Numerical methods for strong solutions of stochastic differential equations: an overview

This paper gives a review of recent progress in the design of numerical methods for computing the trajectories (sample paths) of solutions to stochastic differential equations. We give a brief survey of the area focusing on a number of application areas where approximations to strong solutions are important, with a particular focus on computational biology applications, and give the necessary analytical tools for understanding some of the important concepts associated with stochastic processes. We present the stochastic Taylor series expansion as the fundamental mechanism for constructing effective numerical methods, give general results that relate local and global order of convergence and mention the Magnus expansion as a mechanism for designing methods that preserve the underlying structure of the problem. We also present various classes of explicit and implicit methods for strong solutions, based on the underlying structure of the problem. Finally, we discuss implementation issues relating to maintaining the Brownian path, efficient simulation of stochastic integrals and variable-step-size implementations based on various types of control.

Solvability of singular second order m-point boundary value problems of Dirichlet type

Let f : [0, 1] x R2 -> R be a function satisfying the Caxatheodory conditions and t(1 - t)e(t) epsilon L-1 (0, 1). Let a(i) epsilon R and xi(i) (0, 1) for i = 1,..., m - 2 where 0 < xi(1) < xi(2) < (...) < xi(m-2) < 1 - In this paper we study the existence of C[0, 1] solutions for the m-point boundary value problem [GRAPHICS] The proof of our main result is based on the Leray-Schauder continuation theorem.

Partial regularity of weak solutions of the liquid crystal equilibrium system

For a parameter, we consider the modified relaxed energy of the liquid crystal system. Each minimizer of the modified relaxed energy is a weak solution to the liquid crystal equilibrium system. We prove the partial regularity of minimizers of the modified relaxed energy. We also prove the existence of infinitely many weak solutions for the special boundary value x.

Prediction of the growth of wear-type rail corrugation

The growth behaviour of the vibrational wear phenomenon known as rail corrugation is investigated analytically and numerically using mathematical models. A simplified feedback model for wear-type rail corrugation that includes a wheel pass time delay is developed with an aim to analytically distil the most critical interaction occurring between the wheel/rail structural dynamics, rolling contact mechanics and rail wear. To this end, a stability analysis on the complete system is performed to determine the growth of wear-type rail corrugations over multiple wheelset passages. This analysis indicates that although the dynamical behaviour of the system is stable for each wheel passage, over multiple wheelset passages, the growth of wear-type corrugations is shown to be the result of instability due to feedback interaction between the three primary components of the model. The corrugations are shown analytically to grow for all realistic railway parameters. From this analysis an analytical expression for the exponential growth rate of corrugations in terms of known parameters is developed. This convenient expression is used to perform a sensitivity analysis to identify critical parameters that most affect corrugation growth. The analytical predictions are shown to compare well with results from a benchmarked time-domain finite element model. (C) 2004 Elsevier B.V. All rights reserved.

On multiple solutions of the exterior neumann problem involving critical sobolev exponent

In this paper we consider the exterior Neumann problem involving a critical Sobolev exponent. We establish the existence of two solutions having a prescribed limit at infinity.

Global behavior of positive solutions of nonlinear three-point boundary value problems

We investigate the structure of the positive solution set for nonlinear three-point boundary value problems of the form u('') + h(t) f(u) = 0, u(0) = 0, u(1) = lambdau(eta), where eta epsilon (0, 1) is given lambda epsilon (0, 1/n) is a parameter, f epsilon C ([0, infinity), [0, infinity)) satisfies f (s) > 0 for s > 0, and h epsilon C([0, 1], [0, infinity)) is not identically zero on any subinterval of [0, 1]. Our main results demonstrate the existence of continua of positive solutions of the above problem. (C) 2004 Elsevier Ltd. All rights reserved.

Positive solutions of a Neumann problem with competing critical nonlinearities

We present existence results for a Neumann problem involving critical Sobolev nonlinearities both on the right hand side of the equation and at the boundary condition.. Positive solutions are obtained through constrained minimization on the Nehari manifold. Our approach is based on the concentration 'compactness principle of P. L. Lions and M. Struwe.

