Department of Mathematics - Annual Report 2004

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3. Research in the Department

The Department's research covers Pure Mathematics, Applied Mathematics and Statistics. Research in the traditional areas of Pure Mathematics includes set theory, number theory in linear algebra, group theory, graph theory, combinatorics and logic. A joint appointment with the Department of Computer Science underpins an emphasis on theoretical computer science. The Applied Mathematics group has expertise in wave propagation and analysis, dynamical systems, fluid mechanics, elasticity, partial differential equations, relativity, numerical analysis and variational methods. The Statistics group has research strengths in Bayesian sequential methods, medical statistics, time series, exploratory data analysis, history of statistics and stochastic models in operational research. Several appointments are sponsored by the North Staffordshire District Health Authority.

The research areas of the individual members of staff are described below.

Dr David Bedford

The mathematical study of combinatorial designs grew in importance in the 1940's and 50's with the advocation, by Fisher and Yates, of their use in the planning of certain types of statistical experiment (e.g. agricultural field trials). This is still an active area of research although, more recently, much impetus has been added to the study of designs by their increased application to both coding theory and cryptography.

I have a particular interest in the theory and applications of latin squares. A latin square of order n is an n × n array of symbols from a set of size n such that each symbol occurs exactly once in each row and once in each column. A surprisingly large number of open problems exist concerning the construction of latin squares with additional properties. These properties are often motivated by specific applications such as the design of a statistical experiment, or by connections with other branches of pure mathematics. My main research interests are

generalisations of orthogonality in latin squares,
enumeration of transversals in the Cayley tables of groups,
sequencings for groups,
uniquely completable sets in latin squares,
construction of tournaments which are balanced with respect to carry over effects,
construction of efficient semi-latin squares.

Dr John Belcher

I am interested in developing models for irregularly sampled time series using a state space formulation with medical applications. A multi-disciplinary research group has been working since 1989 in the area of occupational asthma. By analysing long, irregularly spaced peak flow records, continuous time modelling has identified work-related rhythms.

I am also investigating methods for modelling relationships between two or more irregularly sampled series, with sampling possibly carried out at different times. For example the rate of decline of renal function, i.e. the first derivative, over a period of months and years, is a measure of the level of activity of the disease. This may be indirectly measured by blood or urine measurements, but these may be taken at different times. In order to calibrate these measurements, or even to test for an association, a model is required to relate them. A continuous time state model in which the disease level is the prime state variable, with an observation equation for both measured variables, one which represents an integrated response, is envisaged. Development of reliable methods for identifying and estimating such models is needed. Additionally I am examining analysis and repeated measure problems that arise from hospital collaboration. One of the former is assessing the effect of contractions on oxygen saturation in haemoglobin during labour. Modelling binary time series data is a new area where one seeks to explain the relationship between psychiatric episodes, together with applying time and frequency domain methods in areas such as depression.

Finally, I am involved in research into physiotherapy with Dr Mark Poulter, University of Ulster.

Dr C. John Chapman

My research centres around aeroacoustics, especially in relation to the noise produced by high-speed aeroengines. I am interested in mathematical techniques which can be brought to bear on the prediction and control of aeroengine noise, regarded as a practical engineering problem of commercial significance, and I stay up to date with modern developments by maintaining close contact with engineers at Rolls-Royce and with aerodynamicists at NASA in the USA. My current projects include:

Duct aeroacoustics. I have made a detailed study of the diffraction and noise-shielding effects of the duct of a high-speed turbofan aeroengine, and have discovered that the structure of the radiated sound field is controlled by the 'nil-shielding directions', by a set of 'universal diffraction functions', and by the 'open-to-ducted transfer function'. I have also shown that the caustic structure inside a cylindrical duct is determined by the hyperbolic umbilic catastrophe.

Similarity rules for the convected wave equation. I have investigated a type of Lorentz transformation which describes the effect of high-speed motion on acoustic sources.

Aeroacoustic potentials. This project makes use of techniques developed originally in electromagnetism.

Axlets and edgelets in the theory of blade-vortex interaction; the theory of inhomogeneous waves in homogeneous media; energy-tracking in unsteady and inhomogeneous wave fields; the wavenumber surface in diffraction theory; fluid-structure interaction; and error analysis in singular perturbation theory.

The central subject in my research is fluid dynamics, on which I have written the text-book 'High Speed Flow', published by Cambridge University Press in 2000.

