Past Industrial and Applied Mathematics Seminar

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.

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).

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.

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.

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.

Abstract available upon request - Ruth Preston @email

Self-renewal and pluripotency of mouse embryonic stem (ES) cells are controlled by a complex transcriptional regulatory network (TRN) which is rich in positive feedback loops. A number of key components of this TRN, including Nanog, show strong temporal expression fluctuations at the single cell level, although the precise molecular basis for this variability remains unknown. In this talk I will discuss recent work which uses a genetic complementation strategy to investigate genome-wide mRNA expression changes during transient periods of Nanog down-regulation. Nanog removal triggers widespread changes in gene expression in ES cells. However, we found that significant early changes in gene expression were reversible upon re-induction of Nanog, indicating that ES cells initially adopt a flexible “primed” state. Nevertheless, these changes rapidly become consolidated irreversible fate decisions in the continued absence of Nanog. Using high-throughput single cell transcriptional profiling we observed that the early molecular changes are both stochastic and reversible at the single cell level. Since positive feedback commonly gives rise to phenotypic variability, we also sought to determine the role of feedback in regulating ES cell heterogeneity and commitment. Analysis of the structure of the ES cell TRN revealed that Nanog acts as a feedback “linchpin”: in its presence positive feedback loops are active and the extended TRN is self-sustaining; while in its absence feedback loops are weakened, the extended TRN is no longer self-sustaining and pluripotency is gradually lost until a critical “point-of-no-return” is reached. Consequently, fluctuations in Nanog expression levels transiently activate different sub-networks in the ES cell TRN, driving transitions between a (Nanog expressing) feedback-rich, robust, self-perpetuating pluripotent state and a (Nanog-diminished), feedback-depleted, differentiation-sensitive state. Taken together, our results indicate that Nanog- dependent feedback loops play a central role in controlling both early fate decisions at the single cell level and cell-cell variability in ES cell populations.

Motivated by industrial and biological applications, the Waves

Group at Manchester has in recent years been interested in

developing methods for obtaining the effective properties of

complex composite materials. As time allows we shall discuss a

number of issues, such as differences between composites

with periodic and aperiodic distributions of inclusions, and

modelling of nonlinear composites.

I discuss models for the planar spreading of a viscous fluid between an elastic lid and an underlying rigid plane. These have application to the growth of magmatic intrusions, as well as to other industrial and biological processes; simple experiments of an inflated blister will be used for motivation. The height of the fluid layer is described by a sixth order non-linear diffusion equation, analogous to the fourth order equation that describes surface tension driven spreading. The dynamics depend sensitively on the conditions at the contact line, where the sheet is lifted from the substrate and where some form of regularization must be applied to the model. I will explore solutions with a pre-wetted film or a constant-pressure fluid lag, for flat and inclined planes, and compare with the analogous surface tension problems.

When a rubber membrane tube is inflated, a localized bulge will initiate when the internal pressure reaches a certain value known as the initiation pressure. As inflation continues, the bulge will grow in diameter until it reaches a maximum size, after which the bulge will spread in both directions. This simple phenomenon has previously been studied both experimentally, numerically, and analytically, but surprisingly it is only recently that the character of the initiation pressure has been fully understood. In this talk, I shall first show how the entire inflation process can be described analytically, and then apply the ideas to the mathematical modelling of aneurysm initiation in human arteries.

It is generally believed that in order to generate waves, a small object (like an insect) moving at the air-water surface must exceed the minimum wave speed (about 23 centimeters per second). We show that this result is only valid for a rectilinear uniform motion, an assumption often overlooked in the literature. In the case of a steady circular motion (a situation of particular importance for the study of whirligig beetles), we demonstrate that no such velocity threshold exists and that even at small velocities a finite wave drag is experienced by the object. This wave drag originates from the emission of a spiral-like wave pattern. The results presented should be important for a better understanding of the propulsion of water-walking insects. For example, it would be very interesting to know if whirligig beetles can take advantage of such spirals for echolocation purposes.

