The Klein-Gordon equation on the half line: a Riemann-Hilbert approach

We solve an initial-boundary problem for the Klein-Gordon equation on the half line using the Riemann-Hilbert approach to solving linear boundary value problems advocated by Fokas. The approach we present can be also used to solve more complicated boundary value problems for this equation, such as problems posed on time-dependent domains. Furthermore, it can be extended to treat integrable nonlinearisations of the Klein-Gordon equation. In this respect, we briefly discuss how our results could motivate a novel treatment of the sine-Gordon equation.

Spectral analysis of the elliptic sine-Gordon equation in the quarter plane

We study the elliptic sine-Gordon equation in the quarter plane using a spectral transform approach. We determine the Riemann-Hilbert problem associated with well-posed boundary value problems in this domain and use it to derive a formal representation of the solution. Our analysis is based on a generalization of the usual inverse scattering transform recently introduced by Fokas for studying linear elliptic problems.

Initial boundary value problems for the nonlinear Schrödinger equation

A new spectral method for solving initial boundary value problems for linear and integrable nonlinear partial differential equations in two independent variables is applied to the nonlinear Schrödinger equation and to its linearized version in the domain {x≥l(t), t≥0}. We show that there exist two cases: (a) if l″(t)<0, then the solution of the linear or nonlinear equations can be obtained by solving the respective scalar or matrix Riemann-Hilbert problem, which is defined on a time-dependent contour; (b) if l″(t)>0, then the Riemann-Hilbert problem is replaced by a respective scalar or matrix problem on a time-independent domain. In both cases, the solution is expressed in a spectrally decomposed form.

Predictive empirical modelling of Orobanche life cycle and seed ecology

Abstract 1.7.4

Evaluation of NRC, UC Davis and ADAS approaches to estimate the metabolizable energy values of feeds at maintenance energy intake from equations utilizing chemical assays and in vitro determinations

Feed samples received by commercial analytical laboratories are often undefined or mixed varieties of forages, originate from various agronomic or geographical areas of the world, are mixtures (e.g., total mixed rations) and are often described incompletely or not at all. Six unified single equation approaches to predict the metabolizable energy (ME) value of feeds determined in sheep fed at maintenance ME intake were evaluated utilizing 78 individual feeds representing 17 different forages, grains, protein meals and by-product feedstuffs. The predictive approaches evaluated were two each from National Research Council [National Research Council (NRC), Nutrient Requirements of Dairy Cattle, seventh revised ed. National Academy Press, Washington, DC, USA, 2001], University of California at Davis (UC Davis) and ADAS (Stratford, UK). Slopes and intercepts for the two ADAS approaches that utilized in vitro digestibility of organic matter and either measured gross energy (GE), or a prediction of GE from component assays, and one UC Davis approach, based upon in vitro gas production and some component assays, differed from both unity and zero, respectively, while this was not the case for the two NRC and one UC Davis approach. However, within these latter three approaches, the goodness of fit (r(2)) increased from the NRC approach utilizing lignin (0.61) to the NRC approach utilizing 48 h in vitro digestion of neutral detergent fibre (NDF:0.72) and to the UC Davis approach utilizing a 30 h in vitro digestion of NDF (0.84). The reason for the difference between the precision of the NRC procedures was the failure of assayed lignin values to accurately predict 48 h in vitro digestion of NDF. However, differences among the six predictive approaches in the number of supporting assays, and their costs, as well as that the NRC approach is actually three related equations requiring categorical description of feeds (making them unsuitable for mixed feeds) while the ADAS and UC Davis approaches are single equations, suggests that the procedure of choice will vary dependent Upon local conditions, specific objectives and the feedstuffs to be evaluated. In contrast to the evaluation of the procedures among feedstuffs, no procedure was able to consistently discriminate the ME values of individual feeds within feedstuffs determined in vivo, suggesting that the quest for an accurate and precise ME predictive approach among and within feeds, may remain to be identified. (C) 2004 Elsevier B.V. All rights reserved.

Study of the lactation curve in dairy cattle on farms in Central Mexico

Accurate knowledge of lactation curves has an important relevance to management and research of dairy production systems. A number of equations have been proposed to describe the lactation curve, the most widely applied being the gamma equation. The objective of this work was to compare and evaluate candidate functions for their predictive ability in describing lactation curves from central Mexican dairy cows reared under 2 contrasting management systems. Five equations were considered: Gaines ( exponential decay), Wood ( gamma equation), Rook ( Michaelis-Menten x exponential), and 2 more mechanistic ones (Dijkstra and Pollott). A database consisting of 701 and 1283 records of cows in small-scale and intensive systems, respectively, was used in the analysis. Before analysis, the database was divided into 6 groups representing first, second, and third and higher parity cows in both systems. In all cases except second and above parity cows in small-scale systems, all models improved on the Gaines equation. The Wood equation explained much of the variation, but its parameters do not have direct biological interpretation. Although the Rook equation fitted the data well, some of the parameter estimates were not significant. The Dijkstra equation consistently gave better predictions, and its parameters were usually statistically significant and lend themselves to physiological interpretation. As such, the differences between systems and parity could be explained due to variations in theoretical initial milk production at parturition, specific rates of secretory cell proliferation and death, and rate of decay, all of which are parameters in the model. The Pollott equation, although containing the most biology, was found to be over-parameterized and resulted in nonsignificant parameter estimates. For central Mexican dairy cows, the Dijkstra equation was the best option to use in describing the lactation curve.

