Abstracts

Floating and Sinking of a Pair of Spheres at a Liquid-Fluid Interface

Spheres floating at liquid-fluid interfaces cause interfacial deformations such that their weight is balanced by the resultant forces of surface tension and hydrostatic pressure while also satisfying a contact angle condition. Determining the meniscus shape around several floating spheres is a complicated problem because the vertical locations of the spheres and the horizontal projections of the three-phase contact lines are not known a priori. Here, a new computational algorithm is developed to simultaneously satisfy the nonlinear Laplace-Young equation for the meniscus shape, the vertical force balance, and the geometric properties of the spheres. We implement this algorithm to find the shape of the interface around a pair of floating spheres and the horizontal force required to keep them at a fixed center-center separation. Our numerical simulations show that the ability of a pair of spheres to float (rather than sink) depends on their separation. Similar to previous work on horizontal cylinders, sinking may be induced at close range for small spheres that float when isolated. However, we also discover a new and unexpected behaviour: at intermediate inter-particle distances, spheres that would sink in isolation can float as a pair. This effect is more pronounced for spheres of radius comparable to the capillary length, suggesting that this effect is a result of hydrostatic pressure, rather than surface tension. An approximate solution confirms these phenomena and shows that the mechanism is indeed the increased supporting force provided by the hydrostatic pressure. Finally, the horizontal force of capillary attraction between the spheres is calculated using the results of the numerical simulations. These results show that the classic linear superposition approximation (due to Nicolson, 1949) can lose its validity for relatively heavy particles at close range.

Creep, Relaxation, and Thomas Joannes Stieltjes (1856 - 1894)

Lled-Grwpiau Feller wedi'u generadu gan Weithredyddion Differol-Ffug

Bydd y ddarlith yn dechrau gyda chyflwyniad byr i weithredyddion differol-ffug a lled-grwpiau Feller. Mae gweithredyddion differol-ffug wedi cael eu hastudio dros gyfnod hir a chyflwynir rhai o'r prif syniadau yn y ddarlith. Yn ddiweddar, rydym wedi ystyried gweithredyddion differol-ffug ar Zm ac Rn × Zm - ystyrir y rhesymau y tu ôl hyn a'r syniadau cyffredinol. Mae'r darlith yn seiliedig ar waith ar y cyd gyda Niels Jacob, Chenglin Shen ac Owen Morris.

Bayesian Formulations of Inverse Problems

Many mathematical models involve partial differential equations augmented by side conditions (consisting in general of boundary conditions, initial conditions, and functions modelling the system properties) that can be used to make forecasts. If the resulting problem is well posed, then there are many numerical methods for finding an approximate solution. Of course, constructing numerical solutions of high and demonstrable accuracy is often a great challenge. However, in most cases, there is a greater challenge arising from the absence of sufficient side conditions to find even an approximate solution. Instead, only partial, inconsistent information disturbed by measurement error is given. This is of course, an example of an inverse problem. This talk will review some of the main formulations of such inverse problems from the point of view of Bayesian statistics. The talk will describe the main contenders: strong-constraint, weak-constraint and penalty function formulations. The problem of quantifying how much uncertainty there is in any inference about the unknown side conditions will be discussed. To close, we will speculate about future research that might help resolve some of the outstanding difficulties.

Benchmark Problems for Wave Propagation in Layered Media

Accurate methods for the first-order advection equation, used for example in tracking contaminants in fluids, usually exploit the theory of characteristics. Such methods are described and contrasted with methods that do not make use of characteristics.
Then the second-order wave equation, in the form of a first-order system, is considered. A review of the one-dimensional theory using solutions of various Riemann problems will be provided. In the special case that the medium is layered and has the 'Goupillaud' property, that waves take the same time to travel through each layer, one can derive exact solutions. The extension of this method to two-dimensional problems will then be discussed. In two dimensions it is not apparent that exact solutions can be found, however by exploiting a generalised Goupillaud property, it is possible to calculate approximate solutions of high accuracy, perhaps sufficient to be of benchmark quality. Some two-dimensional simulations of wave propagation in heterogeneous layered systems, using exact one-dimensional solutions and operator splitting, will be presented.

On Cherlin's Conjecture for Finite Binary Permutation Groups

In 2000, motivated by questions in model theory, Cherlin formulated a conjecture about finite BINARY permutation groups. Roughly speaking, a permutation group is binary if, by studying its action on pairs of points, one can deduce full information about the action. Cherlin's conjecture asserts that all such permutation groups are known.
I will describe recent work with Francesca Dalla Volta (Milano-Bicocca), Francis Hunt (South Wales) and Pablo Spiga (Milano-Bicocca) that contribute towards a proof of Cherlin's conjecture. Our work makes use of the Classification of Finite Simple Groups, so I will also discuss this classification a little.

Product-free Sets and Filled Groups

A subset S of a group G is product-free if for all x and y in S, the product xy is not in S. This generalises the notion of sum-free sets of integers, where these sets were first studied. In this talk I'll give an overview of what's known about sum-free and product-free sets in groups and describe some recent work with Chimere Anabanti and Grahame Erskine on the related concept of filled groups.

