Diffeomorphism finiteness for manifolds with ricci curvature andL n/2-norm of curvature bounded

Partially supported by NSF Grants.     

Non-perturbative approach to high-index-contrast variations in electromagnetic systems

We present a method that formally calculates exact frequency shifts of an electromagnetic field for arbitrary changes in the refractive index. The possible refractive index changes include both anisotropic changes and boundary shifts. Degenerate eigenmode frequencies pose no problems in the presented method. The approach relies on operator algebra to derive an equation for the frequency shifts, which eventually turn out in a simple and physically sound form. Numerically the equations are well-behaved, easy implementable, and can be solved very fast. Like in perturbation theory a reference system is first considered, which then subsequently is used to solve another related, but different system. For our method precision is only limited by the reference system basis functions and the error induced in frequency is of second order for first-order basis set error. As an example we apply our method to the problem of variations in the air-hole diameter in a photonic crystal fiber.     

Slow-light enhanced optical detection in liquid-infiltrated photonic crystals

Slow-light enhanced optical detection in liquid-infiltrated photonic crystals is theoretically studied. Using a scattering-matrix approach and the Wigner–Smith delay time concept, we show that optical absorbance benefits both from slow-light phenomena as well as a high filling factor of the energy residing in the liquid. Utilizing strongly dispersive photonic crystal structures, we numerically demonstrate how liquid-infiltrated photonic crystals facilitate enhanced light–matter interactions, by potentially up to an order of magnitude. The proposed concept provides strong opportunities for improving existing miniaturized absorbance cells for optical detection in lab-on-a-chip systems.     

Volume comparison and the σk-Yamabe problem

In this paper we study the problem of finding a conformal metric with the property that the kth elementary symmetric polynomial of the eigenvalues of its Weyl-Schouten tensor is constant. A new conformal invariant involving maximal volumes is defined, and this invariant is then used in several cases to prove existence of a solution, and compactness of the space of solutions (provided the conformal class admits an admissible metric). In particular, the problem is completely solved in dimension four, and in dimension three if the manifold is not simply connected.     

A first-principles study of surface and subsurface H on and in Ni(1 1 1): diffusional properties and coverage-dependent behavior

Periodic, self-consistent, density functional theory (GGA-PW91) calculations are performed for both surface and subsurface atomic hydrogen on and in Ni(1 1 1). At a low coverage (θ=0.25 ML), the binding energies (BEs) of a hydrogen atom in surface fcc, subsurface octahedral (first layer), and subsurface octahedral (second layer) sites are −2.89, −2.18, and −2.11 eV, respectively. The activation energy barriers for hydrogen diffusion from the surface to the first subsurface layer and from the first to the second subsurface layer are estimated to be 0.88 and 0.52 eV, respectively. In the entire coverage range studied, hydrogen occupies surface fcc and hcp sites and subsurface octahedral sites. In addition, the magnitude of the BE per hydrogen atom and the magnetization of the nickel slabs both decrease as hydrogen coverage increases. Vibrational frequencies of hydrogen at various surface and subsurface sites are calculated and are in reasonable agreement with experimental data. A phase stability calculation with a 2 × 2 surface unit cell shows that a p(2 × 2)-2H overlayer structure (θ=0.5 ML) and a p(1 × 1)-1H structure (θ=1.0 ML) are stable at low hydrogen pressures, in agreement with numerous experimental results. A very large increase in pressure is required to populate subsurface sites. After such an increase occurs, the first subsurface layer is filled completely.     

Nonparametric estimation of distributions with categorical and continuous data

In this paper we consider the problem of estimating an unknown joint distribution which is defined over mixed discrete and continuous variables. A nonparametric kernel approach is proposed with smoothing parameters obtained from the cross-validated minimization of the estimator's integrated squared error. We derive the rate of convergence of the cross-validated smoothing parameters to their ‘benchmark’ optimal values, and we also establish the asymptotic normality of the resulting nonparametric kernel density estimator. Monte Carlo simulations illustrate that the proposed estimator performs substantially better than the conventional nonparametric frequency estimator in a range of settings. The simulations also demonstrate that the proposed approach does not suffer from known limitations of the likelihood cross-validation method which breaks down with commonly used kernels when the continuous variables are drawn from fat-tailed distributions. An empirical application demonstrates that the proposed method can yield superior predictions relative to commonly used parametric models.     

The response of microstructure to processing in a series of poly(siloxaneimide) copolymers

Poly(siloxaneimide) (PSI) segmented copolymers exhibit organized microdomains if the blocks are sufficiently incompatible. As with neat diblock and triblock copolymers, the processing route employed to prepare films of PSI materials is expected to influence the dimensions and/or morphology of the resultant microstructure. In this work, small-angle neutron scattering (SANS) is utilized to characterize the disordered microstructure found in films of a series of PSI copolymers which are subjected to solvent casting and various thermal treatments. Microstructural dimensions such as the periodicity and correlation length are deduced from the Teubner-Strey (TS) model for disordered microemulsions. The scattering intensity of each copolymer up to q = 5.0 nm^?1, where q is the scattering vector, is found to scale as q^?2.8+?0.1. Results indicate that processing the materials as cast films or as melt-pressed films allowed to cool slowly has a small, but discernible, effect on microstructural characteristics. SANS profiles of films quenched from elevated temperatures reveal a clear transition in microdomain periodicity, which correlates well with the glass transition temperature of the imide microphase in these and other materials of similar chemical structure. © 1993 John Wiley & Sons, Inc.     

On the cover time of random walks on graphs

This article deals with random walks on arbitrary graphs. We consider the cover time of finite graphs. That is, we study the expected time needed for a random walk on a finite graph to visit every vertex at least once. We establish an upper bound ofO(n ²) for the expectation of the cover time for regular (or nearly regular) graphs. We prove a lower bound of (n logn) for the expected cover time for trees. We present examples showing all our bounds to be tight.Mike Saks was supported by NSF-DMS87-03541 and by AFOSR-0271. Jeff Kahn was supported by MCS-83-01867 and by AFOSR-0271.