Research in our group is structured around two main themes, with various projects in each theme:
The main focus here is on several projects relating to the integrability of non-linear partial differential equations using classical, non-classical symmetry methods and more generalised techniques with particular reference to environmental applications and also those occuring in the General Theory of Relativity.
We have many collaborative partners. These include Ildefonso Diaz (Universidad Complutense, Madrid), V Baikov, Gladkov (Ufa, Russia), H Knutsen, (Stavanger, Norway) and Nail Ibragimov (Sweden). In addition Royal Society funding provided the basis for collaborative work with Anatoly Nikitin (Institute of Mathematics, Ukraine) and funding from the University of Cyprus provided the means to collaborate with Christodoulos Sophocleous. In addition to giving invited seminars at many of these institutions Professor Wiltshire has given many seminars in Great Britain .
Recent and current work has focussed on dynamical system modelling, analysis, optimization techniques, game theory, optimal control and estimation theory, intelligent systems and information processing. Some particular projects are listed below
Computational biology is a relatively novel interdisciplinary science that uses mathematical, statistical and computational methods to address biological problems. Main areas of research in our group is computational genomics, bioinformatics, and pharmacokinetics.
The main areas of research focus on modelling microbial populations, either fungi or biofilms, in environments exhibiting nutritional and structural heterogenities.
Additional work has been carried out investigating metapopulations and the impact of habitat fragmentations, the evolution of gigantism in societies, and spatially-dependent enzyme-kinetic reactions.
The research is performed in collaboration with a number of external renowned researchers including Dr. Fordyce Davidson (Mathematical Sciences, University of Dundee), Prof. Geoff Gadd (Biological Sciences, University of Dundee) and Prof. Karl Ritz (National Soil Resources Institute, University of Cranfield). The results of this research have been presented by Dr. Boswell at a number of seminars and both national and international conferences, including an invited slot in the special interest group meeting on Mathematical modeling of fungal growth and function at the 2010 International Mycology Conference.
Social dynamics covers all forms of mathematical and computational modelling of dynamical situations in the social sciences. The main methodologies used are System Dynamics and ordinary differential equations. There are specific projects in church growth, political party activism and the spread of minority languages.
Many areas of recreational mathematics, such as the study of Latin Squares and Rectangles, and Magic Squares, are of great interest within the Combinatorics community in their own right. Moreover, they are also important for their relationship with significant real-world applications, including error correcting codes, timetabling, conflict free wavelength routing in wide band optical networks, and experimental design. The team combines techniques from Combinatorics and Artificial Intelligence (notably search optimization) to reveal and prove important properties of popular puzzles.
Research in this area is focused on the modelling of Newtonian and non-Newtonian fluids. Dr Trevelyan is concerned with thin-film flow, buoyancy-driven instabilities in fluids and reaction-diffusion equations while other interests lie in the mathematical modelling of liquid crystals and other smart materials. Liquid crystals, being anisotropic fluids, are fascinating materials to study and have applications in display technology, bi-layer lipid membranes, ultra-strong materials and much more.
Mathematics underpins much of today’s technological advances and the research in this broad field aims to support and lead experimental research in physics and engineering. One such group of projects is in collaboration with the Centre for Ultrasonic Engineering at the University of Strathclyde. These projects have been concerned with the modelling and design of piezoelectric and electrostatic ultrasonic transducers. Ultrasonics is used in medical imaging, non-destructive evaluation, industrial cleaning, therapeutics and dental ultrasonics. One concern in the manufacture of ultrasonic transducers is their sensitivity and their bandwidth; in particular the maximisation of these aspects through geometrical designs in the electrostatic transducers’ backplates, and of the structural composition of the piezoelectric materials.
Research in this area has focussed on the application of the coupled cluster method (CCM) to the subject of quantum magnetic systems. Lower orders of approximation may be solved analytically for the CCM, whereas higher-order calculations must be carried out computationally. Based on original ideas of Prof. Chen Zeng (George Washington University, USA), computer code has been written in C/C++ that implements the CCM to high orders of approximation for quantum systems. This code has subsequently been extended to run on massively parallelized systems (of many thousands of processors), see:
http://www-e.uni-magdeburg.de/jschulen/ccm/ for details of this code and a (free) GPL licensed version of it. The CCM code has been shown to be very useful at simulating many different types of quantum spin system that are models of magnetic materials (e.g., the 2D “checkerboard” model, aka the planar pyrochlore model – pyrochlore “spin ice” materials have been proposed as effective experimental realisations of magnetic monopoles whose existence was proposed by PAM Dirac.) Future directions of this research include recent extensions to “3D model states” (highly relevant to a host of new magnetic materials), Bose and fermion Hubbard models (relevant, e.g., to graphene), statics and dynamics of highly correlated quantum systems, and measures relevant information theory and quantum computing (e.g., fidelity).
Quality of life following cancer treatment :
Work has been conducted alongside consultant oncologists from Christy Hospital, Manchester (Dr. S. Davidson, Dr. L. Baraclough and Dr. Claire Barker) concerned with collecting and analysing longitudinal data on the quality of patient life following radiotherapy treatment for cancer. Thus, secondary effects of cancer treatment, such as quality of life following various treatment regimes, are being recorded. Indeed, this is the first time that patients have been consulted to such an extent about after effects as a result of radical cancer treatment. The outcome of this ongoing study has the potential to impact on determining suitable treatment regimes for cancer patients on an individual and collective basis and impact on the National Cancer Reform Strategy.
Time Series Modelling & Forecasting:
Research in this area is well established and has spread in a number of directions including academic and industrial contributions. Prof Ameen’s work generated external funding (South Wales Electricity) for new transfer response function models to be developed at doctorate level. The new models are based on the principle of variability decomposition so that indigenous and exogenous influences are quantified and modelled explicitly. Other approaches like Hierarchical Profiling, Dynamic Spatial Disaggregating and Time Series Data Mining are being developed. These approaches offer superior forecasts through providing improved understandings of the underlying dynamics of time series data. Previous projects in this area have involved crime modelling for Cardiff City, in collaboration with Cardiff Police, the modelling and forecasting of Fire and Rescue in South Wales, in collaboration with the South Wales Fire and Rescue services and the use of hybrid models to model and forecast financial data.
Technology Transfer Techno Health Assessment Services (THAS):
This project is a result of PhD projects (directed by Prof. Ameen on modelling the profiles of leg ulcer wounds) and is concerned with the development of a mass marketable product to be used by healthcare nurses for assessing, measuring and monitoring patients’ leg ulcer wounds and pain from their homes. As a tele-medicine device, the tool will record, store and analyse tele-transmittable measures as a result of nurse communication with the patient, transmit the findings to a designated centre of expertise for judgment and recommendations to be transmitted back to the site for decision making. For this stage, the sum of £25,000 was secured from the Welsh Government. A prototype has now been produced and an independent feasibility study was successfully conducted.