Evaluation of a predictive model for Mycosphaerella graminicola for economic and environmental benefits

A method was developed to evaluate crop disease predictive models for their economic and environmental benefits. Benefits were quantified as the value of a prediction measured by costs saved and fungicide dose saved. The value of prediction was defined as the net gain made by using predictions, measured as the difference between a scenario where predictions are available and used and a scenario without prediction. Comparable 'with' and 'without' scenarios were created with the use of risk levels. These risk levels were derived from a probability distribution fitted to observed disease severities. These distributions were used to calculate the probability that a certain disease induced economic loss was incurred. The method was exemplified by using it to evaluate a model developed for Mycosphaerella graminicola risk prediction. Based on the value of prediction, the tested model may have economic and environmental benefits to growers if used to guide treatment decisions on resistant cultivars. It is shown that the value of prediction measured by fungicide dose saved and costs saved is constant with the risk level. The model could also be used to evaluate similar crop disease predictive models.

A predictive model for early-warning of Septoria leaf blotch on winter wheat

Disease-weather relationships influencing Septoria leaf blotch (SLB) preceding growth stage (GS) 31 were identified using data from 12 sites in the UK covering 8 years. Based on these relationships, an early-warning predictive model for SLB on winter wheat was formulated to predict the occurrence of a damaging epidemic (defined as disease severity of 5% or > 5% on the top three leaf layers). The final model was based on accumulated rain > 3 mm in the 80-day period preceding GS 31 (roughly from early-February to the end of April) and accumulated minimum temperature with a 0A degrees C base in the 50-day period starting from 120 days preceding GS 31 (approximately January and February). The model was validated on an independent data set on which the prediction accuracy was influenced by cultivar resistance. Over all observations, the model had a true positive proportion of 0.61, a true negative proportion of 0.73, a sensitivity of 0.83, and a specificity of 0.18. True negative proportion increased to 0.85 for resistant cultivars and decreased to 0.50 for susceptible cultivars. Potential fungicide savings are most likely to be made with resistant cultivars, but such benefits would need to be identified with an in-depth evaluation.

Evaluation of data for atmospheric models: master equation/RRKM calculations on the combination reaction, BrO+NO2 -> BrONO2, a conundrum

Experimental data for the title reaction were modeled using master equation (ME)/RRKM methods based on the Multiwell suite of programs. The starting point for the exercise was the empirical fitting provided by the NASA (Sander, S. P.; Finlayson-Pitts, B. J.; Friedl, R. R.; Golden, D. M.; Huie, R. E.; Kolb, C. E.; Kurylo, M. J.; Molina, M. J.; Moortgat, G. K.; Orkin, V. L.; Ravishankara, A. R. Chemical Kinetics and Photochemical Data for Use in Atmospheric Studies, Evaluation Number 15; Jet Propulsion Laboratory: Pasadena, California, 2006)(1) and IUPAC (Atkinson, R.; Baulch, D. L.; Cox, R. A.: R. F. Hampson, J.; Kerr, J. A.; Rossi, M. J.; Troe, J. J. Phys. Chem. Ref. Data. 2000, 29, 167) 2 data evaluation panels, which represents the data in the experimental pressure ranges rather well. Despite the availability of quite reliable parameters for these calculations (molecular vibrational frequencies (Parthiban, S.; Lee, T. J. J. Chem. Phys. 2000, 113, 145)3 and a. value (Orlando, J. J.; Tyndall, G. S. J. Phys. Chem. 1996, 100,. 19398)4 of the bond dissociation energy, D-298(BrO-NO2) = 118 kJ mol(-1), corresponding to Delta H-0(circle) = 114.3 kJ mol(-1) at 0 K) and the use of RRKM/ME methods, fitting calculations to the reported data or the empirical equations was anything but straightforward. Using these molecular parameters resulted in a discrepancy between the calculations and the database of rate constants of a factor of ca. 4 at, or close to, the low-pressure limit. Agreement between calculation and experiment could be achieved in two ways, either by increasing Delta H-0(circle) to an unrealistically high value (149.3 kJ mol(-1)) or by increasing , the average energy transferred in a downward collision, to an unusually large value (> 5000 cm(-1)). The discrepancy could also be reduced by making all overall rotations fully active. The system was relatively insensitive to changing the moments of inertia in the transition state to increase the centrifugal effect. The possibility of involvement of BrOONO was tested and cannot account for the difficulties of fitting the data.

Application of constrained predictive control on a 3D crane system

This paper describes the SIMULINK implementation of a constrained predictive control algorithm based on quadratic programming and linear state space models, and its application to a laboratory-scale 3D crane system. The algorithm is compatible with Real Time. Windows Target and, in the case of the crane system, it can be executed with a sampling period of 0.01 s and a prediction horizon of up to 300 samples, using a linear state space model with 3 inputs, 5 outputs and 13 states.