The essential information contained in most large data sets is

small when compared to the size of the data set. That is, the

data can be well approximated using relatively few terms in a

suitable transformation. Compressed sensing and matrix completion

show that this simplicity in the data can be exploited to reduce the

number of measurements. For instance, if a vector of length $N$

can be represented exactly using $k$ terms of a known basis

then $2k\log(N/k)$ measurements is typically sufficient to recover

the vector exactly. This can result in dramatic time savings when

k

The periodic orbits of a discrete dynamical system can be described as

permutations. We derive the composition law for such permutations. When

the composition law is given in matrix form the composition of

different periodic orbits becomes remarkably simple. Composition of

orbits in bifurcation diagrams and decomposition law of composed orbits

follow directly from that matrix representation.

We propose a model to reproduce qualitatively and quantitatively the experimental behavior obtained by the AFM techniques for the titin. Via an energetic based minimization approach we are able to deduce a simple analytical formulations for the description of the mechanical behavior of multidomain proteins, giving a physically base description of the unfolding mechanism. We also point out that our model can be inscribed in the led of the pseudo-elastic variational damage model with internal variable and fracture energy criteria of the continuum mechanics. The proposed model permits simple analytical calculations and

to reproduce hard-device experimental AFM procedures. The proposed model also permits the continuum limit approximation which maybe useful to the development of a three-dimensional multiscale constitutive model for biological tissues.

Ultracold atomic gases have recently proven to be enormously rich

systems from the perspective of a condensed matter physicist. With

the advent of optical lattices, such systems can now realise idealised

model Hamiltonians used to investigate strongly correlated materials.

Conversely, ultracold atomic gases can exhibit quantum phases and

dynamics with no counterpart in the solid state due to their extra

degrees of freedom and unique environments virtually free of

dissipation. In this talk, I will discuss examples of such behaviour

arising from spinor degrees of freedom on which my recent research has

focused. Examples will include bosons with artificially induced

spin-orbit coupling and the non-equilibrium dynamics of spinor

condensates.

In this talk I shall describe two rather different, but not entirely unrelated,

problems involving thin-film flow of a viscous fluid which I have found of interest

and which may have some application to a number of practical situations,

including condensation in heat exchangers and microfluidics.

The first problem,

which is joint work with Adam Leslie and Brian Duffy at the University of Strathclyde,

concerns the steady three-dimensional flow of a thin, slowly varying ring of fluid

on either the outside or the inside of a uniformly rotating large horizontal cylinder.

Specifically, we study ``full-ring'' solutions, corresponding to a ring of continuous,

finite and non-zero thickness that extends all the way around the cylinder.

These full-ring solutions may be thought of as a three-dimensional generalisation of

the ``full-film'' solutions described by Moffatt (1977) for the corresponding two-dimensional problem.

We describe the behaviour of both the critical and non-critical full-ring solutions.

In particular,

we show that, while for most values of the rotation speed and the load the azimuthal velocity is

in the same direction as the rotation of the cylinder, there is a region of parameter space close

to the critical solution for sufficiently small rotation speed in which backflow occurs in a

small region on the upward-moving side of the cylinder.

The second problem,

which is joint work with Phil Trinh and Howard Stone at Princeton University,

concerns a rigid plate moving steadily on the free surface of a thin film of fluid.

Specifically, we study two problems

involving a rigid flat (but not, in general, horizontal) plate:

the pinned problem, in which the upstream end of plate is pinned at a fixed position,

the fluid pressure at the upstream end of the plate takes a prescribed value and there is a free surface downstream of the plate, and

the free problem, in which the plate is freely floating and there are free surfaces both upstream and downstream of the plate.

For both problems, the motion of the fluid and the position of the plate

(and, in particular, its angle of tilt to the horizontal) depend in a non-trivial manner on the

competing effects of the relative motion of the plate and the substrate,

the surface tension of the free surface, and of the viscosity of the fluid,

together with the value of the prescribed pressure in the pinned case.

