On a theorem of Laurent Schwartz

We give a proof of a theorem of Schwartz on Borel graphs for linear transforms between Banach spaces, completely different from the original one.     

Ideal rate of collision of cylinders in simple shear flow

The collision of particles influences the behavior of suspensions through the formation of aggregates for adhesive particles or through the contributions of solid-body contacts to the stress for nonadhesive particles. The simplest estimate of the collision rate, termed the ideal collision rate, is obtained when particles translate and rotate with the flow but have no hydrodynamic or colloidal interactions. Smoluchowski calculated the ideal collision frequency of spherical particles in 1917. So far, little work has been done to understand rate of collision for nonspherical particles. In this work, we calculate the ideal collision rate for cylindrical particles over a broad range of particle aspect ratios r defined as the ratio of length to diameter. Monte Carlo simulations are performed with initial relative positions and orientations that model the rate of approach of noninteracting particles following Jeffery orbits with several choices of the orbit distribution. The role of rotational motion of particles on collision frequency is elucidated by comparing the ideal collision rate calculations with similar calculations for nonrotating particles. It is shown that the ratio of the collision rate of cylinders to that of spheres that circumscribe the cylinders is proportional to 1/rr(e) for r ? 1 and r(e) for r ? 1. Here, r(e) is the effective aspect ratio defined as the aspect ratio of a spheroid having the same period of rotation as the cylinder. The effective aspect ratio of the cylindrical particles was determined using finite element calculations of the torque on nonrotating cylinders with their axes parallel to the velocity and velocity gradient directions. In addition to deriving the total collision rate, we categorize collisions as side-side, edge-side, and face-edge based on the initial point of contact. Most collisions are found to be side-edge for r ? 1 and face-edge for r ? 1, suggesting that nonlinear aggregates will develop if particles stick at the point of first contact.     

Large deviations and stochastic flows of diffeomorphisms

Summary Previous results in the theory of large deviations for additive functionals of a diffusion process on a compact manifold M are extended and then applied to the analysis of the Lyapunov exponents of a stochastic flow of diffeomorphisms of M. An approximation argument relates these results to the behavior near the diagonal Δ in M ² of the associated two point motion. Finally it is shown, under appropriate non-degeneracy conditions, that the two-point motion is ergodic on M ²-Δ if the top Lyapunov exponent is positive. At the period when this research was initiated, both authors where guests of the I.M.A. in Minneapolis. The first author was at Aberdeen University, Scotland when this article was prepared. Throughout the period of this research, the second author has been partially supported by N.S.F. grant DMS-8611487 and ARO grant DAAL03-86-K-171     

Membraneless, room-temperature, direct borohydride/cerium fuel cell with power density of over 0.25 W/cm2

The widespread adoption and deployment of fuel cells as an alternative energy technology have been hampered by a number of formidable technical challenges, including the cost and long-term stability of electrocatalyst and membrane materials. We present a microfluidic fuel cell that overcomes many of these obstacles while achieving power densities in excess of 250 mW/cm(2). The poisoning and sluggish reaction rate associated with CO-contaminated H(2) and methanol, respectively, are averted by employing the promising, high-energy density fuel borohydride. The high-overpotential reaction of oxygen gas at the cathode is supplanted by the high-voltage reduction of cerium ammonium nitrate. Expensive, ineffective membrane materials are replaced with laminar flow and a nonselective, porous convection barrier to separate the fuel and oxidant streams. The result is a Nafion-free, room-temperature fuel cell that has the highest power density per unit mass of Pt catalyst employed for a non-H(2) fuel cell, and exceeds the power density of a typical H(2) fuel cell by 50%.     

Non-divergence form operators and variations on Yau's explosion criterion

Let † denote the standard (i.e., Levi-Civita) Laplacian for some non-compact, connected, complete, separable Riemannian manifild M. In a much cited article, Yau [5] proved that when the Ricci curvature is bounded uniformly below, then the only bounded solution to the heat equation ∂ _t μ=Δμ on [0, ∞) × M which vanishes at t=0 is the one which vanishes evarywhere. Equivalently, no matter where it starts, Brownian motion on M never explodes. Yau's original statement was improved or extended in various directions by a long list of authors. With this paper, the present author joins the list.     

On the ergodic properties of Glauber dynamics

We show that if there is an infinite volume Gibbs measure which satisfies a logarithmic Sobolev inequality with local coefficients of moderate growth, then the corresponding stochastic dynamics decays to equilibrium exponentially fast in the uniform norm.