Forthcoming events in this series

Thu, 06 Mar 2014

16:00 - 17:00

The effect of boundary conditions on linear and nonlinear waves

Beatrice Pelloni
In this talk, I will discuss the effect of boundary conditions on the solvability of PDEs that have formally an integrable structure, in the

sense of possessing a Lax pair. Many of these PDEs arise in wave propagation phenomena, and boundary value problems for these models are very important in applications. I will discuss the extent to which general approaches that are successful for solving the initial value problem extend to the solution of boundary value problem.

I will survey the solution of specific examples of integrable PDE, linear and nonlinear. The linear theory is joint work with David Smith. For the nonlinear case, I will discuss boundary conditions that yield boundary value problems that are fully integrable, in particular recent joint results with Thanasis Fokas and Jonatan Lenells on the solution of boundary value problems for the elliptic sine-Gordon equation.

Thu, 27 Feb 2014

16:00 - 17:00

Problems in free boundary Hele-Shaw and Stokes flows

Michael Dallaston
(Oxford University)
Two-dimensional viscous fluid flow problems come about either because of a thin gap geometry (Hele-Shaw flow) or plane symmetry (Stokes flow). Such problems can also involve free boundaries between different fluids, and much has been achieved in this area, including by many at Oxford. In this seminar I will discuss some new results in this field.

Firstly I will talk about some of the results of my PhD on contracting inviscid bubbles in Hele-Shaw flow, in particular regarding the effects of surface tension and kinetic undercooling on the free boundary. When a bubble contracts to a point, these effects are dominant, and lead to a menagerie of possible extinction shapes. This limiting problem is a generalisation of the curve shortening flow equation from the study of geometric PDEs. We are currently exploring properties of this generalised flow rule.

Secondly I will discuss current work on applying a free boundary Stokes flow model to the evolution of subglacial water channels. These channels are maintained by the balance between inward creep of ice and melting due to the flow of water. While these channels are normally modelled as circular or semicircular in cross-section, the inward creep of a viscous fluid is unstable. We look at some simplistic viscous dissipation models and the effect they have on the stability of the channel shape. Ultimately, a more realistic turbulent flow model is needed to understand the morphology of the channel walls.

Thu, 20 Feb 2014

16:00 - 17:00

Mathematical modelling of abnormal beta oscillations in Parkinson’s disease

Rafal Bogacz
(University of Oxford (Neuroscience))
In Parkinson’s disease, increased power of oscillations in firing rate has been observed throughout the cortico-basal-ganglia circuit. In

particular, the excessive oscillations in the beta range (13-30Hz) have been shown to be associated with difficulty of movement initiation. However, on the basis of experimental data alone it is difficult to determine where these oscillations are generated, due to complex and recurrent structure of the cortico-basal-ganglia-thalamic circuit. This talk will describe a mathematical model of a subset of basal-ganglia that is able to reproduce experimentally observed patterns of activity. The analysis of the model suggests where and under which conditions the beta oscillations are produced.

Thu, 13 Feb 2014

16:00 - 17:00

Quasi-solution approach towards nonlinear problems

Saleh Tanveer
(The Ohio State University)
Strongly nonlinear problems, written abstractly in the form N[u]=0, are typically difficult to analyze unless they possess special properties. However, if we are able to find a quasi-solution u_0 in the sense that the residual N[u_0] := R is small, then it is possible to analyze a strongly nonlinear problem with weakly nonlinear analysis in the following manner: We decompose u=u_0 + E; then E satisfies L E = -N_1 [E] - R, where L is the Fre'chet derivative of the operator N and N_1 [E] := N[u_0+E]-N[u_0]-L E contains all the nonlinearity. If L has a suitable inversion property and the nonlinearity N_1 is sufficiently regular in E, then weakly nonlinear analysis of the error E through contraction mapping theorem gives rise to control of the error E. What is described above is quite routine. The only new element is to determine a quasi-solution u_0, which is typically found through a combination of classic orthogonal polynomial representation and exponential asymptotics.

This method has been used in a number of nonlinear ODEs arising from reduction of PDEs. We also show how it can be extended to integro-differential equations that arise in study of deep water waves of permanent form. The method is quite general and can in principle be applied to nonlinear PDEs as well.

