The Halton, Sobol, and Faure sequences and the Braaten-Weller construction of the generalized Halton sequence are studied in order to assess their applicability for the quasi Monte Carlo integration with large number of variates. A modification of the Halton sequence (the Halton sequence leaped) and a new construction of the generalized Halton sequence are suggested for unrestricted number of dimensions and are shown to improve considerably on the original Halton sequence. Problems associated with estimation of the error in quasi Monte Carlo integration and with the selection of test functions are identified. Then an estimate of the maximum error of the quasi Monte Carlo integration of nine test functions is computed for up to 400 dimensions and is used to evaluate the known generators mentioned above and the two new generators. An empirical formula for the error of the quasi Monte Carlo integration is suggested.

There is growing evidence that fine airborne particulates could play the most important role in determining health effects. The aim of this work was to investigate the number concentration and size distributions of particulates in the exhausts of diesel vehicles (mainly buses) of different ages and make, operating under different loads. Particlesizing instruments used were the Scanning Mobility Particle Sizer (SMPS) and Aerodynamic Particle Sizer (APS). The average particle number concentration of the exhausts was in the range (0.7-3.9) × 10 7 cm -3 in the SMPS range (0.0075-0.304 µm) and (0.3-32) × 10 3 cm -3 in the APS range (0.5-30 µm). In most cases, particle number concentrations increased with the increased power output from the engine and, in both SMPS and APS ranges, varied significantly within each group of vehicles, but the differences between the groups were small. For individual vehicles, there was no relation between emissions in the smaller and larger particle ranges. Emission characteristics did not appear to be correlated with engine model or age. The implications of these findings to particle emission testing and control as well as to exposure and risk analysis are discussed.

A possibl e way fro m qua ntum me chanics to classical mechanics can b e achie ved w ith an exp onential substitutio n used in the Schr odin ger equation , and then considerin g the classical limit . T his gives a picture of classical Ûuid and an ensemble of classical tra jectories. In di˜erence from this approach to the classical limit, w hile utilisi ng the same substitution , w e assume a minimum uncertainty w ave packet. It is show n that this approach to the classical limit of quantum mechanics yields a single traj ectory traced by the centroid of the minimum uncertainty w ave packet. T he momentum and the centroid of such packet satisf y the classical H amilton { J acobi equation. PACS numb ers: 03.65.Bz

Monte Carlo and quasi Monte Carlo (ie using low discrepancy sequences) methods (Bratley & Fox 1988, Joe & Sloan 1993, Krommer & Ueberhuber 1994, Lyness 1989, Niederreiter 1978, Sloan & Kachoyan 1987) are used to approximate an integral by the average value of function samples:[EQUATION 1]where v is the volume of integration (taken here to be the unit multi-dimensional cube) and x is a vector with an element for each of the dimensions of the multidimensional space. In the case of Monte Carlo the points x p are chosen at random, while in quasi Monte Carlo the points are chosen to cover the integration volume as uniformly as possible. For numerical integration over a large number of dimensions these two techniques are often the only methods available.

Abstract. It is shown that the semi-classical limit of solutions to the Klein-Gordon equation gives the particle probability density that is in direct proportion to the inverse of the particle velocity. It is also shown that in the case of the Dirac equation a different result is obtained.

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