Specifically, for the pinned problem we show that,

depending on the value of an appropriately defined capillary number and on the value of the

prescribed fluid pressure, there can be either none, one, two or three equilibrium solutions

with non-zero tilt angle.

Furthermore, for the free problem we show that the solutions

with a horizontal plate (i.e.\ zero tilt angle) conjectured by Moriarty and Terrill (1996)

do not, in general, exist, and in fact there is a unique equilibrium solution with,

in general, a non-zero tilt angle for all values of the capillary number.

Finally, if time permits some preliminary results for an elastic plate will be presented.

Part of this work was undertaken while I was a

Visiting Fellow in the Department of Mechanical and Aerospace Engineering

in the School of Engineering and Applied Science at Princeton University, Princeton, USA.

Another part of this work was undertaken while I was a

Visiting Fellow in the Oxford Centre for Collaborative Applied Mathematics (OCCAM),

University of Oxford, United Kingdom.

This publication was based on work supported in part by Award No KUK-C1-013-04,

made by King Abdullah University of Science and Technology (KAUST).

Recent years have seen increasing attention to the subtle effects on

intracellular transport caused when multiple molecular motors bind to

a common cargo. We develop and examine a coarse-grained model which

resolves the spatial configuration as well as the thermal fluctuations

of the molecular motors and the cargo. This intermediate model can

accept as inputs either common experimental quantities or the

effective single-motor transport characterizations obtained through

systematic analysis of detailed molecular motor models. Through

stochastic asymptotic reductions, we derive the effective transport

properties of the multiple-motor-cargo complex, and provide analytical

explanations for why a cargo bound to two molecular motors moves more

slowly at low applied forces but more rapidly at high applied forces

than a cargo bound to a single molecular motor. We also discuss how

our theoretical framework can help connect in vitro data with in vivo

behavior.

We study a thin liquid film on a vertical fibre. Without gravity, there

is a Rayleigh-Plateau instability in which surface tension reduces the

surface area of the initially cylindrical film. Spherical drops cannot

form because of the fibre, and instead, the film forms bulges of

roughly twice the initial thickness. Large bulges then grow very slowly

through a ripening mechanism. A small non-dimensional gravity moves the

bulges. They leave behind a thinner film than that in front of them, and

so grow. As they grow into large drops, they move faster and grow

faster. When gravity is stronger, the bulges grow only to finite

amplitude solitary waves, with equal film thickness behind and in front.

We study these solitary waves, and the effect of shear-thinning and

shear-thickening of the fluid. In particular, we will be interested in

solitary waves of large amplitudes, which occur near the boundary

between large and small gravity. Frustratingly, the speed is only

determined at the third term in an asymptotic expansion. The case of

Newtonian fluids requires four terms.

Some striking, and potentially useful, effects in electrokinetics occur for

bipolar membranes: applications are in medical diagnostics amongst other areas.

The purpose of this talk is to describe the experiments, the dominant features observed

and then model the phenomena: This uncovers the physics that control this process.

Time-periodic reverse voltage bias

across a bipolar membrane is shown to exhibit transient hysteresis.

This is due to the incomplete depletion of mobile ions, at the junction

between the membranes, within two adjoining polarized layers; the layer thickness depends on

the applied voltage and the surface charge densities. Experiments

show that the hysteresis consists of an Ohmic linear rise in the

total current with respect to the voltage, followed by a

decay of the current. A limiting current is established for a long

period when all the mobile ions are depleted from the polarized layer.

If the resulting high field within the two polarized layers is

sufficiently large, water dissociation occurs to produce proton and

hydroxyl travelling wave fronts which contribute to another large jump

in the current. We use numerical simulation and asymptotic analysis

to interpret the experimental results and

to estimate the amplitude of the transient hysteresis and the

water-dissociation current.

