Past Industrial and Applied Mathematics Seminar

I will present a survey of the main results about first and second order models of swarming where repulsion and attraction are modeled through pairwise potentials. We will mainly focus on the stability of the fascinating patterns that you get by random data particle simulations, flocks and mills, and their qualitative behavior.
 

Despite many years of intensive research, the modeling of contact lines moving by spreading and/or evaporation still remains a subject of debate nowadays, even for the simplest case of a pure liquid on a smooth and homogeneous horizontal substrate. In addition to the inherent complexity of the topic (singularities, micro-macro matching, intricate coupling of many physical effects, …), this also stems from the relatively limited number of studies directly comparing theoretical and experimental results, with as few fitting parameters as possible. In this presentation, I will address various related questions, focusing on the physics invoked to regularize singularities at the microscale, and discussing the impact this has at the macroscale. Two opposite “minimalist” theories will be detailed: i) a classical paradigm, based on the disjoining pressure in combination with the spreading coefficient; ii) a new approach, invoking evaporation/condensation in combination with the Kelvin effect (dependence of saturation conditions upon interfacial curvature). Most notably, the latter effect enables resolving both viscous and thermal singularities altogether, without needing any other regularizing effects such as disjoining pressure, precursor films or slip length. Experimental results are also presented about evaporation-induced contact angles, to partly validate the first approach, although it is argued that reality might often lie in between these two extreme cases.

This talk covers two topics: (1) Phenotype change, where we consider the steady-fitness states, in a model developed by Korobeinikov and Dempsey (2014), in which the phenotype is modelled on a continuous scale providing a structured variable to quantify the phenotype state. This enables thresholds for survival/extinction to be established in terms of fitness.

Topic (2) looks at the steady-size distribution of an evolving cohort of cells, such as tumour cells in vitro, and therein establishes thresholds for growth or decay of the cohort. This is established using a new class of non-local (but linear) singular eigenvalue problems which have point spectra, like the traditional Sturm-Liouville problems.  The first eigenvalue gives the threshold required. But these problems are first order unless dispersion is added to incorporate random perturbations. But the same idea will apply here also.  Current work involves binary asymmetrical division of cells, simultaneous with growth. It has implications to cancer biology, helping biologists to conceptualise non-local effects and the part they may play in cancer. This is developed in Zaidi et al (2015).

Acknowledgement. The support of Gravida (NCGD) is gratefully acknowledged.

References

Korobeinikov A & Dempsey C. A continuous phenotype space model of RNA virus evolution within a host. Mathematical Biosciences and Engineering 11, (2014), 919-927.

Zaidi AA, van-Brunt B, & Wake GC. A model for asymmetrical cell division Mathematical Biosciences and Engineering (June 2015).

It is well known that low-Reynolds-number flows ($R_e\ll1$) have unique solutions, but this statement may not be true if complex solutions are permitted.

We begin by considering Stokes series, where a general steady velocity field is expanded as a power series in the Reynolds number. At each order, a linear problem determines the coefficient functions, providing an exact closed form representation of the solution for all Reynolds numbers. However, typically the convergence of this series is limited by singularities in the complex $R_e$ plane. 

We employ a generalised Pade approximant technique to continue analytically the solution outside the circle of convergence of the series. This identifies other solutions branches, some of them complex. These new solution branches can be followed as they boldly go where no flow has gone before. Sometimes these complex solution branches coalesce giving rise to real solution branches. It is shown that often, an unforced, nonlinear complex "eigensolution" exists, which implies a formal nonuniqueness, even for small and positive $R_e$.

