Uniformly convergent finite element and finite difference methods for singularly perturbed ordinary differential equations

This thesis is concerned with uniformly convergent finite element and finite difference methods for numerically solving singularly perturbed two-point boundary value problems. We examine the following four problems: (i) high order problem of reaction-diffusion type; (ii) high order problem of convection-diffusion type; (iii) second order interior turning point problem; (iv) semilinear reaction-diffusion problem. Firstly, we consider high order problems of reaction-diffusion type and convection-diffusion type. Under suitable hypotheses, the coercivity of the associated bilinear forms is proved and representation results for the solutions of such problems are given. It is shown that, on an equidistant mesh, polynomial schemes cannot achieve a high order of convergence which is uniform in the perturbation parameter. Piecewise polynomial Galerkin finite element methods are then constructed on a Shishkin mesh. High order convergence results, which are uniform in the perturbation parameter, are obtained in various norms. Secondly, we investigate linear second order problems with interior turning points. Piecewise linear Galerkin finite element methods are generated on various piecewise equidistant meshes designed for such problems. These methods are shown to be convergent, uniformly in the singular perturbation parameter, in a weighted energy norm and the usual L2 norm. Finally, we deal with a semilinear reaction-diffusion problem. Asymptotic properties of solutions to this problem are discussed and analysed. Two simple finite difference schemes on Shishkin meshes are applied to the problem. They are proved to be uniformly convergent of second order and fourth order respectively. Existence and uniqueness of a solution to both schemes are investigated. Numerical results for the above methods are presented.

Interaction of additive and quantization noises in digital PLLs

Phase-locked loops (PLLs) are a crucial component in modern communications systems. Comprising of a phase-detector, linear filter, and controllable oscillator, they are widely used in radio receivers to retrieve the information content from remote signals. As such, they are capable of signal demodulation, phase and carrier recovery, frequency synthesis, and clock synchronization. Continuous-time PLLs are a mature area of study, and have been covered in the literature since the early classical work by Viterbi [1] in the 1950s. With the rise of computing in recent decades, discrete-time digital PLLs (DPLLs) are a more recent discipline; most of the literature published dates from the 1990s onwards. Gardner [2] is a pioneer in this area. It is our aim in this work to address the difficulties encountered by Gardner [3] in his investigation of the DPLL output phase-jitter where additive noise to the input signal is combined with frequency quantization in the local oscillator. The model we use in our novel analysis of the system is also applicable to another of the cases looked at by Gardner, that is the DPLL with a delay element integrated in the loop. This gives us the opportunity to look at this system in more detail, our analysis providing some unique insights into the variance `dip' seen by Gardner in [3]. We initially provide background on the probability theory and stochastic processes. These branches of mathematics are the basis for the study of noisy analogue and digital PLLs. We give an overview of the classical analogue PLL theory as well as the background on both the digital PLL and circle map, referencing the model proposed by Teplinsky et al. [4, 5]. For our novel work, the case of the combined frequency quantization and noisy input from [3] is investigated first numerically, and then analytically as a Markov chain via its Chapman-Kolmogorov equation. The resulting delay equation for the steady-state jitter distribution is treated using two separate asymptotic analyses to obtain approximate solutions. It is shown how the variance obtained in each case matches well to the numerical results. Other properties of the output jitter, such as the mean, are also investigated. In this way, we arrive at a more complete understanding of the interaction between quantization and input noise in the first order DPLL than is possible using simulation alone. We also do an asymptotic analysis of a particular case of the noisy first-order DPLL with delay, previously investigated by Gardner [3]. We show a unique feature of the simulation results, namely the variance `dip' seen for certain levels of input noise, is explained by this analysis. Finally, we look at the second-order DPLL with additive noise, using numerical simulations to see the effects of low levels of noise on the limit cycles. We show how these effects are similar to those seen in the noise-free loop with non-zero initial conditions.

