MSc in Mathematical Modelling and Scientific Computing

These are some ideas for MSc dissertations. The list is not exclusive and students can always suggest a topic themselves. If you are interested in one of these projects you should get in touch directly with the supervisor.

Industrial Projects

Distinguished limits of nondimensionalised oilfield models - Abingdon Technology Center, Schlumberger: Prof. Chris Farmer and Dr Jim Oliver

Oil reservoir operations are optimised using mathematical models. Both numerical and analytical techniques are used, depending on the model. During optimisation one evalautes quantities of interest as functions of the input and control parameters.

In this project a collection of related models of flow through porous media will be analysed for the purpose of guiding engineers and geoscientists in the selection and use of the best model.

As part of this a reduction in the number of parameters can be made by using the techniques of dimensional analysis, based on a transformation of the model equations. By studying the parameters involved in some actual oil reservoirs, the magnitudes of the dimensionless groups will be found, and recommendations as to which terms in the equations can be ignored will be listed. The method is essentially to supply the tools needed for us to apply the same techniques used for analysing physical, laboratory experiments to numerical experiments.

The aim is to produce a table of models with their associated dimensionless groups, some exact solutions where these can be found, and the simplified models that follow when some terms are small.

Adaptive finite element approximation of kinetic models for dilute polymers - AWE: Peter Johnson and Prof. E Süli

The aim of the project is to develop adaptive finite algorithms for dumbbell-type kinetic models for dilute polymers.

A typical model of this kind (such as FENE - Finitely Extensible Nonlinear Elastic model) involves nonlinear coupling of the incompressible Navier-Stokes equations, including the divergence of the non-Newtonian extra-stress tensor as source term, and a high-dimensional (typically (4+1)-dimensional or (6+1)-dimensional) degenerate parabolic equation (Fokker-Planck equation) with an unbounded drift coefficient.

The project will focus on simplified, low-dimensional, versions of this model, with the aim to understand the amount of computational work that needs to be invested into solving the Fokker-Planck equation for the probability density function, associated with the random fluctuation of the polymer molecules in the solvent (an incompressible Newtonian fluid), in order to obtain accurate numerical approximations of macroscopic quantities, such as the velocity and the pressure.

The project will require familiarity with spectral methods, finite element methods and their mathematical analysis (studied in the Hilary Term course "Finite Element Methods for PDEs"), and strong background in fluid mechanics.

The Nelder-Mead Algorithm with constraints - NAG: Mick Pont and David Sayers, Dr Andy Wathen

If one considers z=f(x,y) as a surface then any three points on this surface define a triangle with a lowest (and highest) vertex or edge. By folding over an edge (taking the mirror image of the highest vertex in a side) and expanding or shrinking such a triangle depending on its current orientation, one can define an algorithm for finding a minimum point of the surface. The extension of this idea of a tumbling and stretching simplex in higher dimensions is the basis of the Nelder-Mead (Simplex) algorithm for unconstrained optimization.

Many optimization problems however require minimization of a function of several variables in the presence of linear (hyperplane) or nonlinear constraints.

The well-known Numerical Algorithms Group software library currently contains a simple implementation of the Nelder-Mead algorithm for unconstrained optimization and NAG have an interest in understanding and implementing an extension of this algorithm to incorporate linear and possibly nonlinear constraints. This MSc project would be to investigate how constraints might be applied and how the dynamic behaviour of such a modified algorithm might differ from the unconstrained version.

Skills required: basically accessible to most students, but best for those with an interest in numerical algorithms.

Radial Basis Function Approximation - NAG: Mick Pont and David Sayers, Dr Andy Wathen

There are considerable advantages (as well as some less obvious disadvantages!) in using approximation spaces for 2- and higher dimensional approximation which do not require a geometric mesh or grid structure. One of the most elegant and powerful set of approximating functions are the so-called "Radial Basis Functions". In a radial basis function approximation one selects a number of "centres" x_i &isin R^m, i = 1, . . . , n and a single function φ:R &rarr Rⁿand seeks an approximation of the form p(x) = ∑ⁿ_i=1 α_i φ ( || x-x_i ||).

That is one seeks an approximation which is an expansion of functions which vary only radially about the centres. In one dimesion (m = 1) for example choosing φ(r) = r gives the functions | x - x_i | which are continuous but not differentiable at x_i: any linear combination is therefore a continuous piecewise linear or linear spline function. One can similarly build cubic spline spaces with φ(r) = | r³ | (which is in C² but not C³ if you think about it!). There are a number of radial functions φ which are used in 2- and 3-dimensions in various different situations.

This MSc project would involve learning about such radial basis function representations for the problems of data interpolation - finding an approximating function that exactly passes through a given set of data points - and to consider how to implement them in software. The particular context would be 2-dimensions for this project and the use of so-called thin plate splines which generalise the idea of a cubic spline from one dimension.

Skills required: basically accessible to most students, but best for those with an interest in numerical algorithms.

A Partial Differential Equation Model of Lipid Metabolism - Unilever: Dr Marcus Tindall

With the incidence of obesity on the increase understanding how the body digests fat (lipid metabolism) is an area of growing research. When the liver digests fatty acids it excretes lipoprotein particles comprised of triglyceride and cholesterol. The ratio of triglyceride and cholesterol content of each particle defines the lipoprotein type and has important consequences on the "healthiness" of the particles. For instance, an abundance of low density lipoproteins (LDL) can lead to coronary artery disease.

We have recently formulated a partial differential equation (PDE) model of lipid metabolism in collaboration with the Systems Biology Group at Unilever Colworth Laboratories. Unlike other models in the literature, our PDE approach allows the distribution of lipoprotein particles to be tracked in a two-dimensional cholesterol-triglyceride space. However, the sparseness of the particle distributions within the cholesterol-triglyceride and the effects of implementing certain biological mechanisms raises some interesting computational issues.

This project will primarily focus on implementing current methods for solving hyperbolic PDEs in two dimensions, using standard computational packages as well as coding of specific numerical routines. New methodologies will be developed, where required, to improve computational efficiency of the model solvers. The project will also involve extending the current model to include new biological mechanisms.

Clinical Assessment of Frailty - Unilever: Ming Li and Dr Patrick McSharry

The assessment of human frailty conventionally involves the observation of individuals constructed to undertake specific tasks and may include physical function measures such as balance, repeated chair stands and short walks. Such tests are frequently interpreted in a categorical way or an assessment made on the time to complete the procedure. Healthy Ageing have been interested in exploring the progressive changes in physical function that may lead ultimately to frailty and are currently engaged in developing and applying new procedures in aged cohorts to explore potential continuous measures of physical function.