Extensive reference will be made to Dean flow in a slowly curved pipe, but also to flows between concentric, differentially rotating spheres, and to convection in a slot. In addition, certain fundamental exact solutions are shown to possess extra complex solutions.

by Jonathan Mestel and Florencia Boshier

In today's interconnected world, the dissemination of an idea, a trend, a rumor through social networks, as well as the propagation of information or cyber-viruses through digital networks are all common phenomena. They are conceptually similar to the spread of infectious diseases among hosts, as common to all these phenomena is the dissemination of a spreading agent on a networked system. A large body of research has been produced in recent years to characterize the spread of epidemics on static connectivity patterns in a wide range of biological and socio-technical systems. In particular, understanding the mechanisms and conditions for widespread dissemination represents a crucial step for its prevention and control (e.g. in the case of diseases) or for its enhancement (e.g. in the case of viral marketing). This task is however further hindered by the temporal nature characterizing the activation of the connections shaping the networked system, for which data has recently become available. As an example, in networks of proximity contacts among individuals, connections represent sequences of contacts that are active for given periods of time. The time variation of contacts in a networked system may fundamentally alter the properties of spreading processes occurring on it, with respect to static networks, and affect the condition at which epidemics become possible. In this talk I will present a novel theoretical framework adopting a multi-layer perspective for the analytical understanding of the interplay between temporal networks and spreading dynamics. The framework is tested on a set of time-varying network models and empirical networks.

Abstract: Motivated loosely by the problem of carbon sequestration in underground aquifers, I will describe computations and analysis of one-sided two-dimensional convection of a solute in a fluid-saturated porous medium, focusing on the case in which the solute decays via a chemical reaction.   Scaling properties of the flow at high Rayleigh number are established and rationalized through an asymptotic model, that addresses the transient stability of a near-surface boundary layer and the structure of slender plumes that form beneath.  The boundary layer is shown to restrict the rate of solute transport to deep domains.  Knowledge of the plume structure enables slow erosion of the substrate of the reaction to be described in terms of a simplified free boundary problem.

Co-authors: KA Cliffe, H Power, DS Riley, TJ Ward

Particle-based stochastic reaction diffusion methods have become a 
popular approach for studying the behavior of cellular processes in 
which both spatial transport and noise in the chemical reaction process 
can be important. While the corresponding deterministic, mean-field 
models given by reaction-diffusion PDEs are well-established, there are 
a plethora of different stochastic models that have been used to study 
biological systems, along with a wide variety of proposed numerical 
solution methods.

In this talk I will motivate our interest in such methods by first 
summarizing several applications we have studied, focusing on how the 
complicated ultrastructure within cells, as reconstructed from X-ray CT 
images, might influence the dynamics of cellular processes. I will then 
introduce our attempt to rectify the major drawback to one of the most 
popular particle-based stochastic reaction-diffusion models, the lattice 
reaction-diffusion master equation (RDME). We propose a modified version 
of the RDME that converges in the continuum limit that the lattice 
spacing approaches zero to an appropriate spatially-continuous model. 
Time-permitting, I will discuss several questions related to calibrating 
parameters in the underlying spatially-continuous model.

In order:

1. Michael Dallaston, "Modelling channelization under ice shelves"

2. Jeevanjyoti Chakraborty, "Growth, elasticity, and diffusion in 
lithium-ion batteries"

3. Roberta Minussi, "Lattice Boltzmann modelling of the generation and 
propagation of action potential in neurons"

The folded structures of proteins display a remarkable variety of three-dimensional forms, and this structural diversity confers to proteins their equally remarkable functional diversity. The accelerating accumulation of experimental structures, and the declining numbers of novel folds among them suggests that a substantial fraction of the protein folds used in nature have already been observed. The physical forces stabilizing the folded structures of proteins are now understood in some detail, and much progress has been made on the classical problem of predicting the structure of a particular protein from its sequence. However, there is as yet no satisfactory theory describing the “morphology” of protein folds themselves. This talk will describe an approach to this problem based on the description of protein folds as geometric objects using the differential geometry of curves and surfaces. Applications of the theory toward modeling of diverse protein folds and assemblies which are refractory to high-resolution structure determination will be emphasized.

Droplet impacts form an important part of many processes and a detailed
understanding of the impact dynamics is critical in determining any
subsequent splashing behaviour. Prior to touchdown a gas squeeze film is
set-up between the substrate and the approaching droplet. The pressure
build-up in this squeeze film deforms the droplet free-surface, trapping
a pocket of gas and delaying touchdown. In this talk I will discuss two
extensions of existing models of pre-impact gas-cushioned droplet
behaviour, to model droplet impacts with textured substrates and droplet
impacts with surfaces hot enough to induce pre-impact phase change.

