Past Industrial and Applied Mathematics Seminar

28 November 2013
16:00
Abstract
The dynamics of networks of interacting systems depend intricately on the interaction topology. Dynamical implications of local topological properties such as the nodes' degrees and global topological properties such as the degree distribution have intensively been studied. Mesoscale properties, by contrast, have only recently come into the sharp focus of network science but have rapidly developed into one of the hot topics in the field. Current questions are: can considering a mesoscale structure such as a single subgraph already allow conclusions on dynamical properties of the network as a whole? And: Can we extract implications that are independent of the embedding network? In this talk I will show that certain mesoscale subgraphs have precise and distinct consequences for the system-level dynamics. In particular, they induce characteristic dynamical instabilities that are independent of the structure of the embedding network.
  • Industrial and Applied Mathematics Seminar
21 November 2013
16:00
Neville Fowkes
Abstract
After an MISG there is time to reflect. I will report briefly on the follow up to two problems that we have worked on. Crack Repair: It has been found that thin elastically weak spray on liners stabilise walls and reduce rock blast in mining tunnels. Why? The explanation seems to be that the stress field singularity at a crack tip is strongly altered by a weak elastic filler, so cracks in the walls are less likely to extend. Boundary Tracing: Using known exact solutions to partial differential equations new domains can be constructed along which prescribed boundary conditions are satisfied. Most notably this technique has been used to extract a large class of new exact solutions to the non-linear Laplace Young equation (of importance in capillarity) including domains with corners and rough boundaries. The technique has also been used on Poisson's, Helmholtz, and constant curvature equation examples. The technique is one that may be useful for handling modelling problems with awkward/interesting geometry.
  • Industrial and Applied Mathematics Seminar
14 November 2013
16:00
Abstract
The problem of hydrodynamic turbulence is reformulated as a heat flow problem along a chain of mechanical systems which describe units of fluid of smaller and smaller spatial extent. These units are macroscopic but have few degrees of freedom, and can be studied by the methods of (microscopic) non-equilibrium statistical mechanics. The fluctuations predicted by statistical mechanics correspond to the intermittency observed in turbulent flows. Speci cally, we obtain the formula $$ \zeta_p = \frac{p}{3} - \frac{1}{\ln \kappa} \ln \Gamma \left( \frac{p}{3} +1 \right) $$ for the exponents of the structure functions ($\left\langle \Delta_{r}v \rangle \sim r^{\zeta_p}$). The meaning of the adjustable parameter is that when an eddy of size $r$ has decayed to eddies of size $r/\kappa$ their energies have a thermal distribution. The above formula, with $(ln \kappa)^{-1} = .32 \pm .01$ is in good agreement with experimental data. This lends support to our physical picture of turbulence, a picture which can thus also be used in related problems.
  • Industrial and Applied Mathematics Seminar
7 November 2013
16:00
David Marshall
Abstract
The ocean is populated by an intense geostrophic eddy field with a dominant energy-containing scale on the order of 100 km at midlatitudes. Ocean climate models are unlikely routinely to resolve geostrophic eddies for the foreseeable future and thus development and validation of improved parameterisations is a vital task. Moreover, development and validation of improved eddy parameterizations is an excellent strategy for testing and advancing our understanding of how geostrophic ocean eddies impact the large-scale circulation. A new mathematical framework for parameterising ocean eddy fluxes is developed that is consistent with conservation of energy and momentum while retaining the symmetries of the original eddy fluxes. The framework involves rewriting the residual-mean eddy force, or equivalently the eddy potential vorticity flux, as the divergence of an eddy stress tensor. A norm of this tensor is bounded by the eddy energy, allowing the components of the stress tensor to be rewritten in terms of the eddy energy and non-dimensional parameters describing the mean "shape" of the eddies. If a prognostic equation is solved for the eddy energy, the remaining unknowns are non-dimensional and bounded in magnitude by unity. Moreover, these non-dimensional geometric parameters have strong connections with classical stability theory. For example, it is shown that the new framework preserves the functional form of the Eady growth rate for linear instability, as well as an analogue of Arnold's first stability theorem. Future work to develop a full parameterisation of ocean eddies will be discussed.
  • Industrial and Applied Mathematics Seminar
31 October 2013
16:00
George Haller
Abstract
We discuss a simple variational principle for coherent material vortices in two-dimensional turbulence. Vortex boundaries are sought as closed stationary curves of the averaged Lagrangian strain. We find that solutions to this problem are mathematically equivalent to photon spheres around black holes in cosmology. The fluidic photon spheres satisfy explicit differential equations whose outermost limit cycles are optimal Lagrangian vortex boundaries. As an application, we uncover super-coherent material eddies in the South Atlantic, which yield specific Lagrangian transport estimates for Agulhas rings. We also describe briefly coherent Lagrangian vortex detection to three-dimensional flows.
  • Industrial and Applied Mathematics Seminar
24 October 2013
16:00
Carl Dettman
Abstract
We consider a random geometric graph model relevant to wireless mesh networks. Nodes are placed uniformly in a domain, and pairwise connections are made independently with probability a specified function of the distance between the pair of nodes, and in a more general anisotropic model, their orientations. The probability that the network is (k-)connected is estimated as a function of density using a cluster expansion approach. This leads to an understanding of the crucial roles of local boundary effects and of the tail of the pairwise connection function, in contrast to lower density percolation phenomena.
  • Industrial and Applied Mathematics Seminar
17 October 2013
16:00
Stephen Coombes
Abstract
Neural field models describe the coarse-grained activity of populations of interacting neurons. Because of the laminar structure of real cortical tissue they are often studied in two spatial dimensions, where they are well known to generate rich patterns of spatiotemporal activity. Such patterns have been interpreted in a variety of contexts ranging from the understanding of visual hallucinations to the generation of electroencephalographic signals. Typical patterns include localised solutions in the form of travelling spots, as well as intricate labyrinthine structures. These patterns are naturally defined by the interface between low and high states of neural activity. Here we derive the equations of motion for such interfaces and show, for a Heaviside firing rate, that the normal velocity of an interface is given in terms of a non-local Biot-Savart type interaction over the boundaries of the high activity regions. This exact, but dimensionally reduced, system of equations is solved numerically and shown to be in excellent agreement with the full nonlinear integral equation defining the neural field. We develop a linear stability analysis for the interface dynamics that allows us to understand the mechanisms of pattern formation that arise from instabilities of spots, rings, stripes and fronts. We further show how to analyse neural field models with linear adaptation currents, and determine the conditions for the dynamic instability of spots that can give rise to breathers and travelling waves. We end with a discussion of amplitude equations for analysing behaviour in the vicinity of a bifurcation point (for smooth firing rates). The condition for a drift instability is derived and a center manifold reduction is used to describe a slowly moving spot in the vicinity of this bifurcation. This analysis is extended to cover the case of two slowly moving spots, and establishes that these will reflect from each other in a head-on collision.
  • Industrial and Applied Mathematics Seminar
30 May 2013
16:00
SangHoon Lee
Abstract
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.
  • Industrial and Applied Mathematics Seminar

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