# Past Differential Equations and Applications Seminar

2 December 2010
16:00
Simon Cotter
Abstract
When modeling biochemical reactions within cells, it is vitally important to take into account the effect of intrinsic noise in the system, due to the small copy numbers of some of the chemical species. Deterministic systems can give vastly different types of behaviour for the same parameter sets of reaction rates as their stochastic analogues, giving us an incorrect view of the bifurcation diagram. Stochastic Simulation Algorithms (SSAs) exist which draw exact trajectories from the Chemical Master Equation (CME). However, these methods can be very computationally expensive, particularly where there is a separation of time scales of the evolution of some of the chemical species. Some of the species may react many times on a time scale for which others are highly unlikely to react at all. Simulating all of these reactions of the fast species is a waste of computational effort, and many different methods exist for reducing the system to one which only contains the slow variables. In this talk we will introduce the conditional Gillespie algorithm, a method for sampling directly from the conditional distribution on the fast variables, given a static value for the slow variables. Using this, we will go on to describe the constrained Gillespie approach, which uses simulations of the CG algorithm to estimate the drift and diffusion terms of the effective dynamics of the slow variables. If there is time at the end, I will briefly describe my work on another project, which involves full sampling of the posterior distributions in various problems in data assimilation using Monte Carlo Markov Chain (MCMC) methods.
• Differential Equations and Applications Seminar
25 November 2010
16:00
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17:30
Andrey Gorbach
Abstract
In my talk I will introduce the concept of spectral discrete solitons (SDSs): solutions of nonlinear Schroedinger type equations, which are localized on a regular grid in frequency space. In time domain such solitons correspond to periodic trains of pulses. SDSs play important role in cascaded four-wave-mixing processes (frequency comb generation) in optical fibres, where initial excitation by a two-frequency pump leads to the generation of multiple side-bands. When free space diffraction is taken into consideration, a non-trivial generalization of 1D SDSs will be discussed, in which every individual harmonic is an optical vortex with its own topological charge. Such excitations correspond to spatio-temporal helical beams.
• Differential Equations and Applications Seminar
18 November 2010
16:00
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17:30
José Antonio Carrillo de la Plata
Abstract
A kinetic theory for swarming systems of interacting individuals will be described with and without noise. Starting from the the particle model \cite{DCBC}, one can construct solutions to a kinetic equation for the single particle probability distribution function using distances between measures \cite{dobru}. Analogously, we will discuss the mean-field limit for these problems with noise. We will also present and analys the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. It will be shown that the solutions concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.
• Differential Equations and Applications Seminar
11 November 2010
17:00
Professor Gui-Qiang G. Chen
Abstract
The Mathematical Institute invites you to attend the Inaugural Lecture of Professor Gui-Qiang G. Chen. Professor in the Analysis of Partial Differential Equations. Examination Schools, 75-81 High Street, Oxford, OX 4BG. There is no charge to attend but registration is required. Please register your attendance by sending an email to events@maths.ox.ac.uk specifying the number of people in your party. Admission will only be allowed with prior registration. --------------------------------------------------------------------------------------------------------------------------------------------------------------------- ABSTRACT While calculus is a mathematical theory concerned with change, differential equations are the mathematician's foremost aid for describing change. In the simplest case, a process depends on one variable alone, for example time. More complex phenomena depend on several variables – perhaps time and, in addition, one, two or three space variables. Such processes require the use of partial differential equations. The behaviour of every material object in nature, with timescales ranging from picoseconds to millennia and length scales ranging from sub-atomic to astronomical, can be modelled by nonlinear partial differential equations or by equations with similar features. The roles of partial differential equations within mathematics and in the other sciences become increasingly significant. The mathematical theory of partial differential equations has a long history. In the recent decades, the subject has experienced a vigorous growth, and research is marching on at a brisk pace. In this lecture, Professor Gui-Qiang G. Chen will present several examples to illustrate the origins, developments, and roles of partial differential equations in our changing world.
• Differential Equations and Applications Seminar
4 November 2010
16:00
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17:30
Abstract
The presence of additives, which may or may not be surface-active, can have a dramatic influence on interfacial flows. The presence of surfactants alters the interfacial tension and drives Marangoni flow that leads to fingering instabilities in drops spreading on ultra-thin films. Surfactants also play a major role in coating flows, foam drainage, jet breakup and may be responsible for the so-called super-spreading" of drops on hydrophobic substrates. The addition of surface-inactive nano-particles to thin films and drops also influences the interfacial dynamics and has recently been shown to accelerate spreading and to modify the boiling characteristics of nanofluids. These findings have been attributed to the structural component of the disjoining pressure resulting from the ordered layering of nanoparticles in the region near the contact line. In this talk, we present a collection of results which demonstrate that the above-mentioned effects of surfactants and nano-particles can be captured using long-wave models.
