A Distribution Relation on Elliptic Curves

In this paper, we give an elementary proof of a curious identityof elliptic functions. It is very similar to a beautiful proofgiven by Coates of a different identity. The result was stronglymotivated by Wildeshaus' generalisation of Zagier's conjecture.1991 Mathematics Subject Classification 11G16, 33E05.     

An experimental and numerical investigation of container filling with viscous liquids

This work is concerned with a study of container filling, with particular reference to the food industry. A computer code was developed and an experimental rig was built, the main purpose being to validate the software. The computational fluid dynamic (CFD) code, called GENSMAC, was specifically designed for relatively slow viscous flow and was capable of capturing multiple free surfaces. This paper focuses on the design of the experimental rig and how it functions. The visual output of the code is then compared with high‐speed photographic shots of glucose syrup being jetted into a tub for a selected number of flow regimes. Copyright © 1999 John Wiley & Sons, Ltd.     

High-speed visualization of flame propagation in explosions

Flow visualization data is presented to describe the structure of flames propagating in methane-air explosions in semi-confined enclosures. The role of turbulence is well established as a mechanism for increasing burning velocity by fragmenting the flame front and increasing the surface area of flames propagating in explosions. This area increase enhances the burning rate and increases the resultant explosion overpressure. In real situations, such as those found in complex process plant areas offshore, the acceleration of a flame front results from a complex interaction between the moving flame front and the local blockage caused by presence of equipment. It is clear that any localised increase in flame burn rate and overpressure would have important implications for any adjacent plant and equipment and may lead to an escalation process internal to the overall event. To obtain the information required to quantify the role of obstacles, it is necessary to apply a range of sophisticated laser-based, optical diagnostic techniques. This paper describes the application of high-speed, laser-sheet flow visualization and digital imaging to record the temporal development of the flame structure in explosions. Data is presented to describe the interaction of the propagating flame with a range of obstacles for both homogeneous and stratified mixtures. The presented image sequences show the importance of turbulent flow structures in the wake of obstacles for controlling the mixing of a stratified concentration field and the subsequent flame propagation through the wake. The data quantifies the flame speed, shape and area for a range of obstacle shapes.     

An elementary proof of a distribution relation¶on elliptic curves

We give another elementary proof of a certain identity of elliptic functions arising from the K-theory of elliptic curves and Wildeshaus's generalisation of Zagier's conjectures. This proof consists of a calculation with the q-expansions, and is offered in the hope that its more explicit flavour may be generalised to other situations. Received: 7 December 1999 / Revised version: 3 July 2000     

Compactification of the universal Picard over the moduli of stable curves

This article provides two different, but closely related, moduli problems, which in characteristic zero provide a type of compactification of the universal Picard over the moduli of stable curves. Although neither is of finite type, both are limits of a sequence of stacks, each of which is a separated algebraic stack of finite type. We discuss relations to previous compactifications and partial compactifications, give a number of examples related to this compactification, and work out the structure of its fibres over certain fixed curves. Some applications are also discussed. Received January 5, 1998; in final form April 1, 1999 / Published online July 3, 2000     

Off resonance ac mode force spectroscopy and imaging with an atomic force microscope

Using an off resonance ac technique in ultrahigh vacuum we have directly measured the force-gradient interaction characteristics of a gold tip and sample and demonstrated a new atomic force microscope imaging mode with the tip located very close to the surface. The method involves the application of a small sinusoidal oscillating force to the tip via a magnetic field created by a conducting coil which interacts with a magnetic particle glued on the backside of the cantilever. By measuring the change in amplitude during the approach and retraction of the sample we have a continuous and accurate measure of the force gradient. The interaction potential is thus found without the need for complex analysis as is necessary in the case of the commonly used technique of measuring frequency shifts.     

The resistivity of thin gold films

Results are obtained on the D.C. electrical resistivity of evaporated gold films on KBr substrates as a function of thickness and temperature. These show that the conduction electrons are not specularly reflected at the sample surface but nevertheless the temperature dependent part of the resistivity is independent of sample thickness. Comparison of the results with Fuchs-Sondheimer theory for the effects of sample thickness upon the resistivity yield a maximum specularity of approximately 0.64 for <100> films on KBr.     

Effects of dung and urine amendments on the isotopic content of N(2)O released from grasslands

The temporal and diurnal changes in nitrous oxide (N(2)O) fluxes were measured between 29(th) September and 2(nd) November 1999 from urine and dung patches from cattle deposited on grazed grassland. The delta(15)N and delta(18)O values of the N(2)O emitted from soil from both treatments were examined on four occasions during this period. The diurnal fluxes of N(2)O were measured by a chamber technique that provides hourly measurement of N(2)O fluxes. The (15)N and (18)O analysis of N(2)O were determined by isotope ratio mass spectrometry. N(2)O fluxes from the excreta patches were large, with peak emissions up to 1893 ng N m(-2) s(-1) occurring after heavy precipitation, measured one month after the treatment applications. Emissions from the urine patches were significantly greater than from the dung. The results showed that excretal patches are an important source of atmospheric N(2)O. The flux pattern showed a strong diurnal variation with maximum fluxes generally occurring in late afternoon or early morning, and generally not in phase with the soil temperature changes. The isotopic content of (15)N and (18)O in the N(2)O showed a similar trend to that of the N(2)O flux. The (15)N and (18)O values of the N(2)O emitted from the soil indicated that denitrification was the major process involved. After heavy precipitation on the 6(th) October, the larger delta(15)N and delta(18)O values suggested a consumption of the N(2)O by total denitrification.     

Nucleophilic displacement reactions on fluorine. I. General phenomenon and lone-pair nucleophiles

S_N2 displacement reactions involving halogen atoms are progressively more facile in going from F to I. Diverse and hitherto largely mechanistically uncorrelated examples of S_N2(F) reactions are presented and its is suggested that they form part of a general phenomenon in fluorine chemistry.