Hadamard's inequality and Trapezoid Rules for the Riemann-Stieltjes integral

We obtain Midpoint and Trapezoid Rules for the Riemann-Stieltjes integral which engender a natural generalization of Hadamard's integral inequality. Error terms are then obtained for this Riemann-Stieltjes Trapezoid Rule and other related quadrature rules.     

A spatially dependent model for washing wool

We analytically model the transport of dirt in the industrial washing of wool using the advection–diffusion equation in two dimensions. Separation of variables leads to a Sturm–Liouville problem where the analytic solution reveals how contamination is distributed both along and down the wool and indicates the operating parameter regimes that optimise the cleaning efficiency.     

Linking [2]rotaxane wheels to create a new type of metal organic rotaxane framework

Three new [2]rotaxanes with aromatic nitrogen donors appended to the crown ether wheel have been synthesized and used as ligands to coordinate Cd(II) ions. One of these yields a new type of 2-periodic, metal organic rotaxane framework in which the wheel rather than the axle is used to link the metal nodes.     

Enantioselective synthesis of aminobenzazepinones

Enantioselective synthesis of constrained trans-aminobenzazepinone utilizing palladium-mediated Jeffery-Heck reaction and rhodium(II) catalyzed asymmetric hydrogenation as key steps are described. Diverse functional groups such as alkyl, aryl, basic or amino acid moieties were introduced with minimal racemization.     

Competitive Tendering Strategies in the Bus Industry

School bus services within the UK are subsidised by the local Passenger Transport Executive and bus operators tender for the contracts, with the lowest price normally winning. Analysis of the bids in one district showed that the companies had pricing strategies, which could be estimated from freely available data when they were simple. The difficulties arose when competitors segmented the market.     

Towards Diocesan Planning

An information system was developed for Blackburn diocese in the early 1970s, using data collected from several sources within the church. These data were then collated and made available to decision-makers within the diocese, at parochial, deanery and diocesan level. Using this information, a mathematical model of a parish as a system was constructed and examples are given of the uses of this in studying the consequences of changes in the characteristics of a parish, in an attempt to utilize fully the limited resources available to the diocese.