The Unilever CR APPT study is collecting triaxial accelerometer data from four body sites (left and right wrists, left and right ankles) on many healthy individuals divided by age decade between twenty and eighty years to establish baseline movement data in the subjects when executing specific tasks from a short physical performance test battery. These tests comprise three separate balance tests (feet side by side, semi-tandem and full tandem stands), a repeated chair stand and a short measured (8 ft) walk. The accelerometers acquire estimates of acceleration (referenced to 'g') from each of the three axis. The objective of the study is to establish baseline data

The exploration of this data for new objective (categorical and continuous) measures of age associated functional change is the starting point for this project.

The exploration of this data for new objective (categorical and continuous) measures of age associated functional change from continuous accelerometer measurements across four body sites.

Using Plastic Self-Organizing Maps for Classifying Radar Data - Thales Group: Ian Ellis (Aerospace Division), Diven Topiwala (Thales Research), Dr Mason Porter and Philip Bond.

The purpose of radar systems, which are ubiquitous for defence purposes, is to detect objects and determine their identities and possible threat levels. One method for collecting intelligence from radars is to simply listen for the signals emitted by other radar systems in the local environment. By identifying which radar systems are in the environment, and what their pattern of behaviour is, it is possible to identify the other vehicles in the area, where they are, and whether or not they pose a threat to you.

A self-organizing map (SOM), or a Kohonnen network, is a static system that is used to classify a set of input vectors (i.e., radar emitter patterns). One can generalize this concept to "plastic SOMs" (PSOMs), which also classify input vectors but learn continuously as new input vectors arrive. (That is, the PSOM is an example of a continuously learning classifier where, instead of there being separate, sequential training and operating phases, these two phases occur simultaneously as part of the same process.) The goal of this project is to use PSOMs for radar emitter classification in non-stationary environments, in which the characteristics of objects or noise can vary with time, new types of objects can arrive, and so on.

This project will use ideas from network theory, data mining, statistics, and perhaps time-series analysis as well as simulated "training data" provided from THALES with which to test and develop these ideas. (This data will come from a specific radar and radar mode, and the objective would be to derive radar type and class of intent.) In a PSOM, one represents the data using a dynamically updating graph in which nodes and edges are added and deleted as new data comes in. In this project, appropriate mathematical tools will be used to analyze and improve the present graph-updating procedures, compare them to various graph clustering (community-detection) algorithms, and develop and test quantitative measures of performance based on graph-theoretic concepts.

The goal of the project is to take PSOM analysis as far as possible in the radar context. For example, the student will attempt to use appropriate mathematical and computational techniques to classify and group detected objects according to class of radar, type, mode of operation, and particular instance of detection. In each case, the reliability of the classification should be as high as possible, so it will be essential to devise and test metrics that measure this reliability.

Adaptive Beamforming - Thales Underwater Systems: Nigel Parsons (TUS) and Dr David Allwright(OCIAM)

A sonar set consists of an array of hydrophones (underwater microphones) and the associated electronics, signal processing and software. The processing is designed to detect contacts, i.e. approximately plane acoustic waves incident on the array from a definite direction, and to distinguish these contacts from noise, i.e. effectively random pressure fluctuations that may be due to turbulence or to acoustic waves coming from the many other sources of underwater noise (e.g. rain, surface waves, distant shipping etc.). The statistics of this background noise are generally unknown, and time-varying, and therefore have to be estimated from the data gathered by the array at the same time as detecting contacts. The process of estimating the background noise at the same time as listening for contacts from definite directions is called adaptive beamforming. When the array has many hydrophones, it is difficult because (among other things) the estimated noise variance matrix is ill-conditioned. TUS is therefore interested in work on Adaptive Beamforming for large acoustic arrays. One approach of interest is the use of the "Support Vector Machine" (SVM) formalism. This is intended to aid in the optimisation while satisfying the required constraints. (These are the constraints on the sidelobes and beamwidth that characterize how well the array distinguishes waves incident from different directions.) The approach is described in "Robust beamforming with sidelobe control using support vector machines", C.C.Gaudes et al. IEEE Transactions on Signal Processing, Vol.55, pp574-584 (2007) and the aim of the project would be to study the approach mathematically, compare it with other ways of optimizing the beamformer, and compare its performance in the circumstances of interest to TUS.

We are specifically interested in applying adaptive beamforming to two dimensional (e.g. bow and flank) sonar arrays in order to improve detection performance in the presence of strong interference sources and to improve the performance in non-isotropic noise fields. A typical 2-D array of interest has about 1000 hydrophones and, therefore, about 1000 adaptive degrees of freedom are available. If the simplest MVDR adaptive beamformer is used, around 3000-10000 data snap-shots (i.e. time samples) will be required to estimate the O(1000 1000) data covariance matrix accurately enough to achieve a reasonably low level of "sidelobe jitter". It is unlikely that the data statistics are sufficiently stationary for this. Furthermore, uncalibrated amplitude and phase mismatches between the hydrophones will tend to cause further problems with target cancellation.

An interesting approach is to apply a large number of desired inequality constraints to the beam pattern in order to reduce the available degrees of freedom. By relaxing the inequality constraints the adaptive beamformer can be reformulated as a support vector machine (SVM). The SVM can be solved using an iterated reweighted least squares (IRWLS) method. We are looking for an efficient method for a solution which evolves gradually with time due to non-stationarity. The method will be coded up in MATLAB and tested on simulated (and possibly real) data.

The aim is to demonstrate that the proposed SVM provides a robust solution when the number of available data snap-shots is inadequate for the standard MVDR approach and to demonstrate that sensor errors have less impact on the energy of the signal of interest. Alternative approaches to this problem could be considered.

Spoking phenomenon for sonar arrays with domes - Thales Underwater Systems: Peter Brazier-Smith (TUS), Dr Gareth Jones and Dr David Allwright (OCIAM)

The objective of this project is to identify and quantify the source of a spoking phenomnenon observed in the response of a sonar array under a sonar dome. Some previous work on this can be seen in the OCIAM MSc dissertation of Gareth Jones. Random forcing (due to turbulence) on the shell is observed to cause an apparent plane wave incident on the hydrophones from a particular direction. Studies at Thales have shown that this effect is due to both the curvature of the shell and the asymmetry of how it is coupled to the surrounding steel casing. So the project will involve studying the vibrational response of a curved shell at an asymmetric junction. An interest in waves would be a good background, and knowledge of elastic wave propagation in a shell would have to be acquired as part of the project.