The use of formal mathematical models in sociology started in the 1940s and attracted mathematicians such as Frank Harary in the 1950s. The idea is to take the rather intuitive ideas described in social theory and express these in formal mathematical terms. Social network analysis is probably the best known of these and it is the area which has caught the imagination of a wider audience and has been the subject of a number of popular books. We shall give a brief over view of the field of social networks and will then look at three examples which have thrown up problems of interest to the mathematical community. We first look at positional analysis techniques and give a formulation that tries to capture the notion of social role by using graph coloration. We look at algebraic structures, properties, characterizations, algorithms and applications including food webs. Our second and related example looks at core-periphery structures in social networks. Our final example relates to what the network community refer to as two-mode data and a general approach to analyzing networks of this form. In all cases we shall look at the mathematics involved and discuss some open problems and areas of research that could benefit from new approaches and insights.

The computational analysis of a mathematical model describing a complex system is often based on the following roadmap: first, an experiment is conceived, in which the measured data are (either directly or indirectly) related to the input data of the model equations; second, such equations are computationally solved to provide iconographic reconstructions of the unknown physical or physiological parameters of the system; third, the reconstructed images are utilized to validate the model or to inspire appropriate improvements. This talk will adopt such framework to investigate three applied problems, respectively in solar physics, neuroscience and physiology. The solar physics problem is concerned with the exploitation of hard X-ray data for the comprehension of energy transport mechanisms in solar flares. The neuroscientific problem is the one to model visual recognition in humans with the help of a magnetocencephalography experiment. Finally, the physiological problem investigates the kinetics of the kidney-bladder system by means of nuclear data.

Complex networks have been used to model almost any

real-world complex systems. An especially important

issue regards how to related their structure and dynamics,

which contributes not only for the better understanding of

such systems, but also to the prediction of important

dynamical properties from specific topological features.

In this talk I revise related research developed recently

in my group. Particularly attention is given to the concept

of accessibility, a new measurement integrating topology

and dynamics, and the relationship between frequency of

visits and node degree in directed modular complex

networks. Analytical results are provided that allow accurate

prediction of correlations between structure and dynamics

in systems underlain by directed diffusion. The methodology

is illustrated with respect to the macaque cortical network.

In this seminar we discuss the gas dynamics of chimneys, solar updraft towers and energy towers. The main issue is to discuss simple fluid dynamic models which still describe the main features of the mentioned applications. We focus first on one dimensional compressible models. Then we apply a small Mach number asymptotics to reduce to complexity and to avoid the known problems

of fully compressible models in the small Mach number regime. In case of the energy tower in addition we have to model the evaporation process.

Finally we obtain a much simpler fluid dynamic model which allows robust and very fast numerical simulations. We discuss the qualitative behaviour and the good agreement with expermental data (in cases such data are available).

The displacement of a liquid by an air finger is a generic two-phase flow that

underpins applications as diverse as microfluidics, thin-film coating, enhanced

oil recovery, and biomechanics of the lungs. I will present two intriguing

examples of such flows where, firstly, oscillations in the shape of propagating

bubbles are induced by a simple change in tube geometry, and secondly, flexible

vessel boundaries suppress viscous fingering instability.

1) A simple change in pore geometry can radically alter the behaviour of a

fluid displacing air finger, indicating that models based on idealized pore

geometries fail to capture key features of complex practical flows. In

particular, partial occlusion of a rectangular cross-section can force a

transition from a steadily-propagating centred finger to a state that exhibits

spatial oscillations via periodic sideways motion of the interface at a fixed

location behind the finger tip. We characterize the dynamics of the

oscillations and show that they arise from a global homoclinic connection

between the stable and unstable manifolds of a steady, symmetry-broken

solution.

2) Growth of complex dendritic fingers at the interface of air and a viscous

fluid in the narrow gap between two parallel plates is an archetypical problem

of pattern formation. We find a surprisingly effective means of suppressing

this instability by replacing one of the plates with an elastic membrane. The

resulting fluid-structure interaction fundamentally alters the interfacial

patterns that develop and considerably delays the onset of fingering. We

analyse the dependence of the instability on the parameters of the system and

present scaling arguments to explain the experimentally observed behaviour.

