Differential Equations and Applications Seminar (past)
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Thu, 27/10/2011 16:00 |
Peter Clarkson (University of Kent) |
Differential Equations and Applications Seminar  |
DH 1st floor SR |
| In this talk I shall discuss special polynomials associated with rational solutions of the Painlevé equations and of the soliton equations which are solvable by the inverse scattering method, including the Korteweg-de Vries, Boussinesq and nonlinear Schrodinger equations. Further I shall illustrate applications of these polynomials to vortex dynamics and rogue waves. The Painlevé equations are six nonlinear ordinary differential equations that have been the subject of much interest in the past thirty years, and have arisen in a variety of physical applications. Further the Painlevé equations may be thought of as nonlinear special functions. Rational solutions of the Painlevé equations are expressible in terms of the logarithmic derivative of certain special polynomials. For the fourth Painlevé equation these polynomials are known as the generalized Hermite polynomials and generalized Okamoto polynomials. The locations of the roots of these polynomials have a highly symmetric (and intriguing) structure in the complex plane. It is well known that soliton equations have symmetry reductions which reduce them to the Painlevé equations, e.g. scaling reductions of the Boussinesq and nonlinear Schrödinger equations are expressible in terms of the fourth Painlevé equation. Hence rational solutions of these equations can be expressed in terms of the generalized Hermite and generalized Okamoto polynomials. I will also discuss the relationship between vortex dynamics and properties of polynomials with roots at the vortex positions. Classical polynomials such as the Hermite and Laguerre polynomials have roots which describe vortex equilibria. Stationary vortex configurations with vortices of the same strength and positive or negative configurations are located at the roots of the Adler-Moser polynomials, which are associated with rational solutions of the Kortweg-de Vries equation. Further, I shall also describe some additional rational solutions of the Boussinesq equation and rational-oscillatory solutions of the focusing nonlinear Schrödinger equation which have applications to rogue waves. | |||
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Thu, 20/10/2011 16:00 |
Alastair Rucklidge (University of Leeds) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Three-wave interactions form the basis of our understanding of many nonlinear pattern forming systems because they encapsulate the most basic nonlinear interactions. In problems with two comparable length scales, such as the Faraday wave experiment with multi-frequency forcing, consideration of three-wave interactions can explain the presence of the spatio-temporal chaos found in some experiments, enabling some previously unexplained results to be interpreted in a new light. The predictions are illustrated with numerical simulations of a model partial differential equation. | |||
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Thu, 13/10/2011 16:00 |
Robert Mackay (University of Warwick) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Isostatic mounts are used in applications like telescopes and robotics to move and hold part of a structure in a desired pose relative to the rest, by driving some controls rather than driving the subsystem directly. To achieve this successfully requires an understanding of the coupled space of configurations and controls, and of the singularities of the mapping from the coupled space to the space of controls. It is crucial to avoid such singularities because generically they lead to large constraint forces and internal stresses which can cause distortion. In this paper we outline design principles for isostatic mount systems for dynamic structures, with particular emphasis on robots. | |||
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Thu, 23/06/2011 16:00 |
Qingchang Zhong (Loughborough University) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Systems with delays frequently appear in engineering. The presence of delays makes system analysis and control design very complicated. In this talk, the standard H-infinity control problem of time-delay systems will be discussed. The emphasis will be on systems having an input or output delay. The problem is solved in the frequency domain via reduction to a one-block problem and then further to an extended Nehari problem using a simple and intuitive method. After solving the extended Nehari problem, the original problem is solved. The solvability of the extended Nehari problem (or the one-block problem) is equivalent to the nonsingularity of a delay-dependent matrix and the solvability conditions of the standard H-infinity control problem with a delay are then formulated in terms of the existence of solutions to two delay-independent algebraic Riccati equations and a delay-dependent nonsingular matrix. | |||
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Thu, 16/06/2011 10:45 |
Oxford / Cambridge Meeting 15th Biennial Event |
Differential Equations and Applications Seminar |
L1 |
