X-ray determination of the mean amplitudes of vibration and debye temperatures of some rare earth monochalcogenides

The x-ray diffraction intensities of Bragg reflections have been measured at room temperature for thulium selenide, samarium sulphide, samarium selenide and samarium telluride. On the basis of a common amplitude approximation, the Debye-Waller factor, the mean amplitude of vibration and the Debye temperature have been evaluated. The values of the Debye temperatures and mean amplitudes of vibration are 176±16°K, 0·185 ± 0·017 Å (TmSe), 155 ± 7°K, 0·244 ± 0·012 Å (SmS), 153 ± 14°K, 0·221 ± 0·020 Å (SmSe) and 151 ± 20°K, 0·204 ± 0·027 Å (SmTe).     

Random Perturbations of Canards

We consider the effect of random perturbations on canards. We find the appropriate size of the random perturbations to produce a random selection of a regular duck versus a headless duck. The appropriate limit theorem, in the appropriate topology, is proved. This material is based upon work supported by the National Science Foundation under Grant Nos. 0305925 and 0604249. The author would also like to thank Professor Jeff Moehlis of the Department of Mechanical and Environmental Engineering at UC Santa Barbara for a number of useful discussions about canards.     

A boundary layer theory for diffusively perturbed transport around a heteroclinic cycle

We consider a two‐dimensional transport equation subject to small diffusive perturbations. The transport equation is given by a Hamiltonian flow near a compact and connected heteroclinic cycle. We investigate approximately harmonic functions corresponding to the generator of the perturbed transport equation. In particular, we investigate such functions in the boundary layer near the heteroclinic cycle; the space of these functions gives information about the likelihood of a particle moving a mesoscopic distance into one of the regions where the transport equation corresponds to periodic oscillations (i.e., a “well” of the Hamiltonian). We find that we can construct such approximately harmonic functions (which can be used as “corrector functions” in certain averaging questions) when certain macroscopic “gluing conditions” are satisfied. This provides a different perspective on some previous work of Freidlin and Wentzell on stochastic averaging of Hamiltonian systems. © 2004 Wiley Periodicals, Inc.     

Targetted stochastic matrix inversion

The solution of linear systems by stochastic techniques is investigated as an alternative to the widely used elimination and relaxation procedures. The Pan-Reif algorithm is used to allow any non-singular matrix to be stochastically inverted. New algorithms are devised to overcome some of the efficiency problems of the original Von Neumann-Ulam method and its refinements. The method is tested on some simple physical problems to gauge its effectiveness.     

Recovery rates in investment-grade pools of credit assets: A large deviations analysis

We consider the effect of recovery rates on a pool of credit assets. We allow the recovery rate to depend on the defaults in a general way. Using the theory of large deviations, we study the structure of losses in a pool consisting of a continuum of types. We derive the corresponding rate function and show that it has a natural interpretation as the favored way to rearrange recoveries and losses among the different types. Numerical examples are also provided.