Feature Selection is a ubiquitous problem in across data mining,

bioinformatics, and pattern recognition, known variously as variable

selection, dimensionality reduction, and others. Methods based on

information theory have tremendously popular over the past decade, with

dozens of 'novel' algorithms, and hundreds of applications published in

domains across the spectrum of science/engineering. In this work, we

asked the question 'what are the implicit underlying statistical

assumptions of feature selection methods based on mutual information?'

The main result I will present is a unifying probabilistic framework for

information theoretic feature selection, bringing almost two decades of

research on heuristic methods under a single theoretical interpretation.

Hollow vortices are vortices whose interior is at rest. They posses vortex sheets on their boundaries and can be viewed as a desingularization of point vortices. We give a brief history of point vortices. We then obtain exact solutions for hollow vortices in linear and nonlinear strain and examine the properties of streets of hollow vortices. The former can be viewed as a canonical example of a hollow vortex in an arbitrary flow, and its stability properties depend. In the latter case, we reexamine the hollow vortex street of Baker, Saffman and Sheffield and examine its stability to arbitrary disturbances, and then investigate the double hollow vortex street. Implications and extensions of this work are discussed.

Nematic liquid crystals (NLCs) are materials that flow like liquids, but have some crystalline features. Their molecules are typically long and thin, and tend to align locally, which imparts some elastic character to the NLC. Moreover at interfaces between the NLC and some other material (such as a rigid silicon substrate, or air) the molecules tend to have a preferred direction (so-called "surface anchoring"). This preferred behaviour at interfaces, coupled with the internal "elasticity", can give rise to complex instabilities in spreading free surface films. This talk will discuss modelling approaches to describe such flows. The models presented are capable of capturing many of the key features observed experimentally, including arrested spreading (with or without instability). Both 2D and 3D spreading scenarios will be considered, and simple ways to model nontrivial surface anchoring patterns, and "defects" within the flows will also be discussed.

In a variety of problems in engineering and applied science, the goal is to design or control a network of dynamical agents so as to achieve some desired asymptotic behaviour. Examples include consensus and rendez-vous problems in robotics, synchronization of generator angles in power grids or coordination of oscillations in bacterial populations. A pressing challenge in all of these problems is to derive appropriate analytical tools to prove convergence towards the target behaviour. Such tools are not only invaluable to guarantee the desired performance, but can also provide important guidelines for the design of decentralized control strategies to steer the collective behaviour of the network of interest in a desired manner. During this talk, a methodology for analysis and design of convergence in networks will be presented which is based on the use of a classical, yet not fully exploited, tool for convergence analysis: contraction theory. As opposed to classical methods for stability analysis, the idea is to look at convergence between trajectories of a system of interest rather that at their asymptotic convergence towards some solution of interest. After introducing the problem, a methodology will be derived based on the use of matrix measures induced by non-Euclidean norms that will be exploited to design strategies to control the collective behaviour of networks of dynamical agents. Representative examples will be used to illustrate the theoretical results.

TALK 1 -- social media for OII:

TITLE: Computational Perspectives on the Structure and Information

Flows in On-Line Networks

ABSTRACT:

With an increasing amount of social interaction taking place in on-line settings, we are accumulating massive amounts of data about phenomena that were once essentially invisible to us: the collective behavior and social interactions of hundreds of millions of people Analyzing this massive data computationally offers enormous potential both to address long-standing scientific questions, and also to harness and inform the design of future social computing applications: What are emerging ideas and trends? How is information being created, how it flows and mutates as it is passed from a node to node like an epidemic?

We discuss how computational perspective can be applied to questions involving structure of online networks and the dynamics of information flows through such networks, including analysis of massive data as well as mathematical models that seek to abstract some of the underlying phenomena.

