Difference equations associated with fully nonlinear boundary value problems for second order ordinary differential equations

We study the continuous problem y"=f(x,y,y'), xc[0,1], 0=G((y(0),y(1)),(y'(0), y'(1))), and its discrete approximation (y(k+1)-2y(k)+y(k-1))/h(2) =f(t(k), y(k), v(k)), k = 1,..., n-1, 0 = G((y(0), y(n)), (v(1), v(n))), where f and G = (g(0), g(1)) are continuous and fully nonlinear, h = 1/n, v(k) = (y(k) - y(k-1))/h, for k =1,..., n, and t(k) = kh, for k = 0,...,n. We assume there exist strict lower and strict upper solutions and impose additional conditions on f and G which are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. We show that the discrete approximation also has solutions which approximate solutions of the continuous problem and converge to the solution of the continuous problem when it is unique, as the grid size goes to 0. Homotopy methods can be used to compute the solution of the discrete approximation. Our results were motivated by those of Gaines.

Modelling of Lamb waves in composite laminated plates excited by interdigital transducers

The technique of permanently attaching interdigital transducers (IDT) to either flat or curved structural surfaces to excite single Lamb wave mode has demonstrated great potential for quantitative non-destructive evaluation and smart materials design, In this paper, the acoustic wave field in a composite laminated plate excited by an IDT is investigated. On the basis of discrete layer theory and a multiple integral transform method, an analytical-numerical approach is developed to evaluate the surface velocity response of the plate due to the IDTs excitation. In this approach, the frequency spectrum and wave number spectrum of the output of IDT are obtained directly. The corresponding time domain results are calculated by applying a standard inverse fast Fourier transformation technique. Numerical examples are presented to validate the developed method and show the ability of mode selection and isolation. A new effective way of transfer function estimation and interpretation is presented by considering the input wave number spectrum in addition to the commonly used input frequency spectrum. The new approach enables the simple physical evaluation of the influences of IDT geometrical features such as electrode finger widths and overall dimension and excitation signal properties on the input-output characteristics of IDT. Finally, considering the convenience of Mindlin plate wave theory in numerical computations as well as theoretical analysis, the validity is examined of using this approximate theory to design IDT for the excitation of the first and second anti-symmetric Lamb modes. (C) 2002 Elsevier Science Ltd. All rights reserved.

Simple operational interpretation of the fidelity of mixed states

This Brief Report presents a corollary to Uhlmann's theorem which provides a simple operational interpretation of the fidelity of mixed states.

Predictor-corrector methods of Runge-Kutta type for stochastic differential equations

In this paper we construct predictor-corrector (PC) methods based on the trivial predictor and stochastic implicit Runge-Kutta (RK) correctors for solving stochastic differential equations. Using the colored rooted tree theory and stochastic B-series, the order condition theorem is derived for constructing stochastic RK methods based on PC implementations. We also present detailed order conditions of the PC methods using stochastic implicit RK correctors with strong global order 1.0 and 1.5. A two-stage implicit RK method with strong global order 1.0 and a four-stage implicit RK method with strong global order 1.5 used as the correctors are constructed in this paper. The mean-square stability properties and numerical results of the PC methods based on these two implicit RK correctors are reported.

A uniform white-noise model for fixed-point roundoff errors in digital systems

Fixed-point roundoff noise in digital implementation of linear systems arises due to overflow, quantization of coefficients and input signals, and arithmetical errors. In uniform white-noise models, the last two types of roundoff errors are regarded as uniformly distributed independent random vectors on cubes of suitable size. For input signal quantization errors, the heuristic model is justified by a quantization theorem, which cannot be directly applied to arithmetical errors due to the complicated input-dependence of errors. The complete uniform white-noise model is shown to be valid in the sense of weak convergence of probabilistic measures as the lattice step tends to zero if the matrices of realization of the system in the state space satisfy certain nonresonance conditions and the finite-dimensional distributions of the input signal are absolutely continuous.

Slowed conduction and ventricular tachycardia after targeted disruption of the cardiac sodium channel gene Scn5a

Voltage-gated sodium channels drive the initial depolarization phase of the cardiac action potential and therefore critically determine conduction of excitation through the heart. In patients, deletions or loss-of-function mutations of the cardiac sodium channel gene, SCN5A, have been associated with a wide range of arrhythmias including bradycardia (heart rate slowing), atrioventricular conduction delay, and ventricular fibrillation. The pathophysiological basis of these clinical conditions is unresolved. Here we show that disruption of the mouse cardiac sodium channel gene, Scn5a, causes intrauterine lethality in homozygotes with severe defects in ventricular morphogenesis whereas heterozygotes show normal survival. Whole-cell patch clamp analyses of isolated ventricular myocytes from adult Scn5a(+/-) mice demonstrate a approximate to50% reduction in sodium conductance. Scn5a(+/-) hearts have several defects including impaired atrioventricular conduction, delayed intramyocardial conduction, increased ventricular refractoriness, and ventricular tachycardia with characteristics of reentrant excitation. These findings reconcile reduced activity of the cardiac sodium channel leading to slowed conduction with several apparently diverse clinical phenotypes, providing a model for the detailed analysis of the pathophysiology of arrhythmias.