The effect of resin type on the sintering of freeformed maraging steel

The role of the resin type on the sintering of maraging steel with boron additions has been investigated. Two different resins were added to the steel mixture and their subsequent debinding was evaluated and sintering responses compared with that of a resin-free alloy. The two resins used, nylon and a mixture of phenolic resin and synthetic wax, possessed different debinding behaviour, with the latter causing significant carbon contamination of the parts. This caused the formation of a Ti-Mo carbide, depleting the matrix of these elements. Consequently, the microstructure consisted of the equilibrium Fe-Fe2B eutectic, as well as a Mo-rich boride. The liquid phase also appeared to contain significant amounts of carbon, which lowered the temperature at which the liquid formed, resulting in high density occurring at a much lower temperature. When nylon was used as the binder, a similar sintering response to the resin-free alloy was observed. (C) 2002 Published by Elsevier Science B.V.

Response of B-type natriuretic pepticle to exercise in hypertensive patients with suspected diastolic heart failure: Correlation with cardiac function, hemodynamics, and workload

Background Diastolic heart failure (DHF) is characterized by dyspnea due to increased left ventricular (LV) filling pressures during stress. We sought the relationship of exercise-induced increases in B-type natriuretic peptide (BNP) to LV filling pressures and parameters of cardiovascular performance in suspected DHF. Methods Twenty-six treated hypertensive patients with suspected DHF (exertional dyspnea, LV ejection fraction >50%, and diastolic dysfunction) underwent maximal exercise echocardiography using the Bruce protocol. BNP, transmitral Doppler, and tissue Doppler for systolic (So) and early (Ea) and late (Aa) diastolic mitral annular velocities were obtained at rest and peak stress. LV filling pressures were estimated with E/Ea ratios. Results Resting BNP correlated with resting pulse pressure (r=0.45, P=0.02). Maximal exercise performance (4.6 +/- 2.5min) was limited by dyspnea. Blood pressure increased with exercise (from 143 +/- 19/88 +/- 8 to 191 +/- 22/90 +/- 10 mm Hg); 13 patients (50%) had a hypertensive response. Peak exercise BNP correlated with peak transmitral E velocity (r = 0.41, P <.05) and peak heart rate (r = -0.40, P <.05). BNP increased with exercise (from 48 57 to 74 97 pg/mL, P =.007), and the increment of BNP with exercise was associated with maximal workload and peak exercise So, Ea, and Aa (P <.01 for all). Filling pressures, approximated by lateral E/Ea ratio, increased with exercise (7.7 +/- 2.0 to 10.0 +/- 4.8, P <.01). BNP was higher in patients with possibly elevated filling pressures at peak exercise (E/Ea >10) compared to those with normal pressures (123 +/- 124 vs 45 +/- 71 pg/mL, P =.027). Conclusions Augmentation of BNP with exercise in hypertensive patients with suspected DHF is associated with better exercise capacity, LV systolic and diastolic function, and left atrial function. Peak exercise BNP levels may identify exercise-induced elevation of filling pressures in DHF.

tRNASer(CGA) differentially regulates expression of wild-type and codon-modified papillomavirus L1 genes

Exogenous transfer RNAs (tRNAs) favor translation of bovine papillomavirus 1 wild-type (wt) L1 mRNA in in vitro translation systems (Zhou et al. 1999, J. Virol., 73, 4972-4982). We, therefore, investigated whether papillomavirus (PV) wt L1 protein expression could be enhanced in eukaryotic cells following exogenous tRNA supplementation. Both Chinese hamster ovary (CHO) and Cos1 cells, transfected with PV1 wt L1 genes, effectively transcribed the genes but did not translate them. However, L1 protein translation was demonstrated following co-transfection with the L1 gene and a gene expressing tRNA(Ser)(CGA). Cell lines, stably transfected with a bovine papillomavirus 1 (BPV1) wt L1 expression construct, produced L1 protein after the transfection of the tRNA(Ser)(CGA) gene, but not following the transfection with basal vectors, suggesting that tRNA(Ser)(CGA) gene enhanced wt L1 translation as a result of endogenous tRNA alterations and phosphorylation of translation initiation factors elF4E and elF2alpha in the tRNA(Ser)(CGA) transfected L1 cell lines. The tRNA(Ser)(CGA) gene expression significantly reduced translation of L1 proteins expressed from codon-modified (HB) PV L1 genes utilizing mammalian preferred codons, but had variable effects on translation of green fluorescent proteins (GFPs) expressed from six serine GFP variants. The changes of tRNA pools appear to match the codon composition of PV wt and HB L1 genes and serine GFP variants to regulate translation of their mRNAs. These findings demonstrate for the first time in eukaryotic cells that translation of the target genes can be differentially influenced by the provision of a single tRNA expression construct.

