UK Nonlinear News, February 2002

Recent thesis

On Continuous Limit Computation and on Bäcklund Transformations

Carl Josson

Institute of Mathematics & Statistic, University of Kent

Supervisor: Professor Peter A. Clarkson


The thesis consists of three chapters. In both chapter 2 and 3 a problem from the field of integrable equations, in particular the Painlevé equations, is addressed.

In the first chapter, a short historical introduction to the continuous Painlevé equations, the concept of the Bäcklund transformation, and the discrete Painlevé equations is given.

In the second chapter, the process of computing the continuous limit of a discrete equation is studied. This computation is situated in the area of the asymptotics of discrete equations, and the `how' and `why' of each step of the computation process is explained. Different types of discrete Painlevé equations illustrate the computation, where the tedious step of arranging terms of expansions of algebraic powers in a variable s is performed by a MAPLE procedure which is presented in an appendix to this chapter.

In the third chapter the group structure of the auto-Bäcklund transformations for the continuous fourth Painlevé equation is investigated. From the known auto-Bäcklund transformations for the canonical form of PIV a complete set of generators together with nontrivial relations is obtained. These form an infinite, yet unidentified group.

Source Peter A. Clarkson

Painlevé analysis and the study of continuous and discrete Painlevé equations

Philip J. Charles

Institute of Mathematics & Statistic, University of Kent

Supervisor: Professor Peter A. Clarkson


The standard and perturbative Painlevé tests are applied to systems of ordinary differential equations. In particular, tests of Darboux-Halphen systems are performed. Closely related to the sixth Painlevé equation, these are fascinating equations which exhibit a seemingly rare property. Specifically, the equations arise as the compatibility conditions of and can be solved via, an associated pair of linear equations (Lax pair) but the general solution is densely branched.

Claimed to be equivalent to the Painlevé test, mirror systems provide an alternative method for detecting specific properties of integrability, and can be used to provide a conceptual proof that the Laurent series obtained in a successful application of the standard Painlevé test is convergent. Mirror systems for some significant systems of equations are found. In applying this analysis to a specific case of a Darboux-Halphen system it is shown that serious drawbacks can apply to their use.

For discrete equations a study is made of the reliability of the singularity confinement, algebraic entropy and `discrete Painléve' tests for integrability by their application to a variety of equations. These tests assume different integrability criteria as their starting point. Results of the tests are shown to be highly contradictory.

New solution hierarchies in terms of Laguerre polynomials are found for the fourth and fifth Painlé equations. As a result, discrete Riccati equations are derived relating solutions both within and between hierarchies of the two equations.

For special function solutions, further relations are derived between the third Painlé equation and the second, fourth and fifth equations. Hierarchies of solutions in terms of Legendre and Jacobi polynomials, together with discrete Riccati relations for the Legendre polynomial solutions, are found for the sixth Painlé equation.

A study is made of the standard Bäcklund transformation of the fifth Painl&eactue; equation. As well as inverses being found not to be unique, solution hierarchies obtained by repeated application of transformations to one-parameter and rational seed solutions are found to produce highly non-trivial structures. A regular structure is shown to exist when the hierarchies are viewed in a certain parameter space.

Source Peter A. Clarkson

Modelling Nonlinear Dynamical Systems: Chaos, Uncertainty, and Error

David Orrell

University of Oxford

Supervisor: Dr Leonard A. Smith


When nonlinear dynamical models are used to approximate physical systems such as the weather, error arises from one of two causes: the initial condition used by the model, and the model itself. Of these two sources, model error is the less well understood; yet a knowledge of model accuracy is essential for reliable error estimates and model optimisation. This thesis develops a technique for measuring model error in the context of nonlinear systems, and explores the link between model error and the ability of the model to shadow the true system. The methods are tested on a variety of model/system pairs in Chapters 2, 3 and 4. In Chapter 5, issues related to longer term behavior are studied, and connections with short term predictability explored. In Chapter 6, the model error techniques are applied to operational weather forecast models. It is seen that the component of forecast error due to model error tends to grow as the square-root of forecast time, and for the days tested is the dominant source of error out to three days. The results are summarised, and the implications further explored, in Chapter 7.

More info: 

Source: David Orell

Modelling T Cell Activation

Cliburn Chan

University College London

Supervisors: Professor Jaroslav Stark and Dr Andrew George


This thesis aims to provide a fuller understanding of how T cells can reliably recognise and respond appropriately to very small densities of foreign antigen on antigen presenting cells. The problem for the T cell is that it has to detect a small signal (foreign peptide-MHC molecules) in the presence of a large amount of noise (self peptide-MHC molecules). Since stochastic processes (e.g., ligand dissociation, recruitment of signalling components) play a large role in the recognition event, it is difficult to see how the T cell can achieve the experimentally documented levels of sensitivity and specificity. In the thesis, mathematical and computational models based on experimental cell signalling data show that feedback and cooperativity for both individual receptors and receptor populations are critical to achieve the observed sensitivity and specificity.

First, the standard model for TCR activation (McKeithan's kinetic proofreading model) is analysed and found to have several biological and theoretical problems, which limit its attractiveness for explaining the specificity and sensitivity of T cell activation. Based on this analysis, a new model that incorporates the essential elements of proofreading (i.e., delay followed by activation) and is more consistent with known TCR signalling biology is constructed. This new model predicts a role for the immune synapse and self ligands in amplifying and sustaining T cell signalling, as well as a possible role for multiple ITAMs to decrease the variance of the activation threshold.

The next model moves from the level of individual TCR to study interactions between a population of receptors. A Monte Carlo simulation of a lattice of TCR interacting with ligands is constructed, which integrates the most important models for T cell specificity (kinetic proofreading) and sensitivity (serial ligation), and incorporates recent evidence for cross-talk between neighbouring receptors. This simulation reveals that the specificity of T cell ligand discrimination can be significantly enhanced with cooperativity. Finally, the model suggests a resolution to the paradox of positive and negative selection on a similar set of ligands, and uses this to explain the surprising repertoire of transgenic mice that express the same peptide on all MHC II molecules.

The PDF can be downloaded from 

Source: Cliburn Chan 


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Last Updated: 5 February 2002.