DH 1st floor SR

In this talk, we elaborate on the notions of no-free-lunch that have proved essential in the theory of financial mathematics---most notably, arbitrage of the first kind. Focus will be given in most recent developments. The precise connections with existence of deflators, numeraires and pricing measures are explained, as well as the consequences that these notions have in the existence of bubbles and the valuation of illiquid assets in the market.

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.

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.