NB. Much of this is joint work with O. Costin and other collaborators.

Thu, 06 Feb 2014

16:00 - 17:00

Urban growth and decay

Hannah Fry
Much of the mathematical modelling of urban systems revolves around the use spatial interaction models, derived from information theory and entropy-maximisation techniques and embedded in dynamic difference equations. When framed in the context of a retail system, the

dynamics of centre growth poses an interesting mathematical problem, with bifurcations and phase changes, which may be analysed analytically. In this contribution, we present some analysis of the continuous retail model and corresponding discrete version, which yields insights into the effect of space on the system, and an understanding of why certain retail centers are more successful than others. This class of models turns out to have wide reaching applications: from trade and migration flows to the spread of riots and the prediction of archeological sites of interest, examples of which we explore in more detail during the talk.

Thu, 30 Jan 2014

16:00 - 17:00

Bottlenecks, burstiness and fat tails regulate mixing times of diffusion over temporal networks

Jean-Charles Delvenne
(Université catholique de Louvain (Belgium))
Many real-life complex systems arise as a network of simple interconnected individual agents. A central question is to determine how network topology and individual agent dynamics combine to create the global dynamics.

In this talk we focus on the case of continuous-time random walks on networks, with a waiting time of the walker on each node assuming arbitrary probability distributions. Such random walks are useful to model diffusion processes over complex temporal networks representing human interactions, often characterized by non-Poissonian contact patterns.

We find that the mixing time of the random walker, i.e. the relaxation time for the process to reach stationarity, is determined by a combination of three factors: the spectral gap, associated to bottlenecks in the underlying topology, burstiness, related to the second moment of the waiting time distribution, and the characteristic time of its exponential tail, which is an indicator of the tail `fatness'. We show

theoretically that a strong modular structure dampens the importance of burstiness, and empirically that either of the three factors may be dominant in real-life data.

These results are available in arXiv:1309.4155

Thu, 23 Jan 2014

16:00 - 17:00

Classifier ensembles: Does the combination rule matter?

Ludmila Kuncheva
Combining classifiers into an ensemble aims at a more accurate and robust classification decision compared to that of a single classifier. For a successful ensemble, the individual classifiers must be as diverse and as accurate as possible. Achieving both simultaneously is impossible, hence compromises have been sought by a variety of ingenious ensemble creating methods. While diversity has been in the focus of the classifier ensemble research for a long time now, the importance of the combination rule has been often marginalised. Indeed, if the ensemble members are diverse, a simple majority (plurality) vote will suffice. However, engineering diversity is not a trivial problem. A bespoke (trainable) combination rule may compensate for the flaws in preparing the individual ensemble members. This talk will introduce classifier ensembles along with some combination rules, and will demonstrate the merit of choosing a suitable combination rule.
Thu, 05 Dec 2013

16:00 - 17:00

Random matrices and the asymptotic behavior of the zeros of the Taylor approximants of the exponential function

Ken McLaughlin
(University of Arizona)
The plan: start with an introduction to several random matrix ensembles and discuss asymptotic properties of the eigenvalues of the matrices, the last one being the so-called "Normal Matrix Model", and the connection described in the title will be explained. If all goes well I will end with an explanation of asymptotic computations for a new normal matrix model example, which demonstrates a form of universality.


Thu, 28 Nov 2013

16:00 - 17:00

Network dynamics and meso-scale structures

Anne-Ly Do
(Max-Planck Institute for the Physics of Complex Systems)
The dynamics of networks of interacting systems depend intricately on the interaction topology. Dynamical implications of local topological properties such as the nodes' degrees and global topological properties such as the degree distribution have intensively been studied. Mesoscale properties, by contrast, have only recently come into the sharp focus of network science but have

rapidly developed into one of the hot topics in the field. Current questions are: can considering a mesoscale structure such as a single subgraph already allow conclusions on dynamical properties of the network as a whole? And: Can we extract implications that are independent of the embedding network? In this talk I will show that certain mesoscale subgraphs have precise and distinct

consequences for the system-level dynamics. In particular, they induce characteristic dynamical instabilities that are independent of the structure of the embedding network.