| 15th Biennial OXFORD / CAMBRIDGE MEETING PROGRAMME FOR THE ‘WOOLLY OWL TROPHY’ Invited Judges John Harper (Victoria University of Wellington, NZ) Arash Yavari (Georgia Tech, Atlanta, USA) Sharon Stephen (University of Birmingham, UK) 10:45 Morning Coffee The Maths Inst Common Room | |||
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Thu, 09/06/2011 16:00 |
Colin B MacDonald (University of Oxford) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Solving partial differential equations (PDEs) on curved surfaces is important in many areas of science. The Closest Point Method is a new technique for computing numerical solutions to PDEs on curves, surfaces, and more general domains. For example, it can be used to solve a pattern-formation PDE on the surface of a rabbit. A benefit of the Closest Point Method is its simplicity: it is easy to understand and straightforward to implement on a wide variety of PDEs and surfaces. In this presentation, I will introduce the Closest Point Method and highlight some of the research in this area. Example computations (including the in-surface heat equation, reaction-diffusion on surfaces, level set equations, high-order interface motion, and Laplace–Beltrami eigenmodes) on a variety of surfaces will demonstrate the effectiveness of the method. | |||
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Thu, 02/06/2011 16:00 |
Chris Bell (Imperial College London) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Voltammetry is a powerful method for interrogating electrochemical systems. A voltage is applied to an electrode and the resulting current response analysed to determine features of the system under investigation, such as concentrations, diffusion coefficients, rate constants and thermodynamic potentials. Here we will focus on ac voltammetry, where the voltage signal consists of a high frequency sine-wave superimposed on a linear ramp. Using multiple scales analysis, we find analytical solutions for the harmonics of the current response and show how they can be used to determine the system parameters. We also include the effects of capacitance due to the double-layer at the electrode surface and show that even in the presence of large capacitance, the harmonics of the current response can still be isolated using the FFT and the Hanning window. | |||
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Thu, 26/05/2011 16:00 |
Demetrios Papageorgiou (Imperial College London) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Flows involving immiscible liquids are encountered in a variety of industrial and natural processes. Recent applications in micro- and nano-fluidics have led to a significant scientific effort whose aim (among other aspects) is to enable theoretical predictions of the spatiotemporal dynamics of the interface(s) separating different flowing liquids. In such applications the scale of the system is small, and forces such as surface tension or externally imposed electrostatic forces compete and can, in many cases, surpass those of gravity and inertia. This talk will begin with a brief survey of applications where electrohydrodynamics have been used experimentally in micro-lithography, and experiments will be presented that demonstrate the use of electric fields in producing controlled encapsulated droplet formation in microchannels. The main thrust of the talk will be theoretical and will mostly focus on the paradigm problem of the dynamics of electrified falling liquid films over topographically structured substrates. Evolution equations will be developed asymptotically and their solutions will be compared to direct simulations in order to identify their practicality. The equations are rich mathematically and yield novel examples of dissipative evolutionary systems with additional effects (typically these are pseudo-differential operators) due to dispersion and external fields. The models will be analysed (we have rigorous results concerning global existence of solutions, the existence of dissipative dynamics and an absorbing set, and analyticity), and accurate numerical solutions will be presented to describe the large time dynamics. It is found that electric fields and topography can be used to control the flow.Time permitting, I will present some recent results on transitions between convective to absolute instabilities for film flows over periodic topography. | |||
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Thu, 19/05/2011 16:00 |
Ralph Kenna (University of Coventry) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| The notion of critical mass in research is one that has been around for a long time without proper definition. It has been described as some kind of threshold group size above which research standards significantly improve. However no evidence for such a threshold has been found and critical mass has never been measured – until now. We present a new, simple, sociophysical model which explains how research quality depends on research-group structure and in particular on size. Our model predicts that there are, in fact, two critical masses in research, the values of which are discipline dependent. Research quality tends to be linearly dependent on group size, but only up to a limit termed the 'upper critical mass'. The upper critical mass is interpreted as the average maximum number of colleagues with whom a given individual in a research group can meaningfully interact. Once the group exceeds this size, it tends to fragment into sub-groups and research quality no longer improves significantly with increasing size. There is also a lower critical mass, which small research groups should strive to achieve for stability. Our theory is tested using empirical data from RAE 2008 on the quantity and quality of research groups, for which critical masses are determined. For pure and applied mathematics, the lower critical mass is about 2 and 6, respectively, while for statistics and physics it is 9 and 13. The upper critical mass, beyond which research quality does not significantly improve with increasing group size, is about twice the lower value. | |||