TALK 2 -- Community detection:

TITLE: Networks, Communities and the Ground-Truth

ABSTRACT: Nodes in complex networks organize into communities of nodes that share a common property, role or function, such as social communities, functionally related proteins, or topically related webpages. Identifying such communities is crucial to the understanding of the structural and functional roles of networks.Current work on overlapping community detection (often implicitly) assumes that community overlaps are less densely connected than non-overlapping parts of communities. This is unnatural as it means that the more communities nodes share, the less likely it is they are linked. We validate this assumption on a diverse set of large networks and find an increasing relationship between the number of shared communities of a pair of nodes and the probability of them being connected by an edge, which means that parts of the network where communities overlap tend to be more densely connected than the non-overlapping parts of communities. Existing community detection methods fail to detect communities with such overlaps. We propose a model-based community detection method that builds on bipartite node-community affiliation networks. Our method successfully detects overlapping, non-overlapping and hierarchically nested communities. We accurately identify relevant communities in networks ranging from biological protein-protein interaction networks to social, collaboration and information networks. Our results show that while networks organize into overlapping communities, globally networks also exhibit a nested core-periphery structure, which arises as a consequence of overlapping parts of communities being more densely connected.

The industrial prilling process is amongst the most favourite technique employed in generating monodisperse droplets. In such a process long curved jets are generated from a rotating drum which in turn breakup and from droplets. In this talk we describe the experimental set-up and the theory to model this process. We will consider the effects of changing the rheology of the fluid as well as the addition of surface agents to modify breakup characterstics. Both temporal and spatial instability will be considered as well as nonlinear numerical simulations with comparisons between experiments.

The inverse acoustic obstacle scattering problem, in its most general

form, seeks to determine the nature of an unknown scatterer from knowl-

edge of its far eld or radiation pattern. The problem which is the main

concern here is:

If the scattering cross section, i.e the absolute value of the radiation

pattern, of an unknown scatterer is known determine its shape.

In this talk we explore the problem from a number of points of view.

These include questions of uniqueness, methods of solution including it-

erative methods, the Minkowski problem and level set methods. We con-

clude by looking at the problem of acoustically invisible gateways and its

connections with cloaking

A central challenge in socio-physics is understanding how groups of self-interested agents make collective decisions. For humans many insights in the underlying opinion formation process have been gained from network models, which represent agents as nodes and social contacts as links. Over the past decade these models have been expanded

to include the feedback of the opinions held by agents on the structure of the network. While a verification of these adaptive models in humans is still difficult, evidence is now starting to appear in opinion formation experiments with animals, where the choice that is being made concerns the direction of movement. In this talk I show how analytical insights can be gained from adaptive networks models and how predictions from these models can be verified in experiments with swarming animals. The results of this work point to a similarity between swarming and human opinion formation and reveal insights in the dynamics of the opinion formation process. In particular I show that in a population that is under control of a strongly opinionated minority a democratic consensus can be restored by the addition of

uninformed individuals.

Brittle failure through multiple cracks occurs in a wide variety of contexts, from microscopic failures in dental enamel and cleaved silicon to geological faults and planetary ice crusts. In each of these situations, with complicated stress geometries and different microscopic mechanisms, pairwise interactions between approaching cracks nonetheless produce characteristically curved fracture paths. We investigate the origins of this widely observed "en passant" crack pattern by fracturing a rectangular slab which is notched on each long side and then subjected to quasistatic uniaxial strain from the short side. The two cracks propagate along approximately straight paths until they pass each other, after which they curve and release a lens-shaped fragment. We find that, for materials with diverse mechanical properties, each curve has an approximately square-root shape, and that the length of each fragment is twice its width. We are able to explain the origins of this universal shape with a simple geometrical model.

This work builds on the foundation laid by Benney & Timson (1980), who

examined the flow near a contact line and showed that, if the contact

angle is 180 degrees, the usual contact-line singularity does not arise.