Modelling Submarine Jet and Wake Interactions on Towed Array Shape Estimation. - Thales Underwater Systems: Peter Brazier-Smith (TUS), Dr Jim Oliver and Dr David Allwright (OCIAM)

A towed sonar array is a long flexible array of hydrophones (underwater microphones) towed (by a cable) behind a surface ship or submarine. In order to interpret correctly the acoustic signals received by the array, its position must be known accurately. The position is estimated by combining data from heading sensors along the array with a mathematical model for the shape of the cable and array called DynaTow. However, DynaTow assumes the array is being towed through water at rest. In practice the tow cable sometimes passes through the flow field aft of the submarine, and this has effects on its position. If the flow field is known, there are accepted models for the average force on the cable. The aim of the project is to establish good approximate parametric models for the average flow field in the region aft of the submarine. This can be considered as consisting of the jet from the propulsor (the main drive of the submarine), the wake from the whole submarine, and the vortices from the fins when turning. Trials in which the array was towed from one of the horizontal tail-fins have shown the effects that can occur when the cable passes through the propulsor jet: in this case the deviation of the actual array shape from the shape predicted by DynaTow only occurs when the submarine turns in a way that makes the cable enter the jet region. A previous study by Ed Tarleton (MSc, OCIAM) used the Sclichting model of a turbulent jet, and showed that the velocity field of the jet was sufficient to cause array deflections of the observed order of magnitude. However, TUS would like to take this further and include either the vortices or wake, whichever is likely to have more effect, and to have a better representation of the jet velocity field during turns. TUS has relevant real data against which model enhancements can be compared.

It is planned that the project will begin with a review of the literature on the flow fields in the jets, wakes and vortices of self-propelled bodies. In particular it is expected that work of Dr Marvin Jones (Southampton) on vortices will be relevant. Although the jet is turbulent, TUS are happy for it to be treated by approximate theories for a slender turbulent jet.

Model for predicting the conditions at the inception of rucking in Towed Array Sonar Hose - Thales Underwater Systems: Steve Randall (TUS), Dr R Whittaker and Dr David Allwright (OCIAM)

A Towed Array Sonar array consists of a long length (several hundred metres) of flexible, viscoelastic PVC hose containing highly sensitive acoustic and non-acoustic sensors plus associated electronics and wiring. In use, the array is pulled through the water by a ship or submarine. Under normal conditions the hose extends due to the hydrodynamic drag over the surface. Under more extreme conditions (high speeds and warmer waters) the hose can extend/creep excessively causing it to collapse. The type of collapse observed is generally referred to as "rucking". Visually, rucking is the type of collapse seen when one pushes the sleeve of a sweater or shirt up ones arm.

To date a relatively simple MATHCAD program has been generated to model the conditions that cause rucking to occur - this was based on Euler Buckling Theory, which is now thought to be inadequately representative of the type of collapse seen in practice. In addition to the type of collapse needing to be modelled in a more representative way, the complexity of the modelling will be increased by the tight fitting internal components that constrain the movement of the PVC hose at intervals along its length.

The array contains a central cable that takes the main tension. The drag force is transmitted from the hose to this cable by bulkheads, rigid discs occupying the cross-section of the hose, connected to the central cable and fitting tight against the hose. These bulkheads are regularly spaced along the length. The project is to analyse this, calculating the stress distribution in the hose and either using existing models for this kind of rucking or developing something more appropriate. The work requires interest in solid mechanics but this can be learned during the project if necessary.

If a model can be developed that will more closely predict the conditions leading to the inception of rucking then it may be possible to take measures in the design of the Towed Array Sonar that would extend the operational parameters of the sonar system. Any relaxation in operational limitations imposed by the sonar equipment will always be welcomed by ship or submarine commanders!

Precise Modelling of Curved Sonar Domes - Thales Underwater Systems: John Halstead (TUS), Dr Peter Howell and Dr David Allwright (OCIAM)

In some sonar sets, acoustic waves travelling through the sea are incident first on a sonar dome (a curved shell, designed to be acoustically transparent) then travel through the still water behind the shell and then are incident on the hydrophones. TUS study the vibration of the shell in response to incident acoustic waves, and generally do this in terms of ray theory and the "tangent plane" approximation. In this, an acoustic ray is considered incident on the shell at a point, and the acoustic field transmitted into the water behind the array is calculated as if the shell were replaced by a flat sheet of the same material tangent to the actual shell at that point. In reality the array is usually a doubly curved shell. TUS have assumed that the errors due to this approximation are small, but would like to quantify the size of the errors in terms of the radii of curvature of the shell and its thickness and the acoustic wavelength. The shell material will be treated as linearly elastic and isotropic. An interest in waves would be a good background, and knowledge of elastic wave propagation in a shell would have to be acquired as part of the project.

Capillary Agglutination Technology - Platform Diagnostics, Prof. Jon Chapman

In medical diagnostic tests, including pregnancy testing and tests for typed red blood cells, a small fluid sample is placed at one end of a capillary channel, which has been lined with a dried reagent. If the sample contains the analyte (the substance being tested for) then an agglutination reaction occurs between it and the reagent in the channel, and the agglutinated complexes progressively slow the flow and may even block the channel, partially or completely, so that the flow only reaches the far end very slowly, or not at all. The aim is that this should give a reliable detection of quite low concentrations of analyte in the sample. Platform Diagnostics would like a mathematical model of the process, so that, for known binding forces in the agglutination complexes, the channel size and shape, and the fluid viscosity, can be designed to maximize the reliable detection of low concentrations. A key question is how the flow time depends on channel size, fluid surface tension and viscosity, (a) in the absence of agglutination, and (b) in the presence of agglutination.

The flow problem will involve lubrication theory for the channel (with different cross-sectional geometries), and capillary statics (Laplace-Young equation) for the driving force. The project will involve a mix of asymptotic analysis and numerical simulation.

Physical Applications Projects

The Fracture of Plasma Crystals by a Laser Beam - Prof. John Allen

Research groups in Germany and Taiwan reported, in 1994, that "Plasma crystals", formed of charged dust particles, could be produced in the laboratory. This subject has since attracted considerable attention from both workers in space science and those interested in laboratory plasmas and their applications. Dust particles in a plasma environment normally become negatively charged due to the collection of electrons. They behave rather like miniature Langmuir probes and attain a certain floating potential such that electron and ion currents become equal in magnitude, the net current to the dust particle being zero. In the laboratory, however, the weight of micron-sized particles pulls the dust out of the plasma and into the adjoining space charge sheath. In this region the gravitational force can be balanced by the electric force on the charged dust particles; other forces, such as ion drag, being less important.

It was first demonstrated in Oxford that such plasma crystals could be fractured by a low power laser beam. A series of measurements has since been made with various crystals and different laser powers. These results certainly merit further analysis. It was found, not surprisingly, that a threshold power is required to fracture the plasma crystal. The object of the present exercise is to calculate, for different models, the forces that hold the crystal together. These are not known at present, although various ideas have been put forward.