Dynamic interfacial tension of aqueous solutions of PVAAs and its role in liquid-liquid dispersion stabilization

In liquid-liquid dispersion systems, the dynamic change of the interfacial properties between the two immiscible liquids plays an important role in both the emulsification process and emulsion stabilization. In this paper, experimentally measured dynamic interfacial tensions of 1-chlorobutane in the aqueous solutions of various random copolymers of polyvinyl acetate and polyvinyl alcohol (PVAA) are presented. Theoretical analyses on these results suggest that the adsorption of the polymer molecules is controlled neither by the bulk diffusion process nor the activation energy barrier for the adsorption but the conformation of polymer molecules. Based on the concept of critical concentration of condensation for polymer adsorption, as well as the observation that the rate at which the dynamic interfacial tension changes does not correlate to the PVAA's ability to stabilize a single drop, it is postulated that the main stabilization mechanism for the PVAAs is by steric hindrance, not the Gibbs-Marangoni effect offered by the small molecule surfactants.

Differential involvement of N-type calcium channels in transmitter release from vasoconstrictor and vasodilator neurons

1 The effects of calcium channel blockers on co-transmission from different populations of autonomic vasomotor neurons were studied on isolated segments of uterine artery and vena cava from guinea-pigs. 2 Sympathetic, noradrenergic contractions of the uterine artery (produced by 200 pulses at 1 or 10 Hz; 600 pulses at 20 Hz) were abolished by the N-type calcium channel blocker omega-conotoxin (CTX) GVIA at 1-10 nM. 3 Biphasic sympathetic contractions of the vena cava (600 pulses at 20 Hz) mediated by noradrenaline and neuropeptide Y were abolished by 10 nM CTX GVIA. 4 Neurogenic relaxations of the uterine artery (200 pulses at 10 Hz) mediated by neuronal nitric oxide and neuropeptides were reduced < 50% by CTX GVIA 10-100 nM. 5 Capsaicin (3 muM) did not affect the CTX GVIA-sensitive or CTX GVIA-resistant neurogenic relaxations of the uterine artery. 6 The novel N-type blocker CTX CVID (100-300 nM), P/Q-type blockers agatoxin IVA (10-100 nM) or CTX CVIB (100 nM), the L-type blocker nifedipine (10 muM) or the 'R-type' blocker SNX-482 (100 nM), all failed to reduce CTX GVIA-resistant relaxations. The T-type channel blocker NiCl2 (100-300 muM) reduced but did not abolish the remaining neurogenic dilations. 7 Release of different neurotransmitters from the same autonomic vasomotor axon depends on similar subtypes of calcium channels. N-type channels are responsible for transmitter release from vasoconstrictor neurons innervating a muscular artery and capacitance vein, but only partly mediate release of nitric oxide and neuropeptides from pelvic vasodilator neurons.

Sequence, circulating levels, and expression of C-type natriuretic peptide in a euryhaline elasmobranch, Carcharhinus leucas

The present study has examined expression and circulating levels of C-type natriuretic peptide (CNP) in the euryhaline bull shark, Carcharhinus leucas. Complementary DNA and deduced amino acid sequence for CNP in C leucas were determined by RACE methods. Homology of CNP amino acid sequence in C. leucas was high both for proCNP and for mature CNP when compared with previously identified elasmobranch CNPs. Mature CNP sequence in C. leucas was identical to that in Triakis seyllia and Seyliorhinus canicula. Levels of expression of CNP mRNA were significantly decreased in the atrium but did not change in either the brain or ventricle following acclimation to a SW environment. However, circulating levels of CNP significantly increased from 86.0 +/- 7.9 fmol ml(-1) in FW to 144.9 +/- 19.5 fmol ml(-1) in SW. The results presented demonstrate that changes in environmental salinity influences both synthesis of CNP from the heart and also circulating levels in C. leucas. Potential stimulus for release and modes of action are discussed. (c) 2005 Elsevier Inc. All rights reserved.