In the first case the substrate will be modelled as a thin porous layer.
This produces additional pathways for some of the gas to escape and
results in less delayed touchdown compared to a flat plate. In the
second case ideas related to the evaporation of heated thin viscous
films will be used to model the phase change. The vapour produced from
the droplet is added to the gas film enhancing the existing cushioning
mechanism by generating larger trapped gas pockets, which may ultimately
prevent touchdown altogether once the temperature enters the Leidenfrost
regime.

It is well known that the Navier-Stokes equations of viscous fluid flow do not give good predictions of when a viscous flow is likely to become unstable.  When classical linearized theory is used to explore the stability of a viscous flow, the Navier-Stokes equations predict that instability will occur at fluid speeds (Reynolds numbers) far in excess of those actually measured in experiments.  In response to this discrepancy, theories have arisen that suggest the eigenvalues computed in classical stability analysis do not give a full account of the behaviour, while others have suggested that fluid instability is a fundamentally non-linear process which is not accessible to linearized stability analyses.

In this talk, an alternative account of fluid instability and turbulence will be explored.  It is suggested that the Navier-Stokes equations themselves might not be entirely appropriate to describe the transition to turbulent flow.  A slightly more general model allows the possibility that the classical viscous fluid flows predicted by Navier-Stokes theory may become unstable at Reynolds numbers much closer to those seen in experiments, and so might perhaps give an account of the physics underlying turbulent behaviour.

Persistent homology is a tool for identifying topological features of (often high-dimensional) data. Typically, the data is indexed by a one-dimensional parameter space, and persistent homology is easily visualized via a persistence diagram or "barcode." Multi-dimensional persistent homology identifies topological features for data that is indexed by a multi-dimensional index space, and visualization is challenging for both practical and algebraic reasons. In this talk, I will give an introduction to persistent homology in both the single- and multi-dimensional settings. I will then describe an approach to visualizing multi-dimensional persistence, and the algebraic and computational challenges involved. Lastly, I will demonstrate an interactive visualization tool, the result of recent work to efficiently compute and visualize multi-dimensional persistent homology. This work is in collaboration with Michael Lesnick of the Institute for Mathematics and its Applications.

LiFePO4 is a commercially available battery material with good theoretical discharge capacity, excellent cycle life and increased safety compared with competing Li-ion chemistries. During discharge, LiFePO4 material can undergo phase separation, between a highly and lowly lithiated form. Discharge of LiFePO4 crystals has traditionally been modelled by one-phase Stefan problems, which assume that phase separation occurs.

Recent work has been using phase-field models based on the Cahn-Hilliard equation, which only phase-separates when thermodynamically favourable. In the past year or two, this work has been having considerable impact in both theoretical and experimental electrochemistry.

Unfortunately, these models are very difficult to solve numerically and involve large, coupled systems of nonlinear PDEs across several different size scales that include a range of different physics and cannot be homogenised effectively.

This talk will give an overview of recent developments in modelling LiFePO4 and the sort of strategies used to solve these systems numerically.

Acto-myosin network growth and remodeling in vivo is based on a large number of chemical and mechanical processes, which are mutually coupled and spatially and temporally resolved. To investigate the fundamental principles behind the self-organization of these networks, we have developed detailed physico-chemical, stochastic models of actin filament growth dynamics, where the mechanical rigidity of filaments and their corresponding deformations under internally and externally generated forces are taken into account. Our work sheds light on the interplay between the chemical and mechanical processes, and also will highlights the importance of diffusional and active transport phenomena. For example, we showed that molecular transport plays an important role in determining the shapes of the commonly observed force-velocity curves. We also investigated the nonlinear mechano-chemical couplings between an acto-myosin network and an external deformable substrate.

We are interested in large systems of agents collectively looking for a

consensus (about e.g. their direction of motion, like in bird flocks). In

spite of the local character of the interactions (only a few neighbours are

involved), these systems often exhibit large scale coordinated structures.