A study of the Hobson and Rogers volatility model

We firstly examine the model of Hobson and Rogers for the volatility of a financial asset such as a stock or share. The main feature of this model is the specification of volatility in terms of past price returns. The volatility process and the underlying price process share the same source of randomness and so the model is said to be complete. Complete models are advantageous as they allow a unique, preference independent price for options on the underlying price process. One of the main objectives of the model is to reproduce the `smiles' and `skews' seen in the market implied volatilities and this model produces the desired effect. In the first main piece of work we numerically calibrate the model of Hobson and Rogers for comparison with existing literature. We also develop parameter estimation methods based on the calibration of a GARCH model. We examine alternative specifications of the volatility and show an improvement of model fit to market data based on these specifications. We also show how to process market data in order to take account of inter-day movements in the volatility surface. In the second piece of work, we extend the Hobson and Rogers model in a way that better reflects market structure. We extend the model to take into account both first and second order effects. We derive and numerically solve the pde which describes the price of options under this extended model. We show that this extension allows for a better fit to the market data. Finally, we analyse the parameters of this extended model in order to understand intuitively the role of these parameters in the volatility surface.

Spin three gauge theory revisited

We study the problem of consistent interactions for spin-3 gauge fields in flat spacetime of arbitrary dimension 3$">n>3. Under the sole assumptions of Poincaré and parity invariance, local and perturbative deformation of the free theory, we determine all nontrivial consistent deformations of the abelian gauge algebra and classify the corresponding deformations of the quadratic action, at first order in the deformation parameter. We prove that all such vertices are cubic, contain a total of either three or five derivatives and are uniquely characterized by a rank-three constant tensor (an internal algebra structure constant). The covariant cubic vertex containing three derivatives is the vertex discovered by Berends, Burgers and van Dam, which however leads to inconsistencies at second order in the deformation parameter. In dimensions 4$">n>4 and for a completely antisymmetric structure constant tensor, another covariant cubic vertex exists, which contains five derivatives and passes the consistency test where the previous vertex failed. © SISSA 2006.

Parity-violating vertices for spin-3 gauge fields

The problem of constructing consistent parity-violating interactions for spin-3 gauge fields is considered in Minkowski space. Under the assumptions of locality, Poincaré invariance, and parity noninvariance, we classify all the nontrivial perturbative deformations of the Abelian gauge algebra. In space-time dimensions n=3 and n=5, deformations of the free theory are obtained which make the gauge algebra non-Abelian and give rise to nontrivial cubic vertices in the Lagrangian, at first order in the deformation parameter g. At second order in g, consistency conditions are obtained which the five-dimensional vertex obeys, but which rule out the n=3 candidate. Moreover, in the five-dimensional first-order deformation case, the gauge transformations are modified by a new term which involves the second de Wit-Freedman connection in a simple and suggestive way. © 2006 The American Physical Society.

Some models for discretized series of events

A discretized series of events is a binary time series that indicates whether or not events of a point process in the line occur in successive intervals. Such data are common in environmental applications. We describe a class of models for them, based on an unobserved continuous-time discrete-state Markov process, which determines the rate of a doubly stochastic Poisson process, from which the binary time series is constructed by discretization. We discuss likelihood inference for these processes and their second-order properties and extend them to multiple series. An application involves modeling the times of exposures to air pollution at a number of receptors in Western Europe.

Temporal modelling of short-term rainfall using Cox processes

We study two marked point process models based on the Cox process. These models are used to describe the probabilistic structure of the rainfall intensity process. Mathematical formulation of the models is described and some second-moment characteristics of the rainfall depth, and aggregated processes are considered. The derived second-order properties of the accumulated rainfall amounts at different levels of aggregation are used in order to examine the model fit. A brief data analysis is presented. Copyright © 1998 John Wiley & Sons, Ltd.

A class of generalized Lévy Laplacians in infinite dimensional calculus

A class of generalized Lévy Laplacians which contain as a special case the ordinary Lévy Laplacian are considered. Topics such as limit average of the second order functional derivative with respect to a certain equally dense (uniformly bounded) orthonormal base, the relations with Kuo’s Fourier transform and other infinite dimensional Laplacians are studied.

A cell-centred finite volume method for modelling viscoelastic flow

An unstructured cell-centred finite volume method for modelling viscoelastic flow is presented. The method is applied to the flow through a planar channel and the 4:1 planar contraction for creeping flow of an Oldroyd-B fluid. The results are presented for a range of Weissenberg numbers. In the case of the planar channel results are compared with analytical solutions. For the 4:1 planar contraction benchmark problem the convection terms in the constitutive equations are approximated using both first and second order differencing schemes to compare the techniques and the effect of mesh refinement on the solution is investigated. This is the first time that a fully unstructured, cell-centredfinitevolume technique has been used to model the Oldroyd-B fluid for the test cases presented in this paper.