Confined-unconfined flow in a horizontal confined aquifer

Mixed confined and unconfined groundwater flow occurs in a bounded initially dry aquifer when the hydraulic head at the side boundary suddenly rises above the elevation of the aquifer's top boundary. The flow problem as modelled by the Boussinesq equation is non-trivial because of the involvement of two moving boundaries. The transformed equation (based on a similarity transformation) can, however, be dealt with more easily. Here, we present an approximate analytical solution for this flow problem. The approximate solution is compared with an 'exact' numerical solution and found to be a very accurate description for describing the mixed confined and unconfined flow in the confined aquifer. (C) 2002 Elsevier Science B.V. All rights reserved.

Repeated three-dimensional magnetic resonance imaging of atherosclerosis development in innominate arteries of low-density lipoprotein receptor-knockout mice

Background-In vivo methods to evaluate the size and composition of atherosclerotic lesions in animal models of atherosclerosis would assist in the testing of antiatherosclerotic drugs. We have developed an MRI method of detecting atherosclerotic plaque in the major vessels at the base of the heart in low-density lipoprotein (LDL) receptor-knockout (LDLR-/-) mice on a high-fat diet. Methods and Results-Three-dimensional fast spin-echo magnetic resonance images were acquired at 7 T by use of cardiac and respiratory triggering, with approximate to140-mum isotropic resolution, over 30 minutes. Comparison of normal and fat-suppressed images from female LDLR-/- mice I week before and 8 and 12 weeks after the transfer to a high-fat diet allowed visualization and quantification of plaque development in the innominate artery in vivo. Plaque mean cross-sectional area was significantly greater at week 12 in the LDLR-/- mice (0.14+/-0.086 mm(2) [mean+/-SD]) than in wild-type control mice on a normal diet (0.017+/-0.031 mm(2), p

Plasma growth hormone and growth hormone-binding protein during development in the marsupial brushtail possum (Trichosurus vulpecula)

Plasma concentrations of growth hormone (GH) were measured in the brushtail possum (Trichosurus vulpecula) pouch young from 25 through to 198 days post-partum (n=71). GH concentrations were highest early in pouch life (around 100 ng/ml), and thereafter declined in an exponential fashion to reach adult concentrations (10.8 +/- 1.8 ng/ml; n=21) by approximately 121-145 days post-partum, one to two months before the young is weaned. Growth hormone-binding protein (GHBP), which has been shown to modify the cellular actions of GH in eutherian mammals, was identified for the first time in a marsupial. Based on size exclusion gel filtration, possum GHBP had an estimated molecular mass of approximate to 65 kDa, similar to that identified in other mammalian species, and binding of I-125-labelled human GH (hGH) was displaced by excess hGH (20 mug). An immunoprecipitation method, in which plasma GHBP was rendered polyethylene glycol precipitable with a monoclonal antibody to the rabbit GHBP/GH receptor (MAb 43) and labelled with I-125-hGH, was used to quantitate plasma GHBP by Scatchard analysis in the developing (pooled plasma samples) and adult (individual animals) possums. Binding affinity (K-a) values in pouch young aged between 45 and 54 and 144 and 153 days post-partum varied between 1.0 and 2.4 x 10(9)/M, which was slightly higher than that in adult plasma (0.96 +/- 0.2 x 10(9)/M, n = 6). Binding capacity (B-max) values increased from non-detectable levels in animals aged 25-38 days post-partum to reach concentrations around half that seen in the adult (1.4 +/- 0.2 x 10(-9) M) by about 117 days post-partum and remained at this level until 153 days post-partum. Therefore, in early pouch life when plasma GH concentrations are highest, the very low concentrations of GHBP are unlikely to be important in terms of competing with GH-receptor for ligand or altering the half-life of circulating GH.

Effects of polyunsaturated fatty acids on voltage-gated K+ and Na+ channels in rat olfactory receptor neurons

Although the polyunsaturated fatty acids arachidonic acid (AA) and docosahexaenoic acid (DHA) are enriched in the olfactory mucosa, their possible contribution to olfactory transduction has not been investigated. This study characterized their effects on voltage-gated K+ and Na+ channels of rat olfactory receptor neurons. Physiological (3-10 mum) concentrations of AA and DHA potently and irreversibly inhibited the voltage-gated K+ current in a voltage-independent manner. In addition, both compounds significantly reduced the inhibitory potency of the odorants acetophenone and amyl acetate at these channels. By comparison, the steady-state effects of both AA and DHA on the voltage-gated Na+ channel were relatively weak, with half-maximal inhibition requiring approximate to 35 mum of either compound. However, a surprising finding was that the initial application of 3 mum AA to a naive neuron caused a strong but transient inhibition of the Na+ current. The channels became almost completely resistant to this inhibition within 1 min, and a 2-min wash in control solution was insufficient to restore the strong inhibitory effect. These observations suggest that polyunsaturated fatty acids have the potential to strongly influence the coding of odorant information by olfactory receptor neurons.

