Well-posed boundary value problems for integrable evolution equations on a finite interval

We consider boundary value problems posed on an interval [0,L] for an arbitrary linear evolution equation in one space dimension with spatial derivatives of order n. We characterize a class of such problems that admit a unique solution and are well posed in this sense. Such well-posed boundary value problems are obtained by prescribing N conditions at x=0 and n–N conditions at x=L, where N depends on n and on the sign of the highest-degree coefficient n in the dispersion relation of the equation. For the problems in this class, we give a spectrally decomposed integral representation of the solution; moreover, we show that these are the only problems that admit such a representation. These results can be used to establish the well-posedness, at least locally in time, of some physically relevant nonlinear evolution equations in one space dimension.

Integrable evolution equations in time-dependent domains

We discuss the implementation of a method of solving initial boundary value problems in the case of integrable evolution equations in a time-dependent domain. This method is applied to a dispersive linear evolution equation with spatial derivatives of arbitrary order and to the defocusing nonlinear Schrödinger equation, in the domain l(t)

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.

The relationship between multi-species catch and effort: among fishery comparisons

The control of fishing mortality via fishing effort remains fundamental to most fisheries management strategies even at the local community or co-management level. Decisions to support such strategies require knowledge of the underlying response of the catch to changes in effort. Even under adaptive management strategies, imprecise knowledge of the response is likely to help accelerate the adaptive learning process. Data and institutional capacity requirements to employ multi-species biomass dynamics and age-structured models invariably render their use impractical particularly in less developed regions of the world. Surplus production models fitted to catch and effort data aggregated across all species offer viable alternatives. The current paper seeks models of this type that best describe the multi-species catch–effort responses in floodplain-rivers, lakes and reservoirs and reef-based fisheries based upon among fishery comparisons, building on earlier work. Three alternative surplus production models were fitted to estimates of catch per unit area (CPUA) and fisher density for 258 fisheries in Africa, Asia and South America. In all cases examined, the best or equal best fitting model was the Fox type, explaining up to 90% of the variation in CPUA. For lake and reservoir fisheries in Africa and Asia, the Schaefer and an asymptotic model fitted equally well. The Fox model estimates of fisher density (fishers km−2) at maximum yield (iMY) for floodplain-rivers, African lakes and reservoirs and reef-based fisheries are 13.7 (95% CI [11.8, 16.4]); 27.8 (95% CI [17.5, 66.7]) and 643 (95% CI [459,1075]), respectively and compare well with earlier estimates. Corresponding estimates of maximum yield are also given. The significantly higher value of iMY for reef-based fisheries compared to estimates for rivers and lakes reflects the use of a different measure of fisher density based upon human population size estimates. The models predict that maximum yield is achieved at a higher fishing intensity in Asian lakes compared to those in Africa. This may reflect the common practice in Asia of stocking lakes to augment natural recruitment. Because of the equilibrium assumptions underlying the models, all the estimates of maximum yield and corresponding levels of effort should be treated with caution.

Research into anxiety of childhood: playing catch-up (to Olympic standard)

This special issue is the culmination of an ESRC seminar series grant awarded to the authors of this editorial. We named the seminar series CATTS (Child Anxiety, Theory and Treatment Seminars) and it took the form of six highly stimulating, one-day seminars on the subject of child anxiety, with participants from clinical and academic backgrounds and from Great Britain, Europe, the USA and Australia. Most of the authors in this publication, and a sister special issue in Cognitions and Emotion (2008), participated in the CATTS series.

Complexity of Monte Carlo algorithms for a class of integral equations

In this work we study the computational complexity of a class of grid Monte Carlo algorithms for integral equations. The idea of the algorithms consists in an approximation of the integral equation by a system of algebraic equations. Then the Markov chain iterative Monte Carlo is used to solve the system. The assumption here is that the corresponding Neumann series for the iterative matrix does not necessarily converge or converges slowly. We use a special technique to accelerate the convergence. An estimate of the computational complexity of Monte Carlo algorithm using the considered approach is obtained. The estimate of the complexity is compared with the corresponding quantity for the complexity of the grid-free Monte Carlo algorithm. The conditions under which the class of grid Monte Carlo algorithms is more efficient are given.