Aspects of the in vitro bioactivity of hydraulic calcium (alumino)silicate cement

In response to a burgeoning interest in the prospective clinical applications of hydraulic calcium (alumino)silicate cements, the in vitro bioactivity and dissolution characteristics of a white Portland cement have been investigated. The formation of an apatite layer within 6 h of contact with simulated body fluid was attributed to the rapid dissolution of calcium hydroxide from the cement matrix and to the abundance of pre-existing Si-OH nucleation sites presented by the calcium silicate hydrate phase. A simple kinetic model has been used to describe the rate of apatite formation and an apparent pseudo-second-order rate constant for the removal of HPO42- ions frorn solultion has been calculated (k(2) = 5.8 x 10(-4) g mg(-1)). Aspects of the chemistry of hydraulic cements are also discussed with respect to their potential use in the remedial treatment of living tissue. (C) 2008 Wiley Periodicals, Inc. J Biomed Mater Res 90A: 166-174, 2009

Ag+- and Zn2+-exchange kinetics and antimicrobial properties of 11 angstrom tobermorites

Ag+- and Zn2+-exchanged zeolites zeolites and clays have been used as coatings and in composites to confer broad-spectrum antimicrobial properties on a range of technical and biomedical materials. 11 angstrom tobermorite is a bioactive layer lattice ion exchanger whose potential as a carrier for Ag+ and Zn2+ ions in antimicrobial formulations has not yet been explored. In view of this, batch Ag+- and Zn2+-exchange kinetics of two structurally distinct synthetic 11 angstrom tobermorites and their subsequent bactericidal action against Staphylococcus aureus and Pseudomonas aeruginosa are reported. During the exchange reactions, Ag+ ions were found to replace labile interlayer cations; whereas, Zn2+ ions also displaced structural Ca2+ ions from the tobermorite lattice. In spite of these different mechanisms, a simple pseudo-second-order model provided a suitable description of both exchange processes (R-2 >= 0.996). The Ag+- and Zn2+-exchanged tobermorite phases exhibited marked bacteriostatic effects against both bacteria, and accordingly, their potential for use as antimicrobial materials for in situ bone tissue regeneration is discussed. (C) 2008 Elsevier Ltd. All rights reserved.

Kinematics and EMG analysis of expert pole vaulter's lower limb during take off phase (P218)

Introduction: The critical phase, in jumping events in track and field, appears to be between touchdown and take-off. Since obvious similarities exist between the take off phase in both long jump and pole vault, numerous 3D kinematics and electromyographic studies have only looked at long jump. Currently there are few detailed kinematics electromyographic data on the pole vault take-off phase. The aim of this study was therefore to characterise kinematics and electromyographic variables during the take-off phase to provide a better understanding of this phase in pole vaulting and its role in performance outcome. Material and methods: Six pole-vaulters took part in the study. Kinematics data were captured with retro reflective markers fixed on the body. Hip, knee and ankle angle were calculated. Differential bipolar surface electrodes were placed on the following muscles of the take-off leg: tibialis anterior, lateral gastrocnemius, vastus lateralis, rectus femoris, bicep femoris and gluteus maximus. EMG activity was synchronously acquired with the kinematic data. EMG data were rectified and smoothed using a second order low pass Butterworth Bidirectional filter (resulting in a 4th order filter) with a cut-off frequency of 14 Hz. Results: Evolution of hip, knee and ankle angle show no significant differences during the last step before touchdown, the take-off phase and the beginning of fly phase. Meanwhile, strong differences in EMG signal are noted inter and intra pole vaulter. However for a same subject the EMG activities seem to converge to some phase locked point. Discussion: All pole vaulters have approximately the same visible coordination This coordination reflects a different muscular control among pole vaulters but also for a considered pole vaulter. These phase locked point could be considered as invariant of motor control i.e. a prerequisite for a normal sequence of the movement and performance realization.

Particulate optical scattering coefficients along an Atlantic Meridional Transect

The particulate optical backscattering coefficient (bbp) is a fundamental optical property that allows monitoring of marine suspended particles both in situ and from space. Backscattering measurements in the open ocean are still scarce, however, especially in oligotrophic regions. Consequently, uncertainties remain in bbp parameterizations as well as in satellite estimates of bbp. In an effort to reduce these uncertainties, we present and analyze a dataset collected in surface waters during the 19th Atlantic Meridional Transect. Results show that the relationship between particulate beam-attenuation coefficient (cp) and chlorophyll-a concentration was consistent with published bio-optical models. In contrast, the particulate backscattering per unit of chlorophyll-a and per unit of cp were higher than in previous studies employing the same sampling methodology. These anomalies could be due to a bias smaller than the current uncertainties in bbp. If that was the case, then the AMT19 dataset would confirm that bbp:cp is remarkably constant over the surface open ocean. A second-order decoupling between bbp and cp was, however, evident in the spectral slopes of these coefficients, as well as during diel cycles. Overall, these results emphasize the current difficulties in obtaining accurate bbp measurements in the oligotrophic ocean and suggest that, to first order, bbp and cp are coupled in the surface open ocean, but they are also affected by other geographical and temporal variations.