Thu, 21 Nov 2013

16:00 - 17:00

Leftovers are just fine

Neville Fowkes
After an MISG there is time to reflect. I will report briefly on the follow up to two problems that we have worked on.

Crack Repair:

It has been found that thin elastically weak spray on liners stabilise walls and reduce rock blast in mining tunnels. Why? The explanation seems to be that the stress field singularity at a crack tip is strongly altered by a weak elastic filler, so cracks in the walls are less likely to extend.

Boundary Tracing:

Using known exact solutions to partial differential equations new domains can be constructed along which prescribed boundary conditions are satisfied. Most notably this technique has been used to extract a large class of new exact solutions to the non-linear Laplace Young equation (of importance in capillarity) including domains with corners and rough boundaries. The technique has also been used on Poisson's, Helmholtz, and constant curvature equation examples. The technique is one that may be useful for handling modelling problems with awkward/interesting geometry.

Thu, 14 Nov 2013

16:00 - 17:00

Hydrodynamic Turbulence as a Problem in Non-Equilibrium Statistical Mechanics

David Ruelle
(Emeritus Professor IHÉS)
The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. These units are macroscopic but have few degrees of freedom, and can be studied by the methods of (microscopic) non-equilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Specically, we obtain the formula

$$ \zeta_p = \frac{p}{3} - \frac{1}{\ln \kappa} \ln \Gamma \left( \frac{p}{3} +1 \right) $$

for the exponents of the structure functions ($\left\langle \Delta_{r}v \rangle \sim r^{\zeta_p}$). The meaning of the adjustable parameter is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(ln \kappa)^{-1} = .32 \pm .01$ is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture which can thus also be used in related problems.

Thu, 07 Nov 2013

16:00 - 17:00

A geometric framework for interpreting and parameterising ocean eddy fluxes

David Marshall
The ocean is populated by an intense geostrophic eddy field with a dominant energy-containing scale on the order of 100 km at midlatitudes. Ocean climate models are unlikely routinely to resolve geostrophic eddies for the foreseeable future and thus development and validation of improved parameterisations is a vital task. Moreover, development and validation of improved eddy parameterizations is an excellent strategy for testing and advancing our understanding of how geostrophic ocean eddies impact the large-scale circulation.

A new mathematical framework for parameterising ocean eddy fluxes is developed that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy fluxes. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. A norm of this tensor is bounded by the eddy energy, allowing the components of the stress tensor to be rewritten in terms of the eddy energy and non-dimensional parameters describing the mean "shape" of the eddies. If a prognostic equation is solved for the eddy energy, the remaining unknowns are non-dimensional and bounded in magnitude by unity. Moreover, these non-dimensional geometric parameters have strong connections with classical stability theory. For example, it is shown that the new framework preserves the functional form of the Eady growth rate for linear instability, as well as an analogue of Arnold's first stability theorem. Future work to develop a full parameterisation of ocean eddies will be discussed.

Thu, 31 Oct 2013

16:00 - 17:00

Coherent Lagrangian vortices: The black holes of turbulence

George Haller
((ETH) Zurich)
We discuss a simple variational principle for coherent material vortices

in two-dimensional turbulence. Vortex boundaries are sought as closed

stationary curves of the averaged Lagrangian strain. We find that

solutions to this problem are mathematically equivalent to photon spheres

around black holes in cosmology. The fluidic photon spheres satisfy

explicit differential equations whose outermost limit cycles are optimal

Lagrangian vortex boundaries. As an application, we uncover super-coherent

material eddies in the South Atlantic, which yield specific Lagrangian

transport estimates for Agulhas rings. We also describe briefly coherent

Lagrangian vortex detection to three-dimensional flows.

Thu, 24 Oct 2013

16:00 - 17:00

Connectivity in confined dense networks

Carl Dettman
We consider a random geometric graph model relevant to wireless mesh networks. Nodes are placed uniformly in a domain, and pairwise connections

are made independently with probability a specified function of the distance between the pair of nodes, and in a more general anisotropic model, their orientations. The probability that the network is (k-)connected is estimated as a function of density using a cluster expansion approach. This leads to an understanding of the crucial roles of

local boundary effects and of the tail of the pairwise connection function, in contrast to lower density percolation phenomena.