Security criteria and locking quantum information - Prof. Artur Ekert

Quantum theory allows atoms, photons and other quantum objects to store information that is not restricted to only the binary 0 or 1, but can also be 0 and 1 at the same time. Such objects are called quantum bits, or qubits. Subtle non-local correlations between qubits, known as entanglement, are responsible for many counter-intuitive properties of quantum information and their potential applications include quantum cryptography and quantum computation. This project is about the correct use of statistical distance and its quantum extension - the trace distance - in defining security of binary strings.

In the classical scenario an adversary (or eavesdropper) may have some partial knowledge about a binary string x via a correlated random variable y, and this knowledge may be quantified by some joint probability distribution p(x,y). In the quantum case x and y are entangled and described by a bipartite density matrix. The questions is how to define security of x and under what conditions one can achieve a prescribed security. In particular, is the definition of security robust. For example, suppose some bits of x are revealed in public, are the remaining bits secure?

Assumed knowledge: formalism of quantum physics including density matrices, probability theory. It is hoped that the project will lead to better understanding of the phenomenon of information locking in quantum information science.

Algebraic quantum control of graphs - Dr Daniel Burgarth

It is often possible to steer a quantum system by acting on a small part of it only. This has important applications in quantum computing and nuclear magnetic resonance.

There is an elegant theory to check controllability by computing the algebra spanned by certain matrices. However it is often not practical, as the dimensionality of these matrices blows up exponentially with the size of the system. This project aims at the development of simpler sufficient criteria for graphs describing the physical interactions. Ideally, these will depend only on classical graph properties.

Requirements: basic knowledge of Lie groups and graph theory. The problem can be approached numerically (matlab) and analytically depending on the preferences of the student.

Mathematical Modelling of Rowing - Dr Marcus Tindall

Rowing remains one of the top medal hopes for Britain in both the 2008 and 2012 Olympics. However, which elements of the sport can be optimised to give Britian the winning edge? This project will focus on extending and developing a recently published simple model of rowing in a single scull to larger sculling and sweep boats. Possible areas for extension include modelling the effects of multiple rowers in a boat, determining the effect that differences in technique have on the overall boat speed and determining how a rower should move in order to achieve a set boat speed. We will also wish to include fluid dynamical aspects affecting boat speed and performance. The project will include undertaking both numerical and analytical modelling.

Swimming in viscous fluids - Dr Paul Dellar

This computational project will study the ability of deforming objects to propel themselves through a viscous fluid. At low Reynolds numbers the fluid motion is reversible in time, so only a complicated non-reversible deformation of the object can produce a systematic motion. The project will involve choosing a time-dependent shape for the boundary of a two-dimensional object embedded in a fluid, and calculating the net hydrodynamic force over the boundary to find the object's velocity.

This project is approachable using MATLAB, but a basic knowledge of a compiled language (Java, C, Fortran etc.) would be an advantage later on.

Motion of contact lines over rough surfaces - Dr Paul Dellar

Treatments of surface tension using surface energies have been criticised because they seem to give the same contact angle for advancing and retreating contact lines. In experiments, the two contact angles are different. However, it has been suggested that the difference between theory and experiment is due to the contact line's motion being obstructed by microscopic surface roughness. The project will test this hypothesis by performing numerical experiments using a surface-energy description for the advancing and retreating motion of a liquid drop on a rough surface.

This project is approachable using MATLAB, but a basic knowledge of a compiled language (Java, C, Fortran etc.) would be an advantage later on.

Citation Networks with Positive and Negative Links - Dr Mason Porter

A network (i.e., graph) consists of nodes connected by links. There has been a considerable amount of research in recent years on network growth models, which have been used to attempt to model paper citation networks, the World Wide Web, and more. Many of the models in the literature use a form of "preferential attachment," in that new nodes are more likely to link to nodes with high degree (papers with many citations or popular web sites like google) than those with low degree. However, citations can either be positive or negative, as illustrated by the case citation network of the United States Supreme Court (a decision on a case can either support or refute prior decisions). Motivated by this example, the goal of this project is to study both numerically and analytically the statistical properties (degree distribution, etc) of network growth mechanics with both positive and negative edges. The results will then be compared with the U.S. Supreme Court case citation network and possibly other appropriate data sets. The project can also involve community detection in this network employing recently-published methods (or generalizations thereof).

Anderson Localization in Granular Lattices - Dr Mason Porter

Consider a granular lattice composed of a one-dimensional chain of beads consisting of different materials of varying material properties (large mass versus small mass, stiff versus soft, and so on). If one hits a chain with a striker particle, a nonlinear wave known as a solitary wave propagates through the system. The purpose of this project is to analyze what happens when the sequence of particles in this chain is "randomized" with different types of particles and/or defects and to investigate "Anderson localization" or an appropriate generalization in this system. (Anderson localization is a general phenomenon that refers to the absence of diffusion - and the concomitant strong localization - of waves in a random medium. It was originally shown to arise for electrons inside a semiconductor, provided the degree of randomness from impurities or defects is sufficiently large.) Time permitting, it will also be interesting to start investigating Anderson localization in two-dimensional granular lattices.

Collisionally Inhomogeneous, Multi-dimensional Bose-Einstein Condensates - Dr Mason Porter

The existence of Bose-Einstein condensates (BECs), a macroscopic quantum phenomenon, was predicted in the 1920s by Bose and Einstein and first observed experimentally in 1995 by Cornell, Wiemann, and Ketterle. A BEC can be achieved via evaporatively cooling when a group of bosons condense into the same (ground) state. The goal of this project is to look for structures in the presence of spatially periodic nonlinearities in two-dimensional BECs, which are described by the Gross-Pitaevskii equation (a cubic nonlinear Schrodinger equation). Relevant questions include whether the periodic nonlinearity might stabilize against collapse (similar to the case of periodic linear potentials), what happens in the defocusing case (to, for example, vortex solutions), and what happens when the sign of the nonlinearity changes as a result of the modulation.

Surfactants in driven thin film fluid flows - Prof. Thomas Witelski

Surfactants (short for "surface active agents") are chemical compounds like detergents or soaps that reduce the surface tension of fluids. Impurities in pure fluids also act as surfactants. Analysing the behaviours of fluid/surfactant systems and comparing against experimentally observed dynamics can be used as a route to developing better mathematical models for pure fluids.