Their local analysis, however, does not allow one to determine the

velocity of the contact line and their expression for the shape of the

free boundary involves undetermined constants - for which they have been

severely criticised by Ngan & Dussan V. (1984). As a result, the ideas

of Benny & Timson (1980) have been largely forgotten.

The present work shows that the criticism of Ngan & Dussan V. (1984)

was, in fact, unjust. We consider a two-dimensional steady Couette flow

with a free boundary, for which the local analysis of Benney & Timson

(1980) can be complemented by an analysis of the global flow (provided

the slope of the free boundary is small, so the lubrication

approximation can be used). We show that the undetermined constants in

the solution of Benney & Timson (1980) can all be fixed by matching

their local solution to the global one. The latter also determines the

contact line's velocity, which we compute among other characteristics of

the global flow.

Understanding the fatigue of metals under cyclic loads is crucial for some fields in mechanical engineering, such as the design of wheels of high speed trains and aero-plane engines. Experimentally it has been found that metal fatigue induced by cyclic loads is closely related to a ladder shape pattern of dislocations known as a persistent slip band (PSB). In this talk, a quantitative description for the formation of PSBs is proposed from two angles: 1. the motion of a single dislocation analised by using asymptotic expansions and numerical simulations; 2. the collective behaviour of a large number of dislocations analised by using a method of multiple scales.

It is an inherent premise in Boltzmann's formulation of linear viscoelasticity, that for shear deformations at constant pressure and constant temperature, every material has a unique continuous relaxation spectrum. This spectrum defines the memory kernel of the material. Only a few models for representing the continuous spectrum have been proposed, and these are entirely empirical in nature.

Extensive laboratory time is spent worldwide in collecting dynamic data from which the relaxation spectra of different materials may be inferred. In general the process involves the solution of one or more exponentially ill-posed inverse problems.

In this talk I shall present rigorous models for the continuous relaxation spectrum. These arise naturally from the theory of continuous wavelet transforms. In solving the inverse problem I shall discuss the role of sparsity as one means of regularization, but there is also a secondary regularization parameter which is linked, as always, to resolution. The topic of model-induced super-resolution is discussed, and I shall give numerical results for both synthetic and real experimental data.

The talk is based on joint work with Neil Goulding (Cardiff University).

Tsunami asymptotics: For most of their propagation, tsunamis are linear dispersive waves whose speed is limited by the depth of the ocean and which can be regarded as diffraction-decorated caustics in spacetime. For constant depth, uniform asymptotics gives a very accurate compact description of the tsunami profile generated by an arbitrary initial disturbance. Variations in depth can focus tsunamis onto cusped caustics, and this 'singularity on a singularity' constitutes an unusual diffraction problem, whose solution indicates that focusing can amplify the tsunami energy by an order of magnitude.

Initial stage of the flow with a free surface generated by a vertical

wall moving from a liquid of finite depth in a gravitational field is

studied. The liquid is inviscid and incompressible, and its flow is

irrotational. Initially the liquid is at rest. The wall starts to move

from the liquid with a constant acceleration.

It is shown that, if the acceleration of the plate is small, then the

liquid free surface separates from the wall only along an

exponentially small interval. The interval on the wall, along which

the free surface instantly separates for moderate acceleration of the

wall, is determined by using the condition that the displacements of

liquid particles are finite. During the initial stage the original

problem of hydrodynamics is reduced to a mixed boundary-value problem

with respect to the velocity field with unknown in advance position of

the separation point. The solution of this

problem is derived in terms of complete elliptic integrals. The

initial shape of the separated free surface is calculated and compared

with that predicted by the small-time solution of the dam break

problem. It is shown that the free surface at the separation point is

orthogonal to the moving plate.

Initial acceleration of a dam, which is suddenly released, is calculated.

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The mechanisms for the selection of the propagation speed of waves

connecting unstable to stable states will be discussed in the

spatially non-homogeneous case, the differences from the very

well-studied homogeneous version being emphasised.