The understanding of how this self-organization emerges at the large scale

is still poorly understood and offer fascinating challenges to the modelling

science. We will discuss a few of these issues on a selection of specific

examples.

The talk is on impacts, penetrations and lift-offs involving bodies and fluids, with applications that range from aircraft and ship safety and our tiny everyday scales of splashing and washing, up to surface movements on Mars. Several studies over recent years have addressed different aspects of air-water effects and fluid-body interplay theoretically. Nonlinear interactions and evolutions are key here and these are to be considered in the presentation. Connections with experiments will also be described.

The synthesis of complex-shaped colloids and nanoparticles has recently undergone unprecedented advancements. It is now possible to manufacture particles shaped as dumbbells, cubes, stars, triangles, and cylinders, with exquisite control over the particle shape. How can particle geometry be exploited in the context of capillarity and surface-tension phenomena? This talk examines this question by exploring the case of complex-shaped particles adsorbed at the interface between two immiscible fluids, in the small Bond number limit in which gravity is not important. In this limit, the "Cheerio's effect" is unimportant, but interface deformations do emerge. This drives configuration dependent capillary forces that can be exploited in a variety of contexts, from emulsion stabilisation to the manufacturing of new materials. It is an opportunity for the mathematics community to get involved in this field, which offers ample opportunities for careful mathematical analysis. For instance, we find that the mathematical toolbox provided by 2D potential theory lead to remarkably good predictions of the forces and torques measured experimentally by tracking particle pairs of cylinders and ellipsoids. New research directions will also be mentioned during the talk, including elasto-capillary interactions and the simulation of multiphase composites.

Molecular modelling has become a valuable tool and is increasingly part of the standard methodology of chemistry, physics, engineering and biology. The power of molecular modelling lies in its versatility: as potential energy functions improve, a broader range of more complex phenomena become accessible to simulation, but much of the underlying methodology can be re-used. For example, the Verlet method is still the most popular molecular dynamics scheme for constant energy molecular dynamics simulations despite it being one of the first to be proposed for the purpose.

One of the most important challenges in molecular modelling remains the computation of averages with respect to the canonical Gibbs (constant temperature) distribution, for which the Verlet method is not appropriate. Whereas constant energy molecular dynamics prescribes a set of equations (Newton's equations), there are many alternatives for canonical sampling with quite different properties. The challenge is therefore to identify formulations and numerical methods that are robust and maximally efficient in the computational setting.

One of the simplest and most effective methods for sampling is based on Langevin dynamics which mimics coupling to a heat bath by the incorporation of random forces and an associated dissipative term. Schemes for Langevin dynamics simulation may be developed based on the familiar principle of splitting. I will show that the invariant measure ('long term') approximation may be strongly affected by a simple re-ordering of the terms of the splitting. I will describe a transition in weak numerical order of accuracy that occurs (in one case) in the t->infty limit.

I will also entertain some more radical suggestions for canonical sampling, including stochastic isokinetic methods that enable the use of greatly enlarged timesteps for expensive but slowly-varying force field components.

A perturbative method is introduced to analyze shear bands formation and

development in ductile solids subject to large strain.

Experiments on discrete systems made up of highly-deformable elements [1]

confirm the validity of the method and suggest that an elastic structure

can be realized buckling for dead, tensile loads. This structure has been

calculated, realized and tested and provides the first example of an

elastic structure buckling without elements subject to compression [2].

The perturbative method introduced for the analysis of shear bands can be

successfuly employed to investigate other material instabilities, such as

for instance flutter in a frictional, continuum medium [3]. In this

context, an experiment on an elastic structure subject to a frictional

contact shows for the first time that a follower load can be generated

using dry friction and that this load can induce flutter instability [4].

The perturbative approach may be used to investigate the strain state near

a dislocation nucleated in a metal subject to a high stress level [5].

Eshelby forces, similar to those driving dislocations in solids, are

analyzed on elastic structures designed to produce an energy release and

therefore to evidence configurational forces. These structures have been

realized and they have shown unexpected behaviours, which opens new

perspectives in the design of flexible mechanisms, like for instance, the

realization of an elastic deformable scale [6].