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Thu, 12/05/2011 16:00 |
Nikolai Brilliantov (University of Leicester) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| We develop a theory of impact of viscoelastic spheres with adhesive interactions. We assume that the collision velocities are not large to avoid the fracture and plastic deformation of particles material and microscopic relaxation time is much smaller than the collision duration. The adhesive interactions are described with the use of Johnson, Kendall and Roberts (JKR) theory, while dissipation is attributed to the viscoelastic behavior of the material. For small impact velocities we apply the condition of a quasi-static collision and obtain the inter-particle force. We show that this force is a sum of four components, having in addition to common elastic, viscous and adhesive force, the visco-adhesive cross term. Using the derived force we compute the coefficient of normal restitution and consider the application of our theory to the collisions of macro and nano-particles. | |||
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Thu, 05/05/2011 16:00 |
Leon Danon (University of Warwick) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Human behaviour can show surprising properties when looked at from a collective point of view. Data on collective behaviour can be gleaned from a number of sources, and mobile phone data are increasingly becoming used. A major challenge is combining behavioural data with health data. In this talk I will describe our approach to understanding behaviour change related to change in health status at a collective level. | |||
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Thu, 10/03/2011 16:00 |
Nick Hill (Glasgow) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| A mathematical model of Olufsen [1,2] has been extended to study periodic pulse propagation in both the systemic arteries and the pulmonary arterial and venous trees. The systemic and pulmonary circulations are treated as separate, bifurcating trees of compliant and tapering vessels. Each model is divided into two coupled parts: the larger and smaller vessels. Blood flow and pressure in the larger arteries and veins are predicted from a nonlinear 1D cross-sectional area-averaged model for a Newtonian fluid in an elastic tube. The initial cardiac output is obtained from magnetic resonance measurements. The smaller blood vessels are modelled as asymmetric structured trees with specified area and asymmetry ratios between the parent and daughter arteries. For the systemic arteries, the smaller vessels are placed into a number of separate trees representing different vascular beds corresponding to major organs and limbs. Womersley's theory gives the wave equation in the frequency domain for the 1D flow in these smaller vessels, resulting in a linear system. The impedances of the smallest vessels are set to a constant and then back-calculation gives the required outflow boundary condition for the Navier–Stokes equations in the larger vessels. The flow and pressure in the large vessels are then used to calculate the flow and pressure in the small vessels. This gives the first theoretical calculations of the pressure pulse in the small `resistance' arteries which control the haemodynamic pressure drop. I will discuss the effects, on both the forward-propagating and the reflected components of the pressure pulse waveform, of the number of generations of blood vessels, the compliance of the arterial wall, and of vascular rarefaction (the loss of small systemic arterioles) which is associated with type II diabetes. We discuss the possibilities for developing clinical indicators for the early detection of vascular disease. References: 1. M.S. Olufsen et al., Ann Biomed Eng. 28, 1281-99 (2000) 2. M.S. Olufsen, Am J Physiol. 276, H257–68 (1999) | |||
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Thu, 03/03/2011 16:00 |
Basile Audoly (CNRS and Ecole Polytechnique) |
Differential Equations and Applications Seminar |
Gibson Grd floor SR |
| The mechanics of thin elastic or viscous objects has applications in e.g. the buckling of engineering structures, the spinning of polymer fibers, or the crumpling of plates and shells. During the past decade the mathematics, mechanics and physics communities have witnessed an upsurge of interest in those issues. A general question is to how patterns are formed in thin structures. In this talk I consider two illustrative problems: the shapes of an elastic knot, and the stitching patterns laid down by a viscous thread falling on a moving belt. These intriguing phenomena can be understood by using a combination of approaches, ranging from numerical to analytical, and based on exact equations or low-dimensional models. | |||