• Differential Equations and Applications Seminar
28 October 2010
16:00
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17:30
Rosemary Dyson
Abstract
Many growing plant cells undergo rapid axial elongation with negligible radial expansion. Growth is driven by high internal turgor pressure causing viscous stretching of the cell wall, with embedded cellulose microfibrils providing the wall with strongly anisotropic properties. We present a theoretical model of a growing cell, representing the primary cell wall as a thin axisymmetric fibre-reinforced viscous sheet supported between rigid end plates. Asymptotic reduction of the governing equations, under simple sets of assumptions about the fibre and wall properties, yields variants of the traditional Lockhart equation, which relates the axial cell growth rate to the internal pressure. The model provides insights into the geometric and biomechanical parameters underlying bulk quantities such as wall extensibility and shows how either dynamical changes in wall material properties or passive fibre reorientation may suppress cell elongation. We then investigate how the action of enzymes on the cell wall microstructure can lead to the required dynamic changes in macroscale wall material properties, and thus demonstrate a mechanism by which hormones may regulate plant growth.
• Differential Equations and Applications Seminar
21 October 2010
16:00
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17:30
Abstract
Whenever we say the words "fluid flows" or "shape changes" we enter the realm of infinite-dimensional geometric mechanics. Water, for example, flows. In fact, Euler's equations tell us that water flows a particular way. Namely, it flows to get out of its own way as adroitly as possible. The shape of water changes by smooth invertible maps called diffeos (short for diffeomorphisms). The flow responsible for this optimal change of shape follows the path of shortest length, the geodesic, defined by the metric of kinetic energy. Not just the flow of water, but the optimal morphing of any shape into another follows one of these optimal paths. The lecture will be about the commonalities between fluid dynamics and shape changes and will be discussed in the language most suited to fundamental understanding -- the language of geometric mechanics. A common theme will be the use of momentum maps and geometric control for steering along the optimal paths using emergent singular solutions of the initial value problem for a nonlinear partial differential equation called EPDiff, that governs metamorphosis along the geodesic flow of the diffeos. The main application will be in the registration and comparison of Magnetic Resonance Images for clinical diagnosis and medical procedures.
• Differential Equations and Applications Seminar
14 October 2010
16:30
Julian Hunt
Abstract
The new model is that the universal small scale structure of high Reynolds number turbulence is determined by the dynamics of thin evolving shear layers, with thickness of the order of the Taylor micro scale,within which there are the familiar elongated vortices .Local quasi-linear dynamics shows how the shear layers act as barriers to external eddies and a filter for the transfer of energy to their interiors. The model is consistent with direct numerical simulations by Ishihara and Kaneda analysed in terms of conditional statistics relative to the layers and also with recent 4D measurements of lab turbulence by Wirth and Nickels. The model explains how the transport of energy into the layers leads to the observed inertial range spectrum and to the generation of intense structures, on the scale of the Kolmogorov micro-scale. But the modelling also explains the important discrepancies between data and the Kolmogorov-Richardson cascade concept ,eg larger amplitudes of the smallest scale motions and of the higher moments ,and why the latter are generally less isotropic than lower order moments, eg in thermal convection. Ref JCRHunt , I Eames, P Davidson,J.Westerweel, J Fernando, S Voropayev, M Braza J Hyd Env Res 2010
• Differential Equations and Applications Seminar
8 July 2010
14:30
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17:30
Frank Dehoog
Abstract
Wound coils or rolls accumulate essentially flat strip compactly without folding or cutting and typically, strip is wound and unwound a number of times before its end use. The variety of material that is wound into coils or rolls is very extensive and includes magnetic tape, paper, cellophane, plastics, fabric and metals such as aluminium and steel. Stresses wound into a coil provide its structural integrity via the frictional forces between the wraps. For a coil with inadequate inter-wrap pressure, the wraps may slip or telescope (causing surface scuffing) or the coil may slump and collapse. On the other hand, large internal stresses can cause increased creep and stress relaxation, collapse at the bore, stress wrinkling and rupture of the material in the coil. Given the range of applications, it is not surprising that the literature on calculating stresses in wound coils is large and has a long history, which goes back at least to the wire winding of gun barrels. However the basic approach of the resulting accretion models, where the residual stress is recalculated each time a layer is added, has remained essentially the same. In this talk, we take a radically different approach in analysing the winding stresses in coils. Instead of the traditional method, we seek to deduce a winding policy that will achieve a target distribution of residual stresses within a coil. In this way, optimising the coiling tension profile is much more straight-forward, by * Specifying the residue stresses required to avoid operational problems, tight-bore collapses, and other issues such as scuffing, then * Determining the winding tension profile to produce the required residue stresses.
• Differential Equations and Applications Seminar
17 June 2010
16:30
Leonid Bunimovich
Abstract
The question in the title seems to be neglected in the studies of open dynamical systems. It occurred though that the features of dynamics may play a role comparable to the one played by the size of a hole. For instance, the escape through the smaller hole could be faster than through the larger one. These studies revealed as well a new role of the periodic orbits in the dynamics which could be exactly quantified in some cases. Moreover, this new approach allows to characterize the elements of networks by their dynamical properties (rather than by static ones like centrality, betweenness, etc.)
• Differential Equations and Applications Seminar