Symmetry, Art and Mathematics

What makes a design or structure beautiful? Often, the objects we find pleasing have a high level of symmetry. We will start by looking at some examples symmetry in art and design, and discuss some of the mathematics that naturally arises from it. Although mathematics underlies symmetry, artists have not necessarily made conscious use of the mathematics. However, one exception is the artist M.C. Escher, whose interactions with the mathematician Donald Coxeter led to a greater understanding and new work on both sides. We'll finish the talk with a look at this correspondence.

Runoff Processes on Trees

The volume of catchment discharge that reaches a stream via the overland flow path is critical for water quality prediction, because it is via this pathway that most particulate pollutants are generated and transported to the stream channel, via surface erosion processes. When it rains, spatial variation in the soil infiltration rate leads to the formation and reabsorption of rivulets on the surface, and local topography determines the coalescence of rivulets.
We consider the question of how coalescence facilitates overland flow using a highly abstracted version of the problem, in which the drainage pattern is represented by a Galton-Watson tree. We show that as the rate of rainfall increases there is a distinct phase-change: when there is no stream coalescence the critical point occurs when the rainfall rate equals the infiltration rate, but when we allow coalescence the critical point occurs when the rainfall rate is less than the infiltration rate, and increasing the amount of coalescence increases the total expected runoff.

Quantum statistical inference departs from its “classical” counterpart in several aspects, the main difference being the extra step of extracting classical data from a quantum state by measuring the system. Hence an important part of any estimation strategy is the choice of the measurements, and this is guided by the Quantum Fisher Information (QFI) associated with the optimal minimum variance bound. We consider two schemes for estimating dynamical parameters in the context of a small quantum system coupled to a large one. In the first scheme, one only observes the small system, which is also controlled so as to optimise the QFI. As as a specific example we consider a Heisenberg spin chain, where information transfer plays a role: the idea is to estimate the magnetic field at one end of the chain via the probe spin at the other end. We find, in particular, that the control step is crucial for enhancing the estimation precision. In the second scheme one instead observes the large system, while the small one remains hidden. Natural examples are quantum hidden Markov processes, and we focus on them; in particular, we derive an information geometric structure and prove local asymptotic Normality for the estimation in this setting.

Rigorous Defect Control in Numerical Solution of IVP ODEs

Modern ODE solvers for the IVP x'(t) = f(t,x(t)), x(t_0) = x_0 construct a continuously differentiable approximate solution. Denoting it by u(t), the defect is defined as Δu(t) = u'(t)-f(t,u(t)). Hence u(t) solves exactly u'(t) = f(t,u(t))+Δu(t). Methods based on defect control (advocated by Enright since 1989) estimate the maximum of the defect on each integration step and ensure that it is within a given tolerance.
We propose a method and an implementation of a rigorous defect control, where the defect is provably within the given tolerance over the whole integration interval. To achieve this, we employ interval arithmetic and Taylor models to bound rigorously the defect on each integration step.
Joint work with John Ernsthausen, McMaster University.

Solving a Mechanical System Directly from Its Lagrangian

We outline a systematic way to describe the kinematics and dynamics of two-dimensional linkage mechanisms, in a cartesian coordinate form rather than the often-used approach that takes rotation angles as coordinates. By an example we show how this maps to a Lagrangian formulation of the equations of motion, and report results of numerical solution directly from the Lagrangian. This uses automatic differentiation without any symbolic manipulation, plus the Nedialkov-Pryce DAETS code for differential-algebraic equation initial-value problems.

Gwlad beirdd a chantorion, mathemategwyr o fri

Mae gan Gymru hanes hir ac anrhydeddus o gyfrannu at ddarganfyddiadau ac at fentrau gwyddonol, mathemategol a thechnolegol. Rhyw ganrif yn ôl ‘diflannodd’ yr hanes hwn o gof y genedl, ond mae'r rhod yn dechrau troi.
Cyfeiriad: Gareth Ffowc Roberts, ‘Unioni'r Recorde’, O'r Pedwar Gwynt, Pasg 2017, tt. 12-13.

Star-Shaped and Convex Sets in The Heisenberg Groups

Convex sets play a significant role in analysis and PDEs, in particular for their relation with star-shaped sets. In the Euclidean Rⁿ there are several equivalent definitions for star-shaped sets. When generalizing these notions to more degenerate geometrical structure they turn out to be not equivalent anymore. We investigate that in more general definitions and construct counterexamples to show non-equivalence. We will also give some applications to capacitary problems.

Grushin Periodicity

We introduce a notion of periodicity using translations induced along vector fields in the Grushin space. We will use these translations to a starting ball to construct periodic sets. These periodic sets depend on the centre and radius of the initial ball.

Mathematical and Computational Modelling of Compressible Non-isothermal Viscoelastic Flow

Compressible and non-isothermal effects are often ignored when modelling flows of non-Newtonian fluids. The additional equations governing density and temperature transport increase the complexity of the governing system of nonlinear partial differential equations which adds to the challenge of devising efficient and stable numerical schemes. Taylor-Galerkin pressure correction schemes coupled with Discrete Elastic Viscous Stress Splitting (DEVSS) stabilisation enable accurate solutions to be generated for a wide range of viscosities and relaxation times. The derivation of the governing equations is described and some numerical results on benchmark problems are presented. In particular, the flow between eccentrically rotating cylinders (the journal bearing problem) is considered for a range of relaxation times and the influence of compressibility and viscoelasticity on torque and load bearing capacity is assessed.