Forthcoming events in this series


Thu, 27 Oct 2011

16:00 - 17:00
DH 1st floor SR

Rogue Waves, Vortices and Polynomials

Peter Clarkson
(University of Kent)
Abstract

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.

Thu, 20 Oct 2011

16:00 - 17:00
DH 1st floor SR

Three-wave interactions, quasipatterns and spatio-temporal chaos in the Faraday Wave experiment

Alastair Rucklidge
(University of Leeds)
Abstract

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.

Thu, 13 Oct 2011

16:00 - 17:00
DH 1st floor SR

Design principles for isostatic mount systems for dynamic structures (Coffee and cake Maths Inst Common Room 05:15 - meet SIAM)

Robert Mackay
(University of Warwick)
Abstract

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.

Thu, 23 Jun 2011

16:00 - 17:00
DH 1st floor SR

H-infinity control of time-delay systems

Qingchang Zhong
(Loughborough University)
Abstract

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.

Thu, 16 Jun 2011

10:45 - 17:30
L1

Woolly Owl - host Oxford

Oxford / Cambridge Meeting 15th Biennial Event
Abstract

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

Thu, 09 Jun 2011

16:00 - 17:00
DH 1st floor SR

Computing on surfaces with the Closest Point Method

Colin B MacDonald
(University of Oxford)
Abstract

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.

Thu, 02 Jun 2011

16:00 - 17:00
DH 1st floor SR

Theory of ac voltammetry for reversible electrochemical systems using multiple scales analysis

Chris Bell
(Imperial College London)
Abstract

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.

Thu, 26 May 2011

16:00 - 17:00
DH 1st floor SR

Electrified multi-fluid film flows

Demetrios Papageorgiou
(Imperial College London)
Abstract

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.

Thu, 19 May 2011

16:00 - 17:00
DH 1st floor SR

Mass and the dependency of research quality on group size

Ralph Kenna
(University of Coventry)
Abstract

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.

Thu, 12 May 2011

16:00 - 17:00
DH 1st floor SR

Collisions of viscoelastic adhesive particles

Nikolai Brilliantov
(University of Leicester)
Abstract

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.

Thu, 05 May 2011

16:00 - 17:30
DH 1st floor SR

Collective human behaviour and epidemics: what (else) can we learn from mobile phone data?

Leon Danon
(University of Warwick)
Abstract

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.

Thu, 10 Mar 2011

16:00 - 17:00
DH 1st floor SR

Modelling the Circulatory System

Nick Hill
(Glasgow)
Abstract

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)

Thu, 03 Mar 2011

16:00 - 17:00
Gibson Grd floor SR

Non-linear Mechanics of Elastic and Viscous Threads

Basile Audoly
(CNRS and Ecole Polytechnique)
Abstract

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.

Thu, 24 Feb 2011

16:00 - 17:00
DH 1st floor SR

Highway Traffic Stability

Eddie Wilson
(Southampton)
Abstract

"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.

Thu, 17 Feb 2011

16:00 - 17:00
DH 1st floor SR

Acoustics of soft solids

Michel Destrade
(National University of Ireland Galway)
Abstract

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).

Thu, 10 Feb 2011

16:00 - 17:00
DH 1st floor SR

Dynamics of aqueous foams

Simon Cox
(Aberystwyth)
Abstract

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.

Thu, 03 Feb 2011

16:00 - 17:00
DH Common Room

CANCELLED

OCIAM Members coffee DH common Room
Thu, 27 Jan 2011

16:00 - 17:00
DH 1st floor SR

Stochastic simulation algorithms for reaction-diffusion systems

Radek Erban
(Oxford)
Abstract

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.

Thu, 20 Jan 2011

16:00 - 17:00
DH 1st floor SR

Gaussian Processes for Active Data Selection, Optimisation, Sequential Exploration and Quadrature

Stephen Roberts
(Oxford)
Abstract

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.

Fri, 17 Dec 2010

15:00 - 16:00
DH 1st floor SR

Random problems

Professor L Mahadevan
(Harvard)
Abstract

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.