Thu, 17 Oct 2013

16:00 - 17:00

Patterns in neural field models

Stephen Coombes
(University of Nottingham)
Neural field models describe the coarse-grained activity of populations of

interacting neurons. Because of the laminar structure of real cortical

tissue they are often studied in two spatial dimensions, where they are well

known to generate rich patterns of spatiotemporal activity. Such patterns

have been interpreted in a variety of contexts ranging from the

understanding of visual hallucinations to the generation of

electroencephalographic signals. Typical patterns include localised

solutions in the form of travelling spots, as well as intricate labyrinthine

structures. These patterns are naturally defined by the interface between

low and high states of neural activity. Here we derive the equations of

motion for such interfaces and show, for a Heaviside firing rate, that the

normal velocity of an interface is given in terms of a non-local Biot-Savart

type interaction over the boundaries of the high activity regions. This

exact, but dimensionally reduced, system of equations is solved numerically

and shown to be in excellent agreement with the full nonlinear integral

equation defining the neural field. We develop a linear stability analysis

for the interface dynamics that allows us to understand the mechanisms of

pattern formation that arise from instabilities of spots, rings, stripes and

fronts. We further show how to analyse neural field models with

linear adaptation currents, and determine the conditions for the dynamic

instability of spots that can give rise to breathers and travelling waves.

We end with a discussion of amplitude equations for analysing behaviour in

the vicinity of a bifurcation point (for smooth firing rates). The condition

for a drift instability is derived and a center manifold reduction is used

to describe a slowly moving spot in the vicinity of this bifurcation. This

analysis is extended to cover the case of two slowly moving spots, and

establishes that these will reflect from each other in a head-on collision.

Thu, 13 Jun 2013

16:00 - 17:00
DH 1st floor SR


(Oxford/Cambridge Meeting to be held in Cambridge)
Thu, 30 May 2013

16:00 - 17:00
DH 1st floor SR

Matchmaker, matchmaker, make me a match: migration of population via marriages in the past

SangHoon Lee
The study of human mobility patterns can provide important information for city planning or predicting epidemic spreading, has recently been achieved with various methods available nowadays such as tracking banknotes, airline transportation, official migration data from governments, etc. However, the dearth of data makes it much more difficult to study human mobility patterns from the past. In the present study, we show that Korean family books (called "jokbo") can be used to estimate migration patterns for the past 500 years. We

apply two generative models of human mobility, which are conventional gravity-like models and radiation models, to quantify how relevant the geographical information is to human marriage records in the data. Based on the different migration distances of family names, we show the almost dichotomous distinction between "ergodic" (spread in the almost entire country) and (localized) "non-ergodic" family names, which is a characteristic of Korean family names in contrast to Czech family names. Moreover, the majority of family names are ergodic throughout the long history of Korea, suggesting that they are stable not only in terms of relative fractions but also geographically.

Thu, 23 May 2013

16:00 - 17:00
DH 1st floor SR

On contact line dynamics with mass transfer

Jim Oliver
We investigate the effect of mass transfer on the evolution of a thin two-dimensional partially wetting drop. While the effects of viscous dissipation, capillarity, slip and uniform mass transfer are taken into account, the effects of inter alia gravity, surface tension gradients, vapour transport and heat transport are neglected in favour of mathematical tractability. Our matched asymptotic analysis reveals that the leading-order outer formulation and contact-line law that is selected in the small-slip limit depends delicately on both the sign and size of the mass transfer flux. We analyse the resulting evolution of the drop and report good agreement with numerical simulations.
Thu, 16 May 2013

16:00 - 17:00
DH 1st floor SR

Modelling size effects in microcantilevers

Ed Tarleton
(Material Science Oxford)
Focused ion beam milling allows small scale single crystal cantilevers to be produced with cross-sectional dimensions on the order of microns which are then tested using a nanoindenter allowing both elastic and plastic materials properties to be measured. EBSD allows these cantilevers to be milled from any desired crystal orientation. Micro-cantilever bending experiments suggest that sufficiently smaller cantilevers are stronger, and the observation is believed to be related to the effect of the neutral axis on the evolution of the dislocation structure. A planar model of discrete dislocation plasticity was used to simulate end-loaded cantilevers to interpret the behaviour observed in the experiments. The model allowed correlation of the simulated dislocation structure to the experimental load displacement curve and GND density obtained from EBSD. The planar model is sufficient for identifying the roles of the neutral axis and source spacing in the observed size effect, and is particularly appropriate for comparisons to experiments conducted on crystals orientated for plane strain deformation. The effect of sample dimensions and dislocation source density are investigated and compared to small scale mechanical tests conducted on Titanium and Zirconium.
Thu, 09 May 2013

