7)R19.1° phase are chiral, by virtue of the arrangement of their achiral components, an observation that may have more general implications.     

Nitrogen adsorption and desorption at iron pyrite FeS2{100} surfaces

We have investigated the interaction of nitrogen with single-crystal iron pyrite FeS(2){100} surfaces in ultra-high vacuum. N(2) adsorbs molecularly at low temperatures, desorbing at 130 K, but does not adsorb dissociatively even at pressures up to 1 bar. Atomic surface N can, however, be obtained with nitrogen ions and/or excited neutral species, generated by passing N(2) through an ion gun. Substantial nitrogen-induced disorder is seen with both ions and neutrals, and no ordered N overlayers form; a decrease in the S/Fe ratio is seen when exposing to nitrogen ions. Recombinative desorption leads to temperature-programmed desorption peaks at 410 and 520-560 K which we associate with interstitial atomic N and substitutional ionic N, respectively, in the surface regions. Thermal repair of sputter damage necessitates segregation of bulk S to the surface, which, over repeated experiments, leads to gross cumulative damage to the bulk crystal. The desorption temperatures associated with recombinative desorption of atomic N from FeS(2){100} are significantly lower than those measured for Fe surfaces. This is linked to the inability of FeS(2){100} to dissociate N(2), but suggests that N(ads) will be significantly more able to react with other species than it is on Fe surfaces.     

All-electron path integral Monte Carlo simulations of warm dense matter: application to water and carbon plasmas

We develop an all-electron path integral Monte Carlo method with free-particle nodes for warm dense matter and apply it to water and carbon plasmas. We thereby extend path integral Monte Carlo studies beyond hydrogen and helium to elements with core electrons. Path integral Monte Carlo results for pressures, internal energies, and pair-correlation functions compare well with density functional theory molecular dynamics calculations at temperatures of (2.5-7.5)×10(5) K, and both methods together form a coherent equation of state over a density-temperature range of 3-12 g/cm(3) and 10(4)-10(9) K.     

Directing Foldamer Self-Assembly with a Cyclopropanoyl Cap

The rational design of self-assembling organic materials is extremely challenging due to the difficulty in precisely predicting solid-state architectures from first principles, especially if synthons are conformationally flexible. A tractable model system to study self-assembly was constructed by appending cyclopropanoyl caps to the N termini of helical α/β-peptide foldamers, designed to form both N−H⋅⋅⋅O and C^α−H⋅⋅⋅O hydrogen bonds, which then rapidly self-assembled to form foldectures (foldamer architectures). Through a combined analytical and computational investigation, cyclopropanoyl capping was observed to markedly enhance self-assembly in recalcitrant substrates and direct the formation of a single intermolecular N−H⋅⋅⋅O/C^α−H⋅⋅⋅O bonding motif in single crystals, regardless of peptide sequence or foldamer conformation. In contrast to previous studies, foldamer constituents of single crystals and foldectures assumed different secondary structures and different molecular packing modes, despite a conserved N−H⋅⋅⋅O/C^α−H⋅⋅⋅O bonding motif. DFT calculations validated the experimental results by showing that the N−H⋅⋅⋅O/C^α−H⋅⋅⋅O interaction created by the cap was sufficiently attractive to influence self-assembly. This versatile strategy to harness secondary noncovalent interactions in the rational design of self-assembling organic materials will allow for the exploration of new substrates and speed up the development of novel applications within this increasingly important class of materials.