Consider slow flow of a thin layer of a viscous fluid down an inclined plane under the influence of gravity. Using lubrication theory for low Reynolds number flows reduces the governing Navier-Stokes equations to an equation for the free-surface height h(x,t). A second equation governing the surfactant concentration Γ(x,t) follows similarly to yield the system of PDEs:
h_t + (¹⁄₃ h³ )_x - (¹⁄₂ Γ_x h² )_x = β (¹⁄₃ h³h_x )_x - σ (¹⁄₃ h³h_xxx )_x
Γ_t + (¹⁄₂ h² Γ )_x - (h Γ_x Γ )_x = β (¹⁄₂ h² Γ h_x )_x- σ (¹⁄₂ h² Γ h_xxx)_x+δ Γ_xx
Of interest are various initial-boundary value problems for this PDE system that would correspond to physically realisable experiments describing adding surfactants to fluid flows. Numerical simulations of the governing equations coupled with perturbation methods and ODE/PDE analysis can be used to yield predictions for classes of behaviours given by traveling waves and self-similar solutions.

Project goals include deriving multiple-scale expansions for the accumulation of the surfactant at the contact line and modifying the model to include saturations effects at the maximum surfactant concentration (Γ_CMC).

Modelling the Post Aligned Bistable Nematic Device - Dr Apala Majumdar

Liquid crystals are an intermediate phase of matter between the commonly observed solid and liquid states. In the simplest liquid crystal phase, the nematic phase, the constituent rod-like molecules translate freely as in a conventional liquid but whilst flowing, tend to align along certain locally preferred directions [1]. The existence of preferred alignment directions results in very interesting optical properties, which make liquid crystals suitable for use in display devices such as liquid crystal displays. Most of the displays in use today are monostable .They have two admissible states - one transparent to incident light and the other opaque, only one of which is stable whereas the other requires a constant source of external power.

Recently, there has been a lot of emphasis on the development of bistable displays that can support multiple stable configurations with contrasting optical properties, resulting in reduced power consumption and higher resolution. Most bistable displays use a combination of complex surface morphologies and surface treatments, in order to stabilize the different configurations. An example of such a bistable display is the Post Aligned Bistable Nematic (PABN) cell, designed by Hewlett-Packard laboratories [2]. The PABN cell consists of a periodic array of rectangular posts between two planar substrates and supports two optically contrasting states (one bright and the other dark), both of which exhibit long-term stability.

The theoretical and numerical modelling of such bistable devices raises a number of interesting mathematical questions and is also of great interest to the research groups based in industry. The main aim of this project is to develop a sound theoretical model for the static stable configurations in the PABN device, particularly in the presence of electric fields and then investigate the switching protocols between the optically contrasting stable configurations. We have already carried out an analytic and numerical study of the static configurations, in the absence of external fields [3]. We now want to include the effect of external electric fields and study the evolution of these systems with time. This project will require a good background in ordinary differential equations, elliptic and parabolic partial differential equations (only up to an undergraduate level) and some experience in numerical methods is also desirable.

Microemulsions - Prof Thomas Witelski and Dr Chris Breward (OCIAM), Prof Bain (Chemistry, University of Durham)

Microemulsions are liquid-liquid mixtures which occur in foodstuffs, many areas of biology, and in cosmetics. In certain temperature ranges, the surface tension between the two phases can be reduced dramatically. This effect makes it possible to use "optical tweezers", devices that use a laser to exert a small force, to deform and manipulate individual droplets. In experiments containing oil, water and a soluble surfactant, a second droplet can be drawn out from a parent which remains connected by a small thread of liquid. This thread can be over 10 times the droplet radius in length but almost imperceptible in thickness. The presence of these nanothreads can be inferred from the dynamics seen when a droplet is released from the tweezers.

In this project we will use fluid dynamics to predict the shape of the objects in the tweezers and the flow in the nanothreads. The skills required will be mathematical modelling, fluid mechanics, perturbation methods and numerical solution of PDEs.

The specific aims are: to determine the shapes of the oil droplets in 3D; to determine the shape of the oil-water interface at the thread-droplet junction; to determine the thinnest nanothread that can be produced; to model the flow along the nanothread.

Numerical Analysis Projects

New Applications of the Chebfun System - Prof. Nick Trefethen

The chebfun system is an extension of Matlab that computes numerically with functions instead of numbers. It's an object-oriented package whose mathematical basis is Chebyshev expansions. For example, to find the roots of a Bessel function in [0,100] to 13 digits of accuracy in a fraction of a second, you can type:

Chebfuns are amazingly fun to play with and the system is moving fast this year in the hands of a four-man team: myself, Ricardo Pachon, Rodrigo Platte, and Toby Driscoll. There are many many avenues to explore, including waveform relaxation, continuous analogues of matrix factorizations, ODE boundary-value problems, ODE initial-value problems, PDEs, multidimensionality, automatic edge detection, approximation theory, analytic continuation, and more. And those are just some methodological matters; in addition there are potential scientific applications in all kinds of areas.

If an MSc student would like to get involved in the chebfun project for a thesis, I would be happy to design a project in line with his or her interests.

The Numerical Solution of the Equations Governing Cardiac Electrophysiology - Dr Jonathan Whiteley

The equations that govern cardiac electrophysiology, namely the monodomain equations or the bidomain equations, arise from modelling electrical conduction and biochemical reactions in cardiac tissue. These equations take the form of an alliptic PDE, a parabolic PDE, and a system of ODEs at each point of cardiac tissue, and model processes that occur on a wide range of timescales.

The governing equations are standard mathematical equations for which reliable solution techniques already exist. Whilst the pros and cons of these techniques are relatively well understood for problems where the computation time if of the order of minutes, the advantages and disadvantages are less clear for cardiac modelling where computation times may be of the order of days. The primary aim of this project is to investigate the most appropriate algorithm for solving these equations. A secondary aim is to investigate whether the efficiency of an algorithm may be heavily dependent on its computational implementation. This project requires an interest in the algorithms that underpin the numerical solution of differential equations.

It is expected that this project will (i) provide an evaluation of some of the numerical techniques currently used, and (ii) suggest improvements to the algorithms currently used.

Biological and Medical Applications

Modelling and optimizing bioreactors for growing bone tissue - Dr Hua (Cathy) Ye and Dr Radek Erban

Millions of bone replacements are needed worldwide every year for different types of bone defects. Surgeons traditionally use either implants made of different synthetic materials (ceramics, metals or polymers), or, in ideal case, patients' own bone from other parts of the body (autografts). The quality of bone tissue is superior to synthetic materials but the availability of autografts is obviously limited. Thus much attention has been recently devoted to the bone tissue engineering, i.e. growing implantable bone tissue outside the human body.