Thu, 02 Dec 2010

16:00 - 17:00
Gibson Grd floor SR

Multiscale stochastic modelling of biochemical reactions

Simon Cotter
(Oxford)
Abstract

When modeling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation diagram.

Stochastic Simulation Algorithms (SSAs) exist which draw exact trajectories from the Chemical Master Equation (CME). However, these methods can be very computationally expensive, particularly where there is a separation of time scales of the evolution of some of the chemical species. Some of the species may react many times on a time scale for which others are highly unlikely to react at all. Simulating all of these reactions of the fast species is a waste of computational effort, and many different methods exist for reducing the system to one which only contains the slow variables.

In this talk we will introduce the conditional Gillespie algorithm, a method for sampling directly from the conditional distribution on the fast variables, given a static value for the slow variables. Using this, we will go on to describe the constrained Gillespie approach, which uses simulations of the CG algorithm to estimate the drift and diffusion terms of the effective dynamics of the slow variables.

If there is time at the end, I will briefly describe my work on another project, which involves full sampling of the posterior distributions in various problems in data assimilation using Monte Carlo Markov Chain (MCMC) methods.

Thu, 25 Nov 2010

16:00 - 17:30
DH 1st floor SR

Spectral discrete solitons: from cnoidal waves to spatio-temporal helical beams

Andrey Gorbach
(University of Bath)
Abstract

In my talk I will introduce the concept of spectral discrete solitons

(SDSs): solutions of nonlinear Schroedinger type equations, which are localized on a regular grid in frequency space. In time domain such solitons correspond to periodic trains of pulses. SDSs play important role in cascaded four-wave-mixing processes (frequency comb generation) in optical fibres, where initial excitation by a two-frequency pump leads to the generation of multiple side-bands. When free space diffraction is taken into consideration, a non-trivial generalization of 1D SDSs will be discussed, in which every individual harmonic is an optical vortex with its own topological charge. Such excitations correspond to spatio-temporal helical beams.

Thu, 18 Nov 2010

16:00 - 17:30
DH 1st floor SR

On some kinetic equations of swarming

José Antonio Carrillo de la Plata
(Universitat Autònoma de Barcelona)
Abstract

A kinetic theory for swarming systems of interacting individuals will be described with and without noise. Starting from the the particle model \cite{DCBC}, one can construct solutions to a kinetic equation for the single particle probability distribution function using distances between measures \cite{dobru}. Analogously, we will discuss the mean-field limit for these problems with noise.

We will also present and analys the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. It will be shown that the solutions concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.

Thu, 11 Nov 2010

17:00 - 18:00

Partial Differential Equations: Origins, Developments and Roles in the Changing World

Professor Gui-Qiang G. Chen
(Oxford)
Abstract

The Mathematical Institute invites you to attend the Inaugural Lecture of Professor Gui-Qiang G. Chen. Professor in the Analysis of Partial Differential Equations. Examination Schools, 75-81 High Street, Oxford, OX 4BG.

There is no charge to attend but registration is required. Please register your attendance by sending an email to @email specifying the number of people in your party. Admission will only be allowed with prior registration.

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ABSTRACT

While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables – perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modelled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace.

In this lecture, Professor Gui-Qiang G. Chen will present several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world.

Thu, 04 Nov 2010

16:00 - 17:30
DH 1st floor SR

Interfacial Dynamics in the Presence of Additives

Omar Matar
(Imperial College London)
Abstract

The presence of additives, which may or may not be surface-active, can have a dramatic influence on interfacial flows. The presence of surfactants alters the interfacial tension and drives Marangoni flow that leads to fingering instabilities in drops spreading on ultra-thin films. Surfactants also play a major role in coating flows, foam drainage, jet breakup and may be responsible for the so-called ``super-spreading" of drops on hydrophobic substrates. The addition of surface-inactive nano-particles to thin films and drops also influences the interfacial dynamics and has recently been shown to accelerate spreading and to modify the boiling characteristics of nanofluids. These findings have been attributed to the structural component of the disjoining pressure resulting from the ordered layering of nanoparticles in the region near the contact line. In this talk, we present a collection of results which demonstrate that the above-mentioned effects of surfactants and nano-particles can be captured using long-wave models.