Approximation by planar elastic curves

We give an algorithm for approximating a given plane curve segment by a planar elastic curve. The method depends on an analytic representation of the space of elastic curve segments, together with a geometric method for obtaining a good initial guess for the approximating curve. A gradient-driven optimization is then used to find the approximating elastic curve.Comment: 18 pages, 10 figures. Version2: new section 5 added (conclusions and discussions

Polar plots of diamond surface energy

Diamond surface energy sigma_hkl determines crystal habit. We discuss three aspects of a paper by Terentiev (1991). Firstly, we compare Terentiev's algorithm for exact sigma_hkl with the analytic solution for h <= k <= l and h + k < l. Secondly, we show that the general formula given by Terentiev should be interpreted probabilistically in order to be self consistent. Finally, we replicate in principle the simulation results for sigma_hkl in a nickel melt using nothing more than Matlab routines

Fe(VI) as a New Chemical Agent for Treatment of Landfill Leachate

More than 50% municipal solid waste (MSW) is being disposed of within landfills in the United States. Despite many benefits, landfills inevitably produce highly contaminated wastewater - landfill leachate. Landfill leachate represents one of the most challengeable wastewaters in the world. Refractory organic matters and ammonia nitrogen are two major persistent traditional contaminants in landfill leachate. The contaminants ought to be properly removed prior to discharge into natural water resources. Ferrate (FeO42-) is a potential water treatment chemical agent in which iron is in its + 6 oxidation state. In engineering practice, Fe(VI) can concurrently function as oxidant, coagulant, adsorbent and disinfectant. Up to date, Fe(VI) has been intensively studied for treatment of drinking water, wastewater, and ballast water. However, few attempts have been made to apply Fe(VI) for treatment of landfill leachate. The overall objective of this thesis was to evaluate ferrate as a new oxidizing agent for treatment of landfill leachate, with an emphasis on removal of two major leachate contaminants - refractory organic matters and ammonia nitrogen. The central hypothesis is that Fe(VI) has an adequately high oxidation capacity to simultaneously degrade refractory organic matters and ammonia in landfill leachate through oxidation, thereby providing a viable alternative for traditional landfill leachate treatment methods. To achieve the overall objective, bench scale tests were conducted in batch mode to pursue four specific objectives: 1) to evaluate the effects of Fe(VI) dose and pH on COD removal and NH3-N removal; 2) to evaluate the effect of chloride on Fe(VI) treatment of landfill leachate; 3) to evaluate the effect of initial NH3-N on Fe(VI) treatment of landfill leachate; and 4) to determine N transformation pathway during Fe(VI) oxidation of NH3- N under different conditions. Results show that Fe(VI) could simultaneously reduce COD and NH3-N under different experimental conditions. In this study, up to 60% COD and 70% NH3-N were removed. Generally, the treatment efficiency was increased with the increasing dimensionless oxidant demand (DOD). COD removal was due to Fe(VI)-induced oxidation and iron sludge-induced coagulation/adsorption, while ammonia was reduced as a result of Fe(VI) oxidation, in addition to volatilization at high pH. The oxidized NH3-N was oxidized to nitrogen gas and NO3-N. High pH and high DOD appeared to favor the NO3-N formation. At any condition, little NO2--N was produced. Cl- (2,100 - 14,880 mg/L) played a complex role in the treatment. It somewhat increased COD removal at pH 4-5, but slightly slowed down the COD reduction at pH 9. At pH 4-5, a low Cl- level (2,100 - 5,000 mg/L) could inhibit the NH3-N removal; however, more Cl-improved NH3-N reduction at a high Cl- range (5,000 - 14,880 mg/L). In contrast, at pH 9, more Cl-favored the NH3-N removal. More initial NH3-N could compete with COD for Fe(VI), thereby inhibiting the COD removal. Our results provide valuable information regarding the treatability of Fe(VI) for landfill leachate, and demonstrate that Fe(VI) is a potential treatment chemical for landfill leachate

Graph colouring for office blocks

The increasing prevalence of WLAN (wireless networks) introduces the potential of electronic information leakage from one company's territory in an office block, to others due to the long-ranged nature of such communications. BAE Systems have developed a system ('stealthy wallpaper') which can block a single frequency range from being transmitted through a treated wall or ceiling to the neighbour. The problem posed to the Study Group was to investigate the maximum number of frequencies ensure the building is secure. The Study group found that this upper bound does not exist, so they were asked to find what are "good design-rules" so that an upper limit exists

B\'ezier curves that are close to elastica

We study the problem of identifying those cubic B\'ezier curves that are close in the L2 norm to planar elastic curves. The problem arises in design situations where the manufacturing process produces elastic curves; these are difficult to work with in a digital environment. We seek a sub-class of special B\'ezier curves as a proxy. We identify an easily computable quantity, which we call the lambda-residual, that accurately predicts a small L2 distance. We then identify geometric criteria on the control polygon that guarantee that a B\'ezier curve has lambda-residual below 0.4, which effectively implies that the curve is within 1 percent of its arc-length to an elastic curve in the L2 norm. Finally we give two projection algorithms that take an input B\'ezier curve and adjust its length and shape, whilst keeping the end-points and end-tangent angles fixed, until it is close to an elastic curve.Comment: 13 pages, 15 figure