Andy Wathen (University of Oxford)
Matrix Iterations, ODEs and Asymptotics. Talk (click to view).  Abstract (click to view).

Mathematical proofs sometimes develop in the most unexpected ways. I will describe in this talk the sequence of unusual, but elementary steps we used in establishing the convergence rate of some popular matrix iterations.

Emma Nichols (University of Manchester)
Field experiments in UK wildfires. Talk (click to view).  Abstract (click to view).

The recent spell of hot, dry weather has seen thousands of wildfires burning across the UK and Ireland, creating immense difficulties for overstretched fire services. One problem is the lack of data on fire behaviour in UK wildfires, making it difficult to estimate how a particular fire will develop or spread. I will give a brief introduction to the mathematical study of wildfires and discuss the field experiments we have been undertaking in Northumberland moorland, where ten gorse plots were burned under various conditions.

Ornella Cominetti (University of Oxford)
Clustering through diffusion processes on graphs. Abstract (click to view).

A spectral fuzzy clustering algorithm, DifFUZZY, applicable to a larger class of clustering problems than other fuzzy clustering algorithms, will be presented. This method yields results comparable as the traditional fuzzy clustering algorithms for “convex-shaped” data sets, but it is better at handling data sets that are curved, elongated or those which contain clusters of different dispersion. DifFUZZY consists of three main steps. It first identifies core clusters (data points that are closely packed together or interconnected when applying a small neighbourhood of each data point), then it identifies the undecided data points, and then it assigns membership values to the undecided points. These steps involve a suitably chosen diffusion process on the graph generated from the data points. DifFUZZY does not require any prior information on the number of clusters, unlike FCM. Examples of data sets for which other algorithms fail to identify the correct clusters will be shown, including computer generated data sets and biological benchmark data sets. Finally recent results obtained using DifFUZZY to study novel clinical malaria data will be presented.

William Parnell (University of Manchester)
Smart materials for acoustic and elastic wave filtering. Talk (click to view).  Abstract (click to view).

Traditional composite materials combine two or more constituent parts to optimize one or more aspects of behaviour such as strength, weight, density, cost, etc. but typically these properties are fixed once the material is designed and built. A new goal is to design smart materials (often termed metamaterials) whose properties can be varied in-situ in a specified manner, by applying an external field (thermal, electric, magnetic, elastic, etc.). In this talk we will focus on applications where we vary the wave propagation characteristics of a material by applying an elastic field which can usually be thought of as a pre-stress. In particular for periodic materials that have a well-defined band-gap structure, the elastic pre-stress acts to shift this band-gap structure so that we can filter out waves in specific frequency ranges.

Tom Shearer (University of Manchester)
Torsional wave propagation in a pre-stressed annular cylinder. Talk (click to view).  Abstract (click to view).

Over the past few decades, much interest has been centred on the effect of pre-stress on the propagation of incremental linear waves in elastic media using the theory of small-on-large, where a small perturbation is applied to a body which has undergone a finite deformation. Since the perturbation is considered to be small in relation to the initial deformation, a linearisation is applied in order to determine the characteristics of wave propagation in the pre-stressed material. Previously, attention has focused mainly on the effect of homogeneous deformations on wave propagation. However, prestress in an inhomogeneous material almost always leads to inhomogeneous deformations. In this talk, we will examine how an initial inhomogeneous pre-stress affects subsequent torsional waves which propagate through an inhomogeneous medium. Hydrostatic pressure is applied to the inner and outer surfaces of an annular cylinder composed of nonlinear-elastic material whose constitutive behaviour is governed by a Mooney-Rivlin strain energy function. The pressure difference creates an inhomogeneous deformation field and modi- es the inner and outer radii of the annular cylinder. Such a deformation leads to a complicated ODE, governing the azimuthal displacement, whose coefficients are spatially dependent. We show the effect this pre-stress, and a prescribed axial stretch, has on the propagation of small-amplitude waves along the cylinder.

Issa Karambal (Heriot-Watt University)
Evans function, Fredholm determinant and the stability of wave. Talk (click to view).  Abstract (click to view).

We discuss two numerical methods for computing the pure-point spectrum associated with the linear stability of coherent structures. These methods are the Evans function defined as the Wronskian of the set of solutions that decay at plus or minus infinity, and the Fredholm determinant, which is the determinant of a trace-class operator that differ by an identity operator. We then show the connection between these methods by providing an example.

Omid Soltan (University of Manchester)
Optimal Trade Execution. Talk (click to view).  Abstract (click to view).

A traders dilemma: trade a large volume of shares fast and incur a negative impact on the trade price, trade too slowly and be exposed to the risk of price depreciation during the trading horizon. In this talk I will give a review of the stochastic control formulation when the trader has a mean variance objective.

Chris Welshman (University of Manchester)
Mapping a Family of Invariant Submanifolds. Talk (click to view).  Abstract (click to view).

A continuous dynamical system can be thought of as a family of vector fields on a state space manifold. A parameter selects a vector field and the resulting flow provides the evolution on the state space. Each flow can produce an interesting low-dimensional structure - an invariant submanifold. We can look to map such a system to a lower-dimensional system in order to describe the family of invariant submanifolds (and their dynamics) in a lower-dimensional ambient space. We introduce this problem in general and consider some of the practical issues associated with constructing such a mapping, particularly the use of finite data sets.

Andrew Hazel (University of Manchester)
Why breathing is good for you! (Two-phase flows in curved tubes). Talk (click to view).  Abstract (click to view).

A simplified model for a single airway of the lung is a rigid, curved pipe containing an air core surrounded by a thin film of watery liquid. If there is no core flow (breath holding), the curvature of the pipe prevents the existence of an axially-uniform steady solution with finite film thickness; in other words, a "dry spot" develops on the inner wall of the bend. We will show that a core flow (steady inhalation or exhalation) can prevent the development of the "dry spot" and, moreover, creates an entire family of steady solutions with finite-film thicknesses. The structure of the solutions is found to be the same in finite-element-based simulations of the (full) Navier--Stokes equations and in a reduced asymptotic model appropriate for the limit of weak centreline curvature when the film is thin. Analysis of the thin-film model will reveal the origin of the solution structure.