Dr Peter Fletcher

I hold a joint appointment in the Mathematics and Computer Science Departments. I have two areas of research.

The first is the intuitionistic philosophy of mathematics. I am working on clarifying the basic notion of a mathematical construction and its use, via the concept of constructive proof, in providing an interpretation of predicate logic, number theory and analysis. I am also concerned to reconcile the intuitionistic view with important insights from Hilbert's formalism and logicism. Recently I have been considering the consequences of intuitionism for the application of mathematics to physics.

The second is neural network models of cognition. I have developed a self-configuring network, which can learn simple iterative geometric structure from example patterns without supervision, and my present work extends the method to cover the parsing of vague and noisy patterns using context-free graph grammars.

Professor Yibin Fu

My main research interest is in characterizing linear and nonlinear waves, especially surface waves and edge waves, in elastic solids, and stability of pre-stressed elastic solids and structures. Studies of such waves have important applications in signal processing, seismology and non-destructive testing, whereas stability analysis has applications in assessing the integrity of structures under stress. Recently, I have obtained a number of results concerning properties of the surface-impedance tensor which plays an important role in a large variety of applications. I have also recently proved the uniqueness of edge waves propagating along the edge of a thin plate. With regards stability analysis, I am particularly interested in the stability of multi-phase states which arise from stress-induced phase transformations. I am a co-author of a recent paper which presents an efficient method for assessing the stability of such multi-phase deformation states. I am currently applying similar ideas to model kink band formation in fibre-reinforced composites. Put differently, my current research interests are

characterisation of linear surface waves and edge waves,
examination of imperfection sensitivity and secondary bifurcations of elastic cylinders and shells with arbitrary thickness,
using asymptotic methods to determine stability curves for various buckling problems,
response of pre-stressed plates and shells to impact,
characterisation of stress-induced phase transformations and kink-bands in fibre-reinforced composites.

Dr Jonathan J. Healey

I am working on the transition of boundary layers from a laminar to a turbulent state. This problem is of importance in the aeronautics industry where the accuracy of drag prediction schemes is compromised by our inability to predict the location of transition accurately. My recent research has been concerned with issues of direction of propagation of disturbances in unstable boundary layers, including determining whether a flow can be classified as absolutely unstable (growing disturbances travel upstream, downstream and in the laboratory frame of reference) or convectively unstable (growth only in frames of reference moving away from a source). The boundary layer that forms over a rotating disk is known to become absolutely unstable at large Reynolds numbers (which means either high flow speeds, low viscosity, or a large distance from the axis of rotation). I have developed an analytic dispersion relation for this absolute instability in the asymptotic large wavelength limit for zero viscosity, which identifies the fundamental mechanisms for the absolute instability in this limit. It also hints at a rather strange type of behaviour in which waves grow exponentially in the wall-normal direction. This is unexpected behaviour because there is no source of energy outside the boundary layer to sustain this growth. Nonetheless, the propagation of energy leading to this wall-normal growth has been elucidated, and the consequences of a range of additional effects, like viscosity, spatial inhomogeneity of the basic flow, proximity of a boundary parallel to the wall (which can interrupt the wall-normal propagation) and nonlinearity, will soon be investigated. It seems likely that this phenomenon can occur in a number of related basic fluid flows, and the same techniques can be applied to these problems to explore this behaviour in a variety of contexts.

Dr Maria A. Heckl

My research interest is in mathematical modelling of acoustic phenomena and applies primarily to noise control and non-destructive testing (NDT).

My latest research project is on noise produced by flames and vortices in combustion systems, such as jet aircraft and industrial burners. Such noise can be extremely loud and even lead to structural damage; it is often caused by acoustic instabilities. I invented a particular form of active control some time ago, which very effectively annuls instabilities in an elementary combustion system. I have recently returned to this project with the aim of developing refined models for more complex combustion systems, taking vortex shedding into account. I also intend to examine the possibility of annulling instabilities in such combustion systems by active control.

I have worked for several years on curve squeal, which is sometimes produced when a train travels round a bend. The squeal noise is due to a structural instability driven by the feedback between the friction force and the lateral sliding velocity of the wheels against the rail. This noise is very annoying, and a problem in inner cities with trams that have to negotiate tight curves. I have developed a mathematical model for the generation of curve squeal and extended the model to predict the performance of an active control system, which can eliminate the squeal by interfering with its generation mechanism. This project was funded by a research grant from the EPSRC.