Dr. Ye's lab (Department of Engineering Science) is using the so called hollow fibre membrane biorectors (HFMBs) to grow bone tissue. HFMBs are capable to supply nutrients and oxygen to cells, to remove waste from cells and to allow the growth of three-dimensional bone structures. In this project, you will model nutrient, oxygen and waste transport in HFMB. You will start with analysis and numerical simulation of previously developed models of HFMB [1,2]. Then you will include a more realistic description of some processes (e.g. consumption/production of nutrients/waste) in the model. The ultimate goal of mathematical modelling is to find an optimal design of HFMBs in order to grow bone tissue most efficiently.

There are no special prerequisite skills needed for this project. Modelling will be based mostly on partial differential equations and their numerical solutions (finite element method or finite differences). Depending on your interest, you might also investigate the applicability of alternative modelling approaches (e.g. stochastic simulations or cellular automata).

Automated Analysis of Large Data Sets (for example photographs or biological data) by Diffusion Maps - Dr Radek Erban

In many applications, data comes as high-dimensional objects. For example, a black-and-white photograph can be represented as a [500 times 500] matrix where 500 is the height/width of the photograph and each matrix entry gives the brightness of the corresponding pixel. Thus the photograph is the vector in the [500 times 500]-dimensional space. Consider that you take a set of photographs of a person's face. Let us suppose that the images differ only by rotation of the head. Then each image can be described by the camera angle (viewpoint), which is a two-dimensional vector. Although the set of images is a sequence in the [500 times 500]-dimensional space, there is also a low-dimensional description of this data set (parametrized by the camera angle). The similar situation occurs in biological systems where the high-dimensional data can often be efficiently described by few coordinates. A natural question is how we can find such a low-dimensional description in an automated way.

In this project, you will study the so called diffusion maps which are useful in detecting the low-dimensional description of high-dimensional systems (e.g. they can detect the camera angle as the appropriate description of the images in the previous example). The diffusion maps approach is relatively new. So the project will involve understanding the theory behind the diffusion maps and an application of the technique to a high-dimensional system (either images, or biological data).

There are no special prerequisite skills needed for this project. The mathematics involved is a mixture of linear algebra, differential equations, graph theory, geometry, mathematical analysis and some computational mathematics (mostly eigenvalue/eigenvector problems).

Oscillatory Behaviour in Biological Systems - Dr Radek Erban

Periodic oscillations are common in many biological systems (e.g. cell cycle which is a series of events that take place in a cell leading to its replication). A typical modelling question in systems with oscillations is to compute the period of oscillations and to understand its dependence on the parameters of the model.

The biochemical systems (e.g. the cell cycle) are traditionally modelled using the ordinary differential equations (ODEs) which describe the time evolution of species concentrations. One advantage of such ODE models is the extensive set of existing theoretical and computational tools for their analysis and efficient simulation. A disadvantage of such ODE models of biological processes is that in the cell there can be a relatively low number of molecules of some of the species involved (e.g. there might only be a single copy of a gene present in the cell). This renders ODE models inaccurate or even inapplicable; individual-based stochastic simulation algorithms then become more appropriate than ODEs. Understanding oscillations in stochastic models is more challenging. It can be done through analysis of certain partial differential equations (PDEs). The goal of this project is to study biological systems with oscillations and the methods for estimating the period of oscillations of their stochastic models.

The mathematics involved is a mixture of differential equations (ODEs, PDEs) and stochastic simulation algorithms. Basic knowledge of probability and Matlab is sufficient for understanding the stochastic algorithms. There are no special prerequisite skills needed.

Receptor-Receptor Interactions and Complex Networks in Bacterial Chemotaxis - Dr Marcus Tindall

Signal detection and amplification in bacterial chemotaxis is a result of interactions between neighbouring groups of receptors. But how do the receptor clusters form to initiate signalling? Bacteria such as E. coli detect and respond to stimuli in their extracellular environment by using an array of internal biochemical signals which signal between the membrane receptors at one of the cell and the flagellar motors at the other. Rotation of the flagella propel the bacteria through their environment. We have recently extended an ordinary differential equation model of receptor-receptor interactions to account for interactions between two receptor types. However, the resultant network is complex and we wish to be able to distinguish between those parts of the network which contribute to the formation of certain receptor components. This project will utilise mathematical techniques (theory of ODEs), complex network theory and experimental work to predict the formation of certain types of receptors and the interactions between them. The modelling work will include both analytical and numerical aspects and the student undertaking the project may also become involved in experiments in association with Prof. Judy Armitage at the Department of Biochemistry, Oxford. The student will be supervised by Dr Marcus Tindall (Mathematical aspects, Centre for Mathematical Biology) and Dr Jukka-Pekka Onnela (Complex network theory, Dept. of Physics).

Modelling Cerebral Control of Breathing - Dr Jonathan Whiteley

An irregular breathing pattern is observed in many subjects with respiratory disease. The volume of gas inhaled, and the rate of breathing, are controlled by sensors at several locations in the body. These sensors are believed to respond to the partial pressure of both oxygen and carbon dioxide in blood. For example, should these sensors detect that the oxygen partial pressure - and therefore oxygen content - is low, this will trigger an increase in the volume of gas inhaled per unit time. As there will be a delay before the increase in oxygen partial pressure is detected by the sensors, it is possible that the body may over-compensate for the original oxygen deficit, and increase the volume of gas inhaled per unit time to a much higher level than is actually needed. This is believed to be the cause of irregular breathing in these subjects.

In this project we will develop the mathematical model described above to take account of a condition known as atelectasis - the collapse of poorly ventilated regions of the lung. Current models of atelectasis that do not take account of respiratory control predict inaccurate times for the region of the lung to collapse. This model will consist of a coupled system of ordinary differential equations and delay differential equations. These equations may be solved in {\sc Matlab}, allowing us to focus on the modelling.

It is expected that this project will provide a novel mathematical model of lung collapse that may be compared with existing experimental data.

Functional Magnetic Resonance Imaging of the Brain - Dr Ivana Drobnjak

Functional Magnetic Resonance Imaging (FMRI) is a completely non-invasive method of imaging brain function in-vivo. However, images produced in an FMRI experiment are corrupted by several forms of imperfections, known as artefacts. These artefacts can result from, for example, rigid-body motion of the head, magnetic field inhomogeneities, chemical shift and unwanted induced currents (eddy currents).

In order to achieve reliable and robust use of FMRI it is necessary to minimise or remove the artefacts. To this end, a computational model of the FMR image acquisition process was built, POSSUM, that can simulate all of the above-mentioned artefacts, which is invaluable for developing methods for reducing these artefacts. POSSUM uses a geometric definition of the object (brain), Maxwell's equations (to model the magnetic field in the scanner), Bloch equations (to model the behaviour of the nuclear magnetisation) and a model for the Blood Oxygen Level Dependent (BOLD) activations. Furthermore, it simulates rigid-body motion of the brain by solving Bloch equations for a continuously moving object.