Difference equations in Banach spaces

Difference equations which discretely approximate boundary value problems for second-order ordinary differential equations are analysed. It is well known that the existence of solutions to the continuous problem does not necessarily imply existence of solutions to the discrete problem and, even if solutions to the discrete problem are guaranteed, they may be unrelated and inapplicable to the continuous problem. Analogues to theorems for the continuous problem regarding a priori bounds and existence of solutions are formulated for the discrete problem. Solutions to the discrete problem are shown to converge to solutions of the continuous problem in an aggregate sense. An example which arises in the study of the finite deflections of an elastic string under a transverse load is investigated. The earlier results are applied to show the existence of a solution; the sufficient estimates on the step size are presented. (C) 2003 Elsevier Science Ltd. All rights reserved.

Modelling the size of the crushed zone around a blasthole

A new model to predict the extent of crushing around a blasthole is presented. The model is based on the back-analysis of a comprehensive experimental program that included the direct measurement of the zone of crushing from 92 blasting tests on concrete blocks using two commercial explosives. The concrete blocks varied from low, medium to high strength and measured 1.5 in in length, 1.0 m in width and 1.1 m in height. A dimensionless parameter called the crushing zone index (CZI) is introduced. This index measures the crushing potential of a charged blasthole and is a function of the borehole pressure, the unconfined compressive strength of the rock material, dynamic Young's modulus and Poisson's ratio. It is shown that the radius of crushing is a function of the CZI and the blasthole radius. A good correlation between the new model and measured results was obtained. A number of previously proposed models could not approximate the conditions measured in the experimental work and there are noted discrepancies between the different approaches reviewed, particularly for smaller diameter holes and low strength rock conditions. The new model has been verified with full scale tests reported in the literature. Results from this validation and model evaluations show its applicability to production blasting. (C) 2003 Elsevier Science Ltd. All rights reserved.

A new method for analysing the equilibrium and time-dependent behaviour of Markovian models

Many large-scale stochastic systems, such as telecommunications networks, can be modelled using a continuous-time Markov chain. However, it is frequently the case that a satisfactory analysis of their time-dependent, or even equilibrium, behaviour is impossible. In this paper, we propose a new method of analyzing Markovian models, whereby the existing transition structure is replaced by a more amenable one. Using rates of transition given by the equilibrium expected rates of the corresponding transitions of the original chain, we are able to approximate its behaviour. We present two formulations of the idea of expected rates. The first provides a method for analysing time-dependent behaviour, while the second provides a highly accurate means of analysing equilibrium behaviour. We shall illustrate our approach with reference to a variety of models, giving particular attention to queueing and loss networks. (C) 2003 Elsevier Ltd. All rights reserved.

Existence of multiple solutions for finite difference approximations to second-order boundary value problems

Error condition detected We consider discrete two-point boundary value problems of the form D-2 y(k+1) = f (kh, y(k), D y(k)), for k = 1,...,n - 1, (0,0) = G((y(0),y(n));(Dy-1,Dy-n)), where Dy-k = (y(k) - Yk-I)/h and h = 1/n. This arises as a finite difference approximation to y" = f(x,y,y'), x is an element of [0,1], (0,0) = G((y(0),y(1));(y'(0),y'(1))). We assume that f and G = (g(0), g(1)) are continuous and fully nonlinear, that there exist pairs of strict lower and strict upper solutions for the continuous problem, and that f and G satisfy additional assumptions that are known to yield a priori bounds on, and to guarantee the existence of solutions of the continuous problem. Under these assumptions we show that there are at least three distinct solutions of the discrete approximation which approximate solutions to the continuous problem as the grid size, h, goes to 0. (C) 2003 Elsevier Science Ltd. All rights reserved.

Implicit vector equilibrium problems with applications

Let X and Y be Hausdorff topological vector spaces, K a nonempty, closed, and convex subset of X, C: K--> 2(Y) a point-to-set mapping such that for any x is an element of K, C(x) is a pointed, closed, and convex cone in Y and int C(x) not equal 0. Given a mapping g : K --> K and a vector valued bifunction f : K x K - Y, we consider the implicit vector equilibrium problem (IVEP) of finding x* is an element of K such that f (g(x*), y) is not an element of - int C(x) for all y is an element of K. This problem generalizes the (scalar) implicit equilibrium problem and implicit variational inequality problem. We propose the dual of the implicit vector equilibrium problem (DIVEP) and establish the equivalence between (IVEP) and (DIVEP) under certain assumptions. Also, we give characterizations of the set of solutions for (IVP) in case of nonmonotonicity, weak C-pseudomonotonicity, C-pseudomonotonicity, and strict C-pseudomonotonicity, respectively. Under these assumptions, we conclude that the sets of solutions are nonempty, closed, and convex. Finally, we give some applications of (IVEP) to vector variational inequality problems and vector optimization problems. (C) 2003 Elsevier Science Ltd. All rights reserved.