16:00 - 16:30
DH 1st floor SR

Discrete nonlinear dynamics and the design of new materials

Chiara Daraio
(ETH, Zurich)
We develop a physical understanding of how stress waves propagate in uniform, heterogeneous, ordered and disordered media composed of discrete granular particles. We exploit this understanding to create experimentally novel materials and devices at different scales, (for example, for application in energy absorption, acoustic imaging and energy harvesting). We control the constitutive behavior of the new materials selecting the particles’ geometry, their arrangement and materials properties. One-dimensional chains of particles exhibit a highly nonlinear dynamic response, allowing a completely new type of wave propagation that has opened the door to exciting fundamental physical observations (i.e., compact solitary waves, energy trapping phenomena, and acoustic rectification). This talk will focus on energy localization and redirection in one-, two- and three-dimensional systems. (For an extended abstract please contact Ruth @email).
Thu, 02 May 2013

16:00 - 17:00
DH 1st floor SR

Consequences of Viscous Anisotropy in Partially Molten Rocks

Richard Katz
In partially molten regions of Earth, rock and magma coexist as a two-phase aggregate in which the solid grains of rock form a viscously deformable matrix. Liquid magma resides within the permeable network of pores between grains. Deviatoric stress causes the distribution of contact area between solid grains to become anisotropic; this causes anisotropy of the matrix viscosity. The anisotropic viscosity tensor couples shear and volumetric components of stress/strain rate. This coupling, acting over a gradient in shear stress, causes segregation of liquid and solid. Liquid typically migrates toward higher shear stress, but under specific conditions, the opposite can occur. Furthermore, in a two-phase aggregate with a porosity-weakening viscosity, matrix shear causes porosity perturbations to grow into a banded structure. We show that viscous anisotropy reduces the angle between these emergent high-porosity features and the shear plane. This is consistent with lab experiments.
Thu, 25 Apr 2013

16:00 - 17:00
Gibson Grd floor SR

A mathematical approach to the mathematical modelling of Lithium-ion batteries

Angel Ramos
(Universidad Complutense de Madrid)
In this talk we will discuss the mathematical modelling of the performance of Lithium-ion batteries. A mathematical model based on a macro-homogeneous approach developed by John Neuman will be presented. The uniqueness and existence of solution of the corresponding problem will be also discussed.
Thu, 07 Mar 2013

16:00 - 17:00
DH 1st floor SR

Theory of equilibria of elastic braids with applications to DNA supercoiling

Gert Van Der Heijden
(UCL London)
We formulate a new theory for equilibria of 2-braids, i.e., structures

formed by two elastic rods winding around each other in continuous contact

and subject to a local interstrand interaction. Unlike in previous work no

assumption is made on the shape of the contact curve. The theory is developed

in terms of a moving frame of directors attached to one of the strands with

one of the directors pointing to the position of the other strand. The

constant-distance constraint is automatically satisfied by the introduction

of what we call braid strains. The price we pay is that the potential energy

involves arclength derivatives of these strains, thus giving rise to a

second-order variational problem. The Euler-Lagrange equations for this

problem (in Euler-Poincare form) give balance equations for the overall

braid force and moment referred to the moving frame as well as differential

equations that can be interpreted as effective constitutive relations

encoding the effect that the second strand has on the first as the braid

deforms under the action of end loads. Hard contact models are used to obtain

the normal contact pressure between strands that has to be non-negative for

a physically realisable solution without the need for external devices such

as clamps or glue to keep the strands together. The theory is first

illustrated by a few simple examples and then applied to several problems

that require the numerical solution of boundary-value problems. Both open

braids and closed braids (links and knots) are considered and current

applications to DNA supercoiling are discussed.