[1] D. Bigoni, Nonlinear Solid Mechanics Bifurcation Theory and Material

Instability. Cambridge Univ. Press, 2012, ISBN:9781107025417.

[2] D. Zaccaria, D. Bigoni, G. Noselli and D. Misseroni Structures

buckling under tensile dead load. Proc. Roy. Soc. A, 2011, 467, 1686.

[3] A. Piccolroaz, D. Bigoni, and J.R. Willis, A dynamical interpretation

of flutter instability in a continuous medium. J. Mech. Phys. Solids,

2006, 54, 2391.

[4] D. Bigoni and G. Noselli Experimental evidence of flutter and

divergence instabilities induced by dry friction. J. Mech. Phys.

Solids,2011,59,2208.

[5] L. Argani, D. Bigoni, G. Mishuris Dislocations and inclusions in

prestressed metals. Proc. Roy. Soc. A, 2013, 469, 2154 20120752.

[6] D. Bigoni, F. Bosi, F. Dal Corso and D. Misseroni, Instability of a

penetrating blade. J. Mech. Phys. Solids, 2014, in press.

Several experimental studies have shown that the abundance distributions of proteins in a population of isogenic cells may display multiple distinct maxima. Each of these maxima may be associated with a subpopulation of a particular phenotype, the quantification of which is important for understanding cellular decision-making. I will present a novel methodology which allows us to quantify multi-modal gene expression distributions and single cell power spectra in gene regulatory networks. The method is based on an extension of the linear noise approximation; in particular we rigorously show that, in the limit of slow promoter dynamics, these distributions can be systematically approximated as a mixture of Gaussian components. The resulting closed-form approximation provides a practical tool for studying complex nonlinear gene regulatory networks that have thus far been amenable only to stochastic simulation. I will demonstrate the applicability of our approach to several examples and discuss some new dynamical characteristics e.g., how the interplay of transcriptional and translational regulation can be exploited to control the multimodality of gene expression distributions in two-promoter networks and how genetic oscillators can display concerted noise-induced bimodality and noise-induced oscillations.

Adaptive network models, in which node states and network topology coevolve, arise naturally in models of social dynamics that incorporate homophily and social influence. Homophily relates the similarity between pairs of agents' states to their network coupling strength, whilst social influence causes the convergence of coupled agents' states. In this talk, I will describe a deterministic adaptive network model of attitude formation in social groups that incorporates these effects, and in which the attitudinal dynamics are represented by an activator-inhibitor process. I will show that consensus, corresponding to all nodes adopting the same attitudinal state and being fully connected, may destabilise via Turing instability, giving rise to chaotic dynamics. For the case where there are just two agents, I will illustrate, using numerical continuation, how such chaotic dynamics arise.

One of the things sustainability applications have in common with industrial applications is their close connection with decision-making and policy. We will discuss how a decision-support viewpoint may inspire new mathematical questions. For example, the concept of resilience (of ecosystems, food systems, communities, economies, etc) is often described as the capacity of a system to withstand disturbance and retain its functional characteristics. This has several familiar mathematical interpretations, probing the interaction between transient dynamics and noise. How does a focus on resilience change the modeling, dynamical and policy questions we ask? I look forward to your ideas and discussion.

Familiar species; humans, mammals, fish, reptiles and plants represent only a razor’s edge of the Earth’s immense biodiversity. Most of the Earth’s multicellular species lie buried in soil, inside of plants, and in the undergrowth, and include millions of unknown species, almost half of which are thought to be fungi. Part of the amazing success of fungi may be the elegant solutions that they have evolved to the problems of dispersing, growing and adapting to changing environments. I will describe how we using both math modeling and experiments to discover some of these solutions. I will focus on (i) how cytoplasmic mixing enables some species to tolerate internal genetic diversity, making them better pathogens and more adaptable, and (ii) how self-organization of these flows into phases of transport and stasis enables cells to function both as transport conduits, and to perform other functions like growth and secretion.

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.

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.

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.

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.

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.

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

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.

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.

(NOTE CHANGE OF VENUE TO L2)

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.