A Quantitative Investigation of Selected Reactions in the Fibrinolytic Cascade

Previous work has shown that thrombin activatable fibrinolysis inhibitor (TAFI) was unable to prolong lysis of purified clots in the presence of Lys-plasminogen (Lys-Pg), indicating a possible mechanism for fibrinolysis to circumvent prolongation mediated by activated TAFI (TAFIa). Therefore, the effects of TAFIa on Lys-Pg activation and Lys-plasmin (Lys-Pn) inhibition by antiplasmin (AP) were quantitatively investigated using a fluorescently labeled recombinant Pg mutant which does not produce active Pn. High molecular weight fibrin degradation products (HMW-FDPs), a soluble fibrin surrogate that models Pn modified fibrin, treated with TAFIa decreased the catalytic efficiency (kcat/Km) of 5IAF-Glu-Pg cleavage by 417-fold and of 5IAF-Lys-Pg cleavage by 55-fold. A previously devised intact clot system was used to measure the apparent second order rate constant (k2) for Pn inhibition by AP over time. While TAFIa was able to abolish the protection associated with Pn modified fibrin in clots formed with Glu-Pg, it was not able to abolish the protection in clots formed with Lys-Pg. However, TAFIa was still able to prolong the lysis of clots formed with Lys-Pg. TAFIa prolongs clot lysis by removing the positive feedback loop for Pn generation. The effect of TAFIa modification of the HMW-FDPs on the rate of tissue type plasminogen activator (tPA) inhibition by plasminogen activator inhibitor type 1 (PAI-1) was investigated using a previously devised end point assay. HMW-FDPs decreased the k2 for tPA inhibition rate by 3-fold. Thus, HMW-FDPs protect tPA from PAI-1. TAFIa treatment of the HMW-FDPs resulted in no change in protection. Vitronectin also did not appreciably affect tPA inhibition by PAI-1. Pg, in conjunction with HMW-FDPs, decreased the k2 for tPA inhibition by 30-fold. Hence, Pg, when bound to HMW-FDPs, protects tPA by an additional 10-fold. TAFIa treatment of the HMW-FDPs completely removed this additional protection provided by Pg. In conclusion, an additional mechanism was identified whereby TAFIa can prolong clot lysis by increasing the rate of tPA inhibition by PAI-1 by eliminating the protective effects of Pn-modified fibrin and Pg. Because TAFIa can suppress Lys-Pg activation but cannot attenuate Lys-Pn inhibition by AP, the Glu- to Lys-Pg/Pn conversion is able to act as a fibrinolytic switch to ultimately lyse the clot.

The inactivation of bovine cathepsin B by novel N-chloro-acetyl-dipeptides. Application of the Houghten 'tea bag' methodology to inhibitor synthesis

In this study, a series of N-chloro-acetylated dipeptides were synthesised by the application of Houghten's methodology of multiple analog peptide syntheses. The peptides, all of which contain a C-terminal free acid, were tested as inactivators of bovine cathepsin B, in an attempt at exploiting the known and, amongst the cysteine proteinases, unique carboxy dipeptidyl peptidase activity of the protease. We have succeeded in obtaining a number of effective inactivators, the most potent of which-chloroacetyl-Leu-Leu-OH, inactivates the enzyme with an apparent second-order rate constant of 3.8 x 10(4) M-1 min(-1). In contrast, the esterified analog, chloroacetyl-Leu-Leu-OMe, inactivates the enzyme some three orders of magnitude less efficiently, lending credence to our thesis that a free carboxylic acid moiety is an important determinant for inhibitor effectiveness. This preliminary study has highlighted a number of interesting features about the specificity requirements of the bovine proteinase and we believe that our approach has great potential for the rapid delineation of the subsite specificities of cathepsin B-like proteases from various species. (c) 2005 Elsevier Inc. All rights reserved.