Applied Mathematics Seminar
University of Leeds
Welcome to the Leeds Applied Mathematics Seminar
Spring 2018
Monday 12 February 2018
Ray Pierrehumbert (University of Oxford)
Atmospheric dynamics of tide-locked exoplanets
A large proportion of the exoplanets discovered so far are in orbits sufficiently close to their host stars that tidal stresses are sufficiently strong that the spin state of planet will become tide-locked to the star on a time scale short compared to the age of the system. This includes all habitable-zone planets orbiting M stars, as well as all lava planets orbiting more massive main-sequence stars. Such planets have a permanent dayside and a permanent nightside, leading to a range of novel atmospheric circulations which have observable consequences in terms of the optical or infrared phase curves of the planets. In this talk, I will discuss the fundamental geophysical fluid dynamics underpinning our understanding of the atmospheric circulation and its ability to transport heat and chemical constituents. The theoretical framework draws on deep analogies with theories used to understand the climate of the Earth’s tropics, but also engages a number of novel phenomena including generation of super-rotating equatorial jets. In addition to having consequences for the interpretation of astronomical observations, these circulations can strongly effect the long term evolution of the atmosphere through their influence on such things as the runaway greenhouse threshold, volatile exchange with the crust, and sequestration of condensible substances on the nightside of the planet.
Monday 19 February 2018
Hadi Susanto (University of Essex)
Homoclinic snaking in discrete systems
‘Snaking’ bifurcations, describing localised solutions existing within a small region of parameter space, are widely observed in in numerous physical applications. In this talk, I will present some of our recent works on the snaking of localised patterns in discrete systems. Particularly I am going to consider three different equations: the discrete Swift-Hohenberg equation (i.e., obtained from discretizing the spatial derivatives of the Swift-Hohenberg equation using central finite differences), coupled discrete nonlinear Schrodinger equations with parity-time symmetric potential, and two-dimensional Allen-Cahn equations. The coupled Schrodinger equations are proposed as a classical model of the parity-time symmetric quantum physics. Our study shows that discrete systems can yield bifurcation diagrams that are different from their continuum counterparts. In particular, we provide analytical methods for the width of the snaking region in the weak and strong coupling region.
Monday 12 March 2018
Martin Lopez-Garcia (University of Leeds)
A unified stochastic modelling framework for the spread of nosocomial infections
Joint work with Theodore Kypraios, University of Nottingham. Over the last years, a number of stochastic models have been proposed for analysing the spread of nosocomial infections in hospital settings. These models often account for a number of factors governing the spread dynamics: spontaneous patient colonization, patient-staff contamination/colonization, environmental contamination, patient cohorting, or health-care workers (HCWs) hand-washing compliance levels. For each model, tailor-designed methods are implemented in order to analyse the dynamics of the nosocomial outbreak, usually by means of studying quantities of interest such as the reproduction number of each agent in the hospital ward, which is usually computed by means of stochastic simulations or deterministic approximations. In this work, we propose a highly versatile stochastic modelling framework that can account for all these factors simultaneously, and analyse the reproduction number of each agent at the hospital ward during a nosocomial outbreak, in an exact and analytical way. By means of five representative case studies, we show how this unified modelling framework comprehends, as particular cases, many of the existing models in the literature. We implement various numerical studies via which we: i) highlight the importance of maintaining high hand-hygiene compliance levels by HCWs, ii) support infection control strategies including to improve environmental cleaning during an outbreak, and iii) show the potential of some HCWs to act as super-spreaders during nosocomial outbreaks.
Monday 16 April 2018
Beth Wingate (University of Exeter)
Time after time: oscillations in fluids and their role in the creation of low frequency dynamics
Working in the framework of fast singular limits (Bogoliubov and Mitropolsky [1961], Klainerman and Majda [1981], Shochet [1994], Embid and Majda [1996] and others) I will discuss the role of dispersive waves on the creation of long-time dynamics for simple geometries in rotating and stratified fluids. In particular, asking the question: what can be learned about finite versus infinite frequencies? I will discuss how this type of thinking impacts time stepping methods, and if there is time I will also discuss generalizations of the method of cancellations of oscillations of Schochet for two distinct fast time scales, i.e. which fast time scale (rotation and gravity) is fastest?
Monday 23 April 2018
Jijun Liu (Southeast University, Nanjing, P. R. China)
On fluorescence imaging: diffusion equation model and non-uniqueness of the inverse problem
Fluorescence imaging is a type of wave spectroscopy that extracts the quantitative property of fluorescence from some measurable data of the sample. This process in the randomly inhomogeneous medium is governed by the radiative transfer equation for excitation and emission fields. By introducing the average of angularly reserved wave energy density, we derive an imaging model by a coupled diffusion system for the average fields. This nonlinear inverse problem is linearized with an error estimate on the excitation field indicating the model approximation. Then we give the explicit expression for emission fields, which provide the fundamentals for the efficient realizations for fluorescence imaging by the iterative schemes. Finally, in terms of the representations of the solution to the diffusion equation, the imaging of fluorophore is implemented by solving a linear integral equation of the first kind. The uniqueness and non-uniqueness of this inverse problem are rigorously analyzed for boundary measurement data, which reveals the essence of the imaging model.
Monday 30 April 2018
Gianne Derks (University of Surrey)
Existence and stability of fronts in inhomogeneous wave equations
Models describing waves in anisotropic media or media with imperfections usually have inhomogeneous terms. Examples of such models can be found in many applications, for example in nonlinear optical waveguides, water waves moving over a bottom with topology, currents in nonuniform Josephson junctions, DNA-RNAP interactions etc. Homogeneous nonlinear wave equations are Hamiltonian partial differential equations with the homogeneity providing an extra symmetry in the form of the spatial translations. Inhomogeneities break the translational symmetry, though the Hamiltonian structure is still present. When the spatial translational symmetry is broken, travelling waves are no longer natural solutions. Instead, the travelling waves tend to interact with the inhomogeneity and get trapped, reflected, or slowed down.
In this talk, wave equations with finite length inhomogeneities will be considered, assuming that the spatial domain can be written as the union of disjoint intervals, such that on each interval the wave equation is homogeneous. The underlying Hamiltonian structure allows for a rich family of stationary front solutions and the values of the energy (Hamiltonian) in each intermediate interval provide natural parameters for the family of orbits. Criteria for the existence of and stability of stationary solutions will be discussed. The results will be illustrated with an example related to a Josephson junction system with a finite length inhomogeneity associated with variations in the Josephson tunneling critical current and an application to DNA-RNAP interactions.