Spectral asymptotics of hypersurfaces

Pseudodifferential operator techniques are employed to obtain higher-order spectral asymptotic results for integral operators on hypersurfaces in Euclidean space.Supported by a grant from the National Science Foundation.     

Wide-band intracavity microwave cells for laser-microwave double resonance spectroscopy

The design and use of two wide-band microwave cells suitable for double resonance experiments within the resonator of a cw CO₂-laser are described. The high sensitivity of the intracavity arrangement allowed very strong double-resonance and two-photon transition signals to be observed with high microwave-frequency scan-rates (1 GHz/min), and low microwave pumping powers (≦10mW/cm²). Both cells were used over a range 15 to 63 GHz, with transmission and VSWR measurements made over the range 18 to 26.5 GHz.     

A better step-off algorithm for the knapsack problem

The knapsack problem, maximize Σ^m_{i = 1}c_ix_i when Σ^m_{i = 1}a_ix_i?b for integers x_i?0, can be solved by the classical step-off algorithm. The algorithm develops a series of feasible solutions with ever-increasing objective values. We make a change in the problem so that the step-off algorithm produces a series of solutions of not necessarily increasing objective values. A point is reached when no better solutions can be found and the calculation is stopped.     

Shape memory and pseudoelasticity in metal nanowires

Structural reorientations in metallic fcc nanowires are controlled by a combination of size, thermal energy, and the type of defects formed during inelastic deformation. By utilizing atomistic simulations, we show that certain fcc nanowires can exhibit both shape memory and pseudoelastic behavior. We also show that the formation of defect-free twins, a process related to the material stacking fault energy, nanometer size scale, and surface stresses is the mechanism that controls the ability of fcc nanowires of different materials to show a reversible transition between two crystal orientations during loading and thus shape memory and pseudoelasticity.     

Differential Equations for Dyson Processes

We call a Dyson process any process on ensembles of matrices in which the entries undergo diffusion. We are interested in the distribution of the eigenvalues (or singular values) of such matrices. In the original Dyson process it was the ensemble of n×n Hermitian matrices, and the eigenvalues describe n curves. Given sets X₁,...,X_m the probability that for each k no curve passes through X_k at time _k is given by the Fredholm determinant of a certain matrix kernel, the extended Hermite kernel. For this reason we call this Dyson process the Hermite process. Similarly, when the entries of a complex matrix undergo diffusion we call the evolution of its singular values the Laguerre process, for which there is a corresponding extended Laguerre kernel. Scaling the Hermite process at the edge leads to the Airy process (which was introduced by Prähofer and Spohn as the limiting stationary process for a polynuclear growth model) and in the bulk to the sine process; scaling the Laguerre process at the edge leads to the Bessel process.In earlier work the authors found a system of ordinary differential equations with independent variable whose solution determined the probabilitieswhere A() denotes the top curve of the Airy process. Our first result is a generalization and strengthening of this. We assume that each X_k is a finite union of intervals and find a system of partial differential equations, with the end-points of the intervals of the X_k as independent variables, whose solution determines the probability that for each k no curve passes through X_k at time _k. Then we find the analogous systems for the Hermite process (which is more complicated) and also for the sine process. Finally we find an analogous system of PDEs for the Bessel process, which is the most difficult.Dedicated to Freeman Dyson on the occasion of his eightieth birthdayAcknowledgement We thank Kurt Johansson for sending us his unpublished notes on the extended Hermite kernel. This work was supported by National Science Foundation under grants DMS-0304414 (first author) and DMS-0243982 (second author).     

Reduced basis set for the gold atom in cluster complexes

To extend the metal cluster size used in interfacing between bulk metals and molecules in ab initio studies of molecular electronics and chemisorption, a reduced size atomic orbital basis set for the gold atom has been generated. Based on the SKBJ relativistic effective core potential set, the three component 5d Gaussian orbital basis set is completely contracted. Comparisons between the full and reduced basis set in Au atom clusters and cluster complexes for geometry, bond distances, dipole moments, atomic charges, spin, bond dissociation energies, lowest energy harmonic frequencies, electron affinities, ionization energies, and density of states distributions show the contracted set to be a viable replacement for the full basis set. This result is obtained using both the B3LYP and BPW91 exchange-correlation potentials in density functional theory.