POSSUM forms a part of the FMRIB Software Library (FSL) that is a very widely used analysis package for FMRI (used in over 600 laboratories worldwide). Currently, simulations can only be used for imaging sequences that are based on gradient-echo sequences, which excludes several interesting imaging methods (e.g. fast spin-echo, SSFP, DWI). This project aims to extend the current software to make simulations of spin-echo based imaging sequences possible.

This project would involve finding fast, accurate and efficient numerical/analytical solutions to a set of ODE's and then implementing those in C++ language into the already existing software environment. The project would also involve learning the basics of Medical Imaging for functional MRI.

Multiscale modelling of cell motility in neurogenesis - Dr Sarah Waters and Dr Jim Oliver (OCIAM), Dr Francis Szele (Dept. Physiology, Anatomy and Genetics)

Neurogenesis is the process by which neurons are created, and is responsible for populating the growing brain. Neurogenesis persists in adult mammals: in particular, tens of thousands of neuroblasts generated in the sub-ventricular zone (SVZ) migrate daily in the rostal migratory stream (RMS) to the olfactory bulb (OB) (see Figure 1a and Comparative Neurology for movies). Moreoever, neuroblasts have been observed to migrate into regions of cell damage, and thus they may eventually be used as a source of cells for brain repair.

Many fundamental features of neuroblast migration are still poorly understood. Using two-photon time-laspe microscopy, Dr Szele has been able to dynamically visualise the migration of neuroblasts in situ in a non-damaging fashion. Neuroblasts are observed to migrate in longitudinal arrays, referred to as chain migration (see Figure 1a). In addition to this gross macroscale behaviour, neuroblasts move locally in complex exploratory patterns, and at average speeds slower than the long distance macroscale movement (see Figure 1b). The key questions that this project will address are

The project will begin with an analysis of the experimental data on individual cell motility: assuming each cell executes a random walk, we will ascertain quantities such as the distributions of step lengths and turning angles. This information will then be fed into a "velocity-jump" transport equation, which has been used previously to model motile cells and bacteria [1,2]. The resulting equations will then be analysed on length and time-scales much greater than those of individual steps. This will then enable us to obtain a partial differential equations for the probability density of walkers, which can be solved to determine the macroscale behaviour of the system. If time permits, this analytical approach will be complemented by Monte Carlo simulations.

No previous biological knowlege is required for this project. There is also the opportunity to spend time in Dr Szele's laboratory.

Figure 1: (a) Chain migration of neuroblasts. (b) Local movement of individual neuroblasts.

Mathematical modelling of vascular disease - Dr Sarah Waters

The most common arterial disease is atherosclerosis, which is characterised by atherosclerotic plaques. Figure 2a shows the cross-section of a healthy artery, in which there is a wide open lumen through which the blood can flow, and a healthy arterial wall. Figure 2b shows the cross-section of a diseased artery, in which there is significant plaque deposit in the artery wall, and the lumen has significantly narrowed.

It is now widely accepted that the sites at which plaques are initiated, and their subsequent development, is correlated with the wall shear stress exerted by the flowing blood on the artery wall. In particular, plaques tend to develop where the shear stress is low or where it changes direction during the course of a cardiac cycle.

A feature of large arteries is that they have significant curvature. To understand the flow in curved arteries, and the associated shear stress distribution, we have modelled the artery as a tube of uniform circular cross-section, having a centreline which lies on the arc of a circle [1]. To mimic pumping of blood by the heart, the flow is driven by a prescribed pulsatile pressure gradient [2]. The flow is governed by four key dimensionless parameters: the curvature, steady and unsteady effective Reynolds numbers, and the amplitude of the pressure gradient.

The aim of this project will be to determine how the wall shear stress distribution depends on the key parameters. In certain parameter regimes, it will be possible to construct asymptotic solutions. These will be complemented by numerical solutions (an existing code is available that can be developed as necessary) in regions of parameter space where it is not possible to obtain analytical solutions.

Figure 2: (a) Cross-section of a healthy artery. (b) Cross-section of an artery with significant atherosclerotic plaque. Images from http://www-medlib.med.utah.edu.

No previous biological knowledge is required for this project. This work will be in collaboration with Dr Jennifer Siggers, Bioengineering, Imperial College London.

Lab on a Chip: design of a microfluidic device for cell interrogations - Dr Jim Oliver and Dr Sarah Waters (OCIAM, University of Oxford), Dr Tracy Melvin (Optoelectronics Research Centre, University of Southampton), Dr Rudolfo Repetto (Department of Structures, Water and Soil, University of L'Aquila)

Advances in manufacturing technologies and the control of liquid flow on the micron-scale have made possible devices that can be used to great advantage in miniaturizing and automating testing processes in chemical and biological experiments. The paradigm has been to reduce an entire laboratory to the size of a computer chip (a "lab on a chip"). This project involves working with experimentalists to develop a mathematical model to improve the design of a microfluidic device.

We will consider a device that has the configuration illustrated in Figure 3, in which a series of parallel S-shaped channels feed a Hele-Shaw cell called the "interrogator region". The cells are seeded on spots on the base of the interrogator region, and respond to both the mechanical and biochemical environment provided by the media which flows over them.

Motivated by the proposed experimental design, we will develop models for the fluid flow and chemical transport and uptake in the interrogator region. We will exploit the existence of disparate length and time scales to simplify the resulting governing equations, which we will solve using a range of analytic and numerical techniques. The results will be used to provide insight into the experimental design.

Keywords: Hele-Shaw flow, Taylor dispersion, cell receptor kinetics, ray theory for high-Peclet number convection diffusion, numerical methods for 2D parabolic/elliptic PDEs.

No previous biological knowledge is required for this project. At least two visits to Dr Melvin's laboratory are anticipated. This project follows up work begun at the 7th Mathematics in Medicine Study Group, University of Southampton, 10-14 September 2007.

Figure 3: Schematic of the Hele-Shaw microfluidic device (courtesy of Dr Marvin Jones, University of Southampton).

One drop measurement: surface wettability characterisation using picolitre liquid drops - Dr Jim Oliver and Prof Tom Witelski (OCIAM, University of Oxford) Dr Morgan Alexander (School of Pharmacy, University of Nottingham)

The dynamic contact angle θ is the angle between a moving liquid/vapour interface and a solid surface, measured within the liquid at the contact line where the three phases (liquid, solid, gas) meet. There is significant empirical evidence that the dynamic contact angle depends on the velocity V of the contact line, with the dynamics being well described by the contact-line law illustrated in Figure 4. Placement of a drop onto a surface and measurement of its advancing contact angle θ_A is the oldest and probably still the most commonly performed surface analysis experiment, since it is simple and indicates the degree of wettability of the surface. The less-frequently measured receding contact angle θ_R provides additional important information on the surface of materials.

