Fifteen consecutive integers with exactly four prime factors

We describe a successful search for a sequence of fifteen consecutive integers, each the product of exactly four prime factors. Fifteen is best possible.

     

Pulsed laser deposition of nanostructured Ag films

Ultra-thin (0.5-5 nm) films of Ag have been prepared by pulsed laser deposition in vacuum using a 26 ns KrF excimer laser at 1 J cm⁻². The deposition was controlled using a Langmuir ion probe and a quartz crystal thickness monitor. Transmission electron microscopy showed that the films are not continuous, but are structured on nanometer size scales. Optical absorption spectra showed the expected surface plasmon resonance feature, which shifted to longer wavelength and increased in strength as the equivalent film thickness was increased. It is shown that Maxwell Garnett effective medium theory can be used to calculate the main features of optical absorption spectra.     

Dynamical fermions in lattice gauge theories: Metropolis and Langevin techniques in two dimensions

A variety of techniques for the inclusion of dynamical fermions in lattice gauge theory is examined. Three pseudo-fermionic techniques that have the characteristics desirable for an unquenched simulation of four-dimensional QCD are studied in detail. Langevin and Metropolis pseudo-fermionic techniques are implemented for a 64×64 lattice on the Distributed Array Processor and their relative merits examined both for free fermions and the lattice Schwinger model.     

On a nonlinear multigrid algorithm with primal relaxation for the image total variation minimisation

Digital image restoration has drawn much attention in the recent years and a lot of research has been done on effective variational partial differential equation models and their theoretical studies. However there remains an urgent need to develop fast and robust iterative solvers, as the underlying problem sizes are large. This paper proposes a fast multigrid method using primal relaxations. The basic primal relaxation is known to get stuck at a ‘local’ non-stationary minimum of the solution, which is usually believed to be ‘non-smooth’. Our idea is to utilize coarse level corrections, overcoming the deadlock of a basic primal relaxation scheme. A further refinement is to allow non-regular coarse levels to correct the solution, which helps to improve the multilevel method. Numerical experiments on both 1D and 2D images are presented.     

Immobilized polysiloxanes as stationary phases for high-performance liquid chromatography and solid phase extraction

Polysiloxanes immobilized onto the surfaces of porous silica particles have proven to be good stationary phases for the separation of multiresidues of pesticides and their metabolic/degradation products by reversed-phase high-performance liquid chromatography (RP-HPLC). Similar materials have proven effective for pre-concentration and clean-up procedures using solid phase extraction. The present paper describes the preparation and some applications of several of these packing materials.     

Critical evaluation of gas standards based on hydrogen

Summary Six commercial hydrogen standards containing helium, oxygen, nitrogen, methane and carbon dioxide at trace levels were analyzed by gas chromatography, for periods up to 105 days. The concentrations, in the range of 0–120 μmol mol⁻¹, were stable (with the exception of oxygen) but often significantly different from the certified values provided by the suppliers, especially for helium, oxygen and nitrogen. Concurrently, some experiments were carried out to verify the stability of gas mixtures based on hydrogen stored in cylinders submitted to different chemical and physical treatments. The causes that led to the deviations observed, as well as the decreases in oxygen, are discussed. Presented at the 21^st ISC held in Stuttgart, Germany, 15^th–20^th September, 1996.     

Modified Equations for Stochastic Differential Equations

We describe a backward error analysis for stochastic differential equations with respect to weak convergence. Modified equations are provided for forward and backward Euler approximations to Itô SDEs with additive noise, and extensions to other types of equation and approximation are discussed.