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Thu, 24/02/2011 16:00 |
Eddie Wilson (Southampton) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| "Most drivers will recognize the scenario: you are making steady progress along the motorway when suddenly you come to a sudden halt at the tail end of a lengthy queue of traffic. When you move off again you look for the cause of the jam, but there isn't one. No accident damaged cars, no breakdown, no dead animal, and no debris strewn on the road. So what caused everyone to stop?" RAC news release (2005) The (by now well-known) answer is that such "phantom traffic jams" exist as waves that propagate upstream (opposite to the driving direction) - so that the vast majority of individuals do not observe the instant at which the jam was created - yet what exactly goes on at that instant is still a matter of debate. In this talk I'll give an overview of empirical data and models to describe such spatiotemporal patterns. The key property we need is instability: and using the framework of car-following (CF) models, I'll show how different sorts of linear (convective and absolute) and nonlinear instability can be used to explain empirical patterns. | |||
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Thu, 17/02/2011 16:00 |
Michel Destrade (National University of Ireland Galway) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Rubbers and biological soft tissues undergo large isochoric motions in service, and can thus be modelled as nonlinear, incompressible elastic solids. It is easy to enforce incompressibility in the finite (exact) theory of nonlinear elasticity, but not so simple in the weakly nonlinear formulation, where the stress is expanded in successive powers of the strain. In linear and second-order elasticity, incompressibility means that Poisson's ratio is 1/2. Here we show how third- and fourth-order elastic constants behave in the incompressible limit. For applications, we turn to the propagation of elastic waves in soft incompressible solids, a topic of crucial importance in medical imaging (joint work with Ray Ogden, University of Aberdeen). | |||
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Thu, 10/02/2011 16:00 |
Simon Cox (Aberystwyth) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Predicting the dynamics of foams requires input from geometry and both Newtonian and non-Newtonian fluid mechanics, among many other fields. I will attempt to give a flavour of this richness by discussing the static structure of a foam and how it allows the derivation of dynamic properties, at least to leading order. The latter includes coarsening due to gas diffusion, liquid drainage under gravity, and the flow of the bubbles themselves. | |||
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Thu, 03/02/2011 16:00 |
OCIAM Members coffee DH common Room |
Differential Equations and Applications Seminar |
DH Common Room |
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Thu, 27/01/2011 16:00 |
Radek Erban (Oxford) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this talk, two commonly used SSAs will be studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. The connections between SSAs and the deterministic models (based on reaction-diffusion PDEs) will be presented. I will consider chemical reactions both at a surface and in the bulk. I will show how the "microscopic" parameters should be chosen to achieve the correct "macroscopic" reaction rate. This choice is found to depend on which SSA is used. I will also present multiscale algorithms which use models with a different level of detail in different parts of the computational domain. | |||
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Thu, 20/01/2011 16:00 |
Stephen Roberts (Oxford) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| This talk will focus on a family of Bayesian inference algorithms built around Gaussian processes. We firstly introduce an iterative Gaussian process for multi-sensor inference problems. Extensions to our algorithm allow us to tackle some of the decision problems faced in sensor networks, including observation scheduling. Along these lines, we also propose a general method of global optimisation, Gaussian process global optimisation (GPGO). This paradigm is extended to the Bayesian decision problem of sequential multi-scale observation selection. We show how the hyperparameters of our system can be marginalised by use of Bayesian quadrature and frame the selection of the positions of the hyperparameter samples required by Bayesian quadrature as a sequential decision problem, with the aim of minimising the uncertainty we possess about the values of the integrals we are approximating. | |||
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Fri, 17/12/2010 15:00 |
Professor L Mahadevan (Harvard) |
Differential Equations and Applications Seminar |
DH 1st floor SR |
| I will discuss a few problems that involve randomness , chosen randomly (?) from the following : (i) the probability of a coin landing on a side (ii) optimal strategies for throwing accurately, (iii) the statistical mechanics of a ribbon, (iv) the intermittent dynamics of a growing polymeric assembly (v) fat tails from feedback. | |||