Figure 4: (a) Schematic of a liquid droplet on a solid surface showing the dynamic contact angle θ and outward normal velocity V of the contact line; (b) schematic of a contact-line law showing the advancing (θ_A) and receding (θ_R) contact angles.

Typically two separate experiments are required to measure the advancing and receding contact angles (by forcing the contact line to advance or recede, respectively). However, at room temperature a sufficiently small drop may evaporate on the time scale of contact-line motion, so that the contact line is forced to exhibit both types of dynamic behaviour. Hence, by fitting a dynamic model it is anticipated that both the advancing and receding contact angles may be extracted from a single experiment.

An additional advantage of using a picolitre drop is that it is sufficiently small to characterise variations in the wettability of a biomaterial surface on a length scale pertinent to the motion of biological cells. This has been demonstrated recently in [1] in which a series of high throughput experiments using picolitre droplets were used to characterise the gradient in the wettability of a chemically modified biomaterial surface. Once characterised, cells are placed on the biomaterial sample, and a typical long-time response is shown in Figure 5. The sensitivity of the cell response to the local surface wettability has exciting implications for tissue engineering and medical device design.

Figure 5: Fibroblasts adhere and proliferate preferentially on hydrophilic plasma polymerised allylamine RIGHT (θ_A ~ 60 degrees) compared to the hydrophobic plasma polymerised hexane LEFT (θ_A ~ 93 degrees) two days after seeding.

This project will first use image-processing techniques to extract the contact angle and the volume of an evaporating picolitre drop from experimental data provided by Dr Alexander. We will then fit the simplest possible theoretical model (in which the drop is assumed to have constant mean curvature, evaporation is proportional to drop surface area, and the motion of the contact line is governed by a contact line law of the type shown in Figure 4b), and explore its domain of applicability. Extra physics will then be incorporated as necessary. This study will enable static contact angles to be extracted from dynamic experiments, and it is hoped that a simple MATLAB or Java based GUI will be written for use in the experimentalists' lab.

Keywords: Surface tension, advancing/receding contact lines, droplet spreading and evaporation on heterogeneous surfaces; image processing; lubrication theory, the thin-film equation.

No previous biological knowledge is required for this project. At least two visits to Dr Morgan's laboratory are anticipated.

Visco-poroelastic modelling for movement of cerebrospinal fluid - Dr Ian Sobey

Cerbrospinal fluid (CSF) is generated near the centre of the brain (choroid plexus) and flows out around and through the brain and spinal chord before being absorbed into a large vein (superior sagittal sinus). There are a number of models for the behaviour of the brain tissue and the movement of the CSF. A test that can be applied to patients to try to determine some properties of the CSF system is an infusion test where fluid is injected into the system through a lumbar puncture for a short period (effectively a step test). I have developed poroelastic models for this situation but am now intend to add viscoelasticity to the poro-elastic model. A poroelastic model has an elastic sponge-like structure with fluid in interstitial spaces and part of any load applied to the structure is taken up by the fluid pressure. The question posed here is: what happens if the underlying solid matrix is a viscoelastic material, and not just an elastic material and time dependent (step or periodic) forcing is applied?

A search on google scholar for visco-poroelasticity reveals very few references (half a dozen over the two forms visco-poroelasticity, viscoporoelasticity).

Constitutive modelling of tendons and ligaments - Dr Mark Thompson, Prof. Jon Chapman

Disorders of tendons are common and debilitating, causing individual pain and imposing a large societal burden due to healthcare costs and lost days at work. Tendons play a key role in locomotion, transmitting the forces developed by muscles along limbs, across joints and finally to bones, while ligaments provide essential stability for joints by limiting and guiding movement. The mechanical function of these complex tissues depends in ways not well understood upon a hierarchy of structures, starting from the triple helical tropo-collagen molecule and packed in progressively larger helical bundles[1] up to macroscopic tissue units[2], all embedded in a ground substance of proteoglycans with a high affinity for water[3]. Deeper mechanical understanding of these tissues is essential for the design of new regenerative therapies and in order to provide baseline information from which to judge clinical outcomes of intervention.

The aim of this project is the development of 3-D microstructural constitutive models of tendon and ligament that can account for behaviour under static and dynamic loading. The main objectives are:

In representing elastic behaviour of tendon and ligament both microstructural models, explicitly representing observed features of collagen fibrils, and phenomenological models, with no direct physical interpretation for parameters, have proven successful. 1-D microstructural models took into account sequential recruitment of elastic fibres [4] or directly modelled their hinging geometry[5]. Some phenomenological models fit polynomials to uniaxial test data[6], while a more enduring approach used exponential functions[7]. Extending constitutive models to 3-D is vital to enable the representation of anisotropy and shear loading. Although linear approaches developed for composite materials[8] have been used, non-linear strain energy methods[9] provide a comprehensive description and a Mooney-Rivlin model in particular successfully predicted uniaxial data obtained under different loading conditions[7]. 3-D nonlinear constitutive models including explicit representation of fibre contributions have been developed[10]. However, although detailed molecular models for mechanical interactions between collagen fibrils and the ground substance, the matrix in which they are embedded, have been proposed[11], their significance remains highly controversial as it is claimed that there is a lack of evidence for the ends of the fibrils that are required for the shear lag hypothesis[12].

Ranking competitors in sport - Prof. Jon Chapman

Time Series

Time series analysis of baroclinic wave data - Dr Irene Moroz

Data assimilation in a 3-D nonlinear system - Dr Irene Moroz

An exloratory study on whispering - Dr Christina Orphanidou

Even though there is a clear understanding of the acoustic properties of speech which characterise voicing not much is known about what it is that changes when people whisper. This knowledge is essential when developing any kind of automatic speech/speaker recognition system. We propose to explore the acoustic correlates of whispering and propose techniques for modelling it. An experimental protocol will be designed and implemented and classic signal processing techniques will be incorporated in order to gain an understanding of the differences between "normal" speech and whispering.

Voice Pitch Detection using Wavelet Processing - Dr Christina Orphanidou

The pitch of a person's voice is one of the fundamental characteristics defining its timbre (and consequently "voice identity") and is essential for the development of any speaker-specific application. Wavelets have been recently successfully used in speech processing because of their properties with respect to analyzing the different frequencies of the speech signal. We propose to develop a new algorithm for detecting the pitch of the speech signal by using simple signal processing algorithms and testing it against already existing pitch-detection algorithms. Currently existing speech corpora will be used as well as speech processing software.

Brief Descriptions of Possible M.Sc. Projects