I have also been involved in a number of other projects. I have developed mathematical models for the acoustic behaviour of profiled cladding, in order to find cladding designs that minimise the noise transmitted from industrial buildings into the environment. I have modelled rolling noise from railways, to predict the vibrations and noise of a variety of track configurations, and to study, for example, the effect of irregular sleeper arrangements. I have contributed to the development non-destructive testing methods for nuclear reactors by my research into sound propagation across tube bundles; this project was funded by AEA Technology and an EPSRC research grant.

Dr J. Mary Jones

Nutritional screening, trauma and clinical audit are main areas of research. A review of the methodology of 44 nutritional screening and assessment tools revealed that these have been published with insufficient regard given to important aspects of design, development and evaluation. Papers have since been written to give guidance regarding the design and analysis of studies for evaluating the reliability and validity of a tool. Trauma research has centred around the inappropriateness of the present practice of using American trauma models for evaluating trauma care in Britain and elsewhere. A visit to Australian and New Zealand trauma centres has led to increased international co-operation aimed at establishing relevant methods for assessing trauma care. My participation in clinical audit has been to increase the awareness of the importance of statistical concepts in the design and analysis of clinical audit studies. I am also actively involved in a number of research projects involving the application of statistical techniques in various fields of medicine in collaboration with colleagues at the North Staffordshire Hospital Centre. Major research collaborations in primary care have included an epidemiological investigation of dyspepsia in the local community, and a prospective study of patients with musculoskeletal pain in order to identify predictive factors for chronicity.

Professor Peter W. Jones

My research interests are medical statistics and Bayesian sequential estimation and optimisation, especially bandit problems with applications to clinical trials.

I have had a long and productive association with Clinical Geneticists (Professor Richard Strange�s group) at the North Staffordshire Hospital Centre where we have been looking at large scale case control studies to detect genetic susceptibility and genetic bases of severity in a number of diseases (various cancers and rheumatoid arthritis for example) as well as interactions between genes and gene and environment. This has involved developing models for time to events, counts and binary data.

I have been closely involved with the rheumatologists lead by Dr Peter Dawes at the Haywood Hospital in Burslem, where we, together with a research student, developed and evaluated an index of disease activity (the Stoke Index) in rheumatoid arthritis. I have worked on several other indices (OSRA: Overall Status in RA, SASS: Stoke Ankylosing Spondilytis Score and the body chart for pain assessment) also with clinicians at the Haywood.

A results of number of RCT�s (randomised controlled trials) in obstetrics have been published. I have been developing an interest in systematic reviews and meta analysis in this area.

Linear mixed models have been applied to a number of clinical problems.

I have been interested in bandit problems for a number of years, in particular multiobjective Bayesian bandits, suboptimal schemes and the effects of changing the methods of sampling. These have grown from my interest in producing usable adaptive designs for clinical trials with reasonable properties when compared with one observation at a time designs, which are optimal for a single objective. Group sequential sampling schemes have been considered which have fixed and variable numbers of observations.

Dr P. Lovie

My current research is mainly on the history of statistical thinking and methods in the late nineteenth and first half of the twentieth centuries. Areas of especial interest include the origins of factor analysis, aspects of mathematical psychology and the conflicts, which arose in Britain and the USA between the traditional and biometric schools of statistics. In addition, over the last few years I have been working on biographical essays for several major reference works including the New Dictionary of National Biography, the American National Biography and the Encyclopedia of Psychology. However, I retain more general links with contemporary statistical methodology through its applications to the social sciences and statistical computing. After serving for almost six years as Editor of the British Journal of Mathematical and Statistical Psychology, I have now serve as Consultant Editor. I am also an associate editor to Statistics and Computing, a consulting editor to History of Psychology and a member of the editorial boards of History and Philosophy of Psychology and Understanding Statistics.

Professor Gilbert MacKenzie

My methodological interests lie principally in statistical modelling - mainly in stochastic processes involving medical phenomena. I am particularly interested in survival analysis and I have continued to develop the Generalised Time-Dependent Logistic (GTDL) family of survival models, which contains a parametric competitor for the proportional hazard model of Cox (1972). Currently, research is focused on the canonical form of the GTDL (the CTDL) which has now been generalised to handle multivariate survival data with structured dispersions - a development which links multivariate survival analysis and covariance modelling in longitudinal studies. Recent developments in this latter area include the publication of a new algorithm for identifying the optimal covariance structure in a joint mean-covariance model, which arises frequently in longitudinal trials.

