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 effects of a pre-movement delay on the kinematics of prehension in middle childhood

The present study examined the effects of a pre-movement delay on the kinematics of prehension in middle childhood. Twenty-five children between the ages of 5 and 11 years made visually open-loop reaches to two different sized objects at two different distances along the midline. Reaches took place either (i) immediately, or (ii) 2 s after the occlusion of the stimulus. In all age groups, reaches following the pre-movement delay were characterised by longer movement durations, lower peak velocities, larger peak grip apertures and longer time spent in the final slow phase of the movement. This pattern of results suggests that the representations that control the transport and grasp component are affected similarly by delay, and is consistent with the results previously reported for adults. Such representations therefore appear to develop before the age of 5. (C) 2004 Elsevier B.V. All rights reserved.

An on-line algorithm of uncertain time delay estimation in a continuous system

In this paper, we present an on-line estimation algorithm for an uncertain time delay in a continuous system based on the observational input-output data, subject to observational noise. The first order Pade approximation is used to approximate the time delay. At each time step, the algorithm combines the well known Kalman filter algorithm and the recursive instrumental variable least squares (RIVLS) algorithm in cascade form. The instrumental variable least squares algorithm is used in order to achieve the consistency of the delay parameter estimate, since an error-in-the-variable model is involved. An illustrative example is utilized to demonstrate the efficacy of the proposed approach.

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.

Global asymptotic stability in a class of Putnam-type equations

In this paper, we initiate the study of a class of Putnam-type equation of the form x(n-1) = A(1)x(n) + A(2)x(n-1) + A(3)x(n-2)x(n-3) + A(4)/B(1)x(n)x(n-1) + B(2)x(n-2) + B(3)x(n-3) + B-4 n = 0, 1, 2,..., where A(1), A(2), A(3), A(4), B-1, B-2, B-3, B-4 are positive constants with A(1) + A(2) + A(3) + A(4) = B-1 + B-2 + B-3 + B-4, x(-3), x(-2), x(-1), x(0) are positive numbers. A sufficient condition is given for the global asymptotic stability of the equilibrium point c = 1 of such equations. (c) 2005 Elsevier Ltd. All rights reserved.

On two rational difference equations

In this paper, we study the oscillating property of positive solutions and the global asymptotic stability of the unique equilibrium of the two rational difference equations [GRAPHICS] and [GRAPHICS] where a is a nonnegative constant. (c) 2005 Elsevier Inc. All rights reserved.

On the system of rational difference equations x(n) = A+(1)over-bar y(n-p), y(n) = A+(gamma(n-1))over-bar x(n-r)y(n-5)

In this paper, we study the behavior of the positive solutions of the system of two difference equations [GRAPHICS] where p >= 1, r >= 1, s >= 1, A >= 0, and x(1-r), x(2-r),..., x(0), y(1-max) {p.s},..., y(0) are positive real numbers. (c) 2005 Elsevier Inc. All rights reserved.