A variety of publications based on our multi-centre randomised controlled clinical trial (funded by the MRC) of teletherapy in age-related macular degeneration have appeared. Overall, this trial has proved to be a fertile test bed for developing new analytical methods, which are now being implemented in other areas of application, such as modelling the impact of heterogeneity in trauma survival.

Dr Killian O'Brien

My research is in the area of knot theory and the development of algorithms for the calculation of certain knot invariants. Firstly, I have developed an implementation of Seifert's algorithm. This is a well known algorithm for the construction of an oriented surface bounded by a given knot.

Secondly, I have developed an algorithm for dealing with ideals of the ring of Laurent polynomials in one variable with integer coefficients. This is useful because the Alexander module of a knot, which is the first homology group of the infinite cyclic cover of the exterior of a knot, is a module over this ring and many knot invariants can be calculated from a presentation of the Alexander module. The algorithm is an adaptation of previous algorithms by Buchberger and it allows one to calculate nice generators of the so-called higher Alexander ideals. I am particularly interested in how these higher ideals can differentiate between mutant knots. Mutant knots are pairs of knots that are related by a simple cut and paste type operation but which many knot invariants fail to differentiate between.

Dr Anthony D. Osborne

My main area of interest lies in real and complex analysis of one form or another. I have published an undergraduate textbook on Complex Variables and I am currently working on a textbook on real analysis. I have worked in the area of systematic methods for solving nonlinear partial differential equations, in particular in the method of separation and its connection with similarity methods. I have also worked in the area of the qualitative theory of ordinary differential equations, applied particularly to geodesic equations in general relativity. At the present time I am also writing a book on relativity.

I am also interested in natural philosophy and, in particular, I have worked with N.V. Pope on a new philosophical approach to the classical physics and relativity interface over the past twenty years. This collaboration has led to our �angular momentum synthesis� which falls under Pope�s Normal Realist philosophy. Our work is ongoing and addresses a number of issues of current concern, particularly action-at-distance.

In pure mathematics, I am also interested in the use of number theory in algebra. I have worked with Dr Hans Liebeck, now retired from the Department, on the problem of constructing orthonormal bases for the vector space Qⁿ and the more general problem of rational equivalence of matrices involving p-excess.

Dr John Preater

My work is in applied probability and operational research. Particular interests are stochastic dynamic programming, queuing models and combinatorial problems.

I have studied a variety of sequential selection problems related to the celebrated "secretary problem", especially those involving multiple procurement or partially ordered objects.

Dynamic sequential selection problems led to the study of congestion models based on the M/M/∞ queue, including variations in which either the servers or the customers are ranked. I also work on applications of queuing theory to health care.

In combinatorics I am interested in discrete probability and aspects of graph theory.

Recently I have been engaged in modelling and software development in regard to a percolation problem in oil reservoir exploration.

Professor Douglas A. Quinney

Currently, my major interest is the development of courseware for teaching undergraduate mathematics particularly, though not exclusively, at the interface between secondary and higher education. This work includes developing web and other computer based tools for supporting students and assessing their progress in the context of their pedagogical development.

I am also involved with the Calculus "reform" project based at Harvard and Arizona Universities in the US and have been involved in writing a number of text books for the US market. These include Calculus, which covers the first two semesters of the standard Calculus sequence in the US and Multivariable Calculus, which covers the third semester. These books have also been re-purposed and produced as Applied Calculus for Business, Social Sciences and Life Sciences and Brief Calculus. Currently my interests are focused on computer based assessment using e-Grade Plus (John Wiley, Inc) and WebWork.

Additionally, on a more fundamental level I am interested in developing techniques for improving the accuracy and rate of convergence of numerical techniques in Numerical Analysis by using extrapolation methods based on the E-Algorithm an L and T transformation of Levin.

Professor Victor I. Shrira

My main area of research is concerned with nonlinear waves of different nature, most often in the context of environmental dynamics, and, in particular, physical oceanography.

Over the last ten years, nonlinear water waves were the subject of my prime interest. Specifically, I am interested in the formation of three-dimensional patterns, wave instabilities, Hamiltonian and "almost-Hamiltonian" systems, wave interactions with shear currents and turbulence, fundamental problems of statistical description of random wave fields, formation of wave groups and of Langmuir circulations, and electromagnetic scattering by wind wave patterns.

My main interest in physical oceanography (the central theme) is in understanding the basic mechanisms of manifestations of interior processes on the ocean surface, but also includes internal waves, coastal processes, Rossby waves, wave-current interactions, and nonlinear flows over topography. I am also interested in remote sensing of physical processes in the upper ocean, remote sensing of shear currents near the surface, developing new methods of measurements of turbulence below the air-water interface and cross-interface fluxes.

The theoretical topic I am most actively involved in now is concerned with the effects of the horizontal component of the Earth's rotation on the dynamics of inertial waves.

In the area of basic fluid mechanics, I am currently working on issues of boundary layer dynamics and laminar-turbulent transition in the flows where there are no linearly unstable modes. the key idea is that transition might occur as the result of accumulation of nonlinear effects by the least decaying modes.

I am also working on the classical problem of formation of singularities in the inviscid incompressible fluid.

Professor Graham Wilks

My long-standing interests are in the area of fluid mechanics. In particular I have concentrated on boundary layer theory. Much of my work involves heat transfer and I have strong interests in forced, free and mixed convection boundary layer flows. More recently I have taken an additional interest in laminar film condensation flow. The work requires expertise in computational and analytic techniques. These skills have been applied over a range of fundamental to industrial problems. Inundation and drainage flows against horizontal plates or over horizontal cylinders have been examined for a variety of flow rates and impingement velocities. Heat transfer characteristics have been established. Mixed convection flows against moving surfaces have also been examined for both favourable and adverse buoyancy. Comprehensive details of velocity and temperature distributions have been obtained numerically and synthesised with series extension results. Further mixed convection studies have considered non-uniform external streams against vertical plates and free convection flow in the vicinity of step discontinuities in wall heat flux has been studied. The assimilation of jets into uniform and non-uniform streams has been studied, including the associated heat transfer characteristics when the jet is a source of heat as well as momentum.

I am currently taking an active interest in nonlinear dynamics as applied to cardiac arrhythmias and would wish to explore the relevance of chaos control in such settings. The mathematical aspects of chaos control and the relevance to the prospects of life-saving intervention at the onset of fibrillation are the focus of these studies.

Professor Andrew J. Willmott

I am the lead Principal Investigator for two NERC research projects. One of the projects is supported by the Thematic Programme AUTOSUB-under-ice, while the other is supported by the Rapid Climate Change (RAPID) programme. A common theme linking these projects is modelling the water mass transformation taking place in major recurrent polynya systems. In the former project, dense water formed in the polynya adjacent to the floating Filchner-Ronne ice shelf drives under-ice-shelf-circulation. The AUTOSUB (an autonomous underwater vehicle) will be used to measure physical oceanographic variables beneath the ice shelf in February 2005. In terms of risk and reward, operating the AUTOSUB in this inaccessible and hostile environment is analogous to sending unmanned probes to Mars. Data from the AUTOSUB will be incorporated into the ice shelf circulation models developed by Dr Gareth Owen and myself at Keele.

My involvement with RAPID includes collaboration with Drs Holt and Proctor at the Proudman Oceanographic Laboratory (POL). Our team is modelling the dense water formation in the Barents Sea that contribute to driving the global thermohaline circulation. At Keele, Dr Ian Walkington and myself are developing models for dense water production in coastal polynyas, including parameterisations of these processes into large-scale ocean circulation models. Our parameterisations will be tested in a Barents Sea ocean circulation model that is being developed at POL.

A NERC supported student, Mr Martin Stott, is also examining the feedback between brine induced currents in polynyas and the opening/closing of these features.

With Professor David Abrahams (Manchester) and Professor Victor Shrira and Dr Gareth Owen at Keele, I am studying baroclinic Rossby wave scattering by topography and wave refraction by large-scale currents.

The NERC, via a small grant, has supported my research collaboration with Professor B. Cushman-Roisin at Dartmouth Dartmouth College, USA, to study the impact of stratification on the period of seiches in semi-enclosed seas. This project is relevant to seiche mode propagation in the Adriatic Sea. The seiches are responsible for the disastrous flooding events in Venice.