# Applied Mathematics Seminar

Seminars are held at 2:00 pm on Wednesdays in the Access Grid Room ( Carslaw Building, 8th floor, room 829), unless otherwise noted.

For more information and to be added to the mailing list, please contact Eduardo G. Altmann .

## Upcoming Seminars

** Wednesday ** October 10, 2pm in the AGR room

Dr. Minh-Ngoc Tran (Business School, University of Sydney)

** Title:** Bayesian Deep Net GLM and GLMM

** Abstract: ** Deep feedforward neural networks (DFNNs) are a powerful tool for functional approximation. We describe flexible versions of generalized linear and generalized linear mixed models incorporating basis functions formed by a DFNN. Efficient computational methods for high-dimensional Bayesian inference are developed using Gaussian variational approximation, with a parsimonious but flexible factor parametrization of the covariance matrix. We implement natural gradient methods for the optimization, exploiting the factor structure of the variational covariance matrix in computation of the natural gradient. Our flexible DFNN models and Bayesian inference approach lead to a regression and classification method that has a high prediction accuracy, and is able to quantify the prediction uncertainty in a principled and convenient way. We also describe how to perform variable selection in our deep learning method. The proposed methods are illustrated in a wide range of simulated and real-data examples, and the results compare favourably to a state of the art flexible regression and classification method in the statistical literature, the Bayesian additive regression trees (BART) method. User-friendly software packages in Matlab and R implementing the proposed methods are available at https://github.com/VBayesLab.

## Previous seminars

### 2018

#### Second Semester

** Wednesday ** September 19, 2pm in the AGR room

Dr. Lachlan Smith (University of Sydney)

** Title:** Chaos and the flow capture problem: Polluting is easy, cleaning is hard

** Abstract: ** Where do you place pollutant capture units? When objects move through heterogeneous flow environments, such as oceanic micro-plastics, the answer is not obvious. We formulate flow capture problems, involving flows and sinks, and, using dynamical systems techniques, show that blindly positioning capture units carries high risk of failure. Capture efficiency depends on capture rate: long-term efficiency decreases as the number of capture units increases, whereas short-term efficiency increases. Doubling numbers of capture units can more than double the capture rate. The formal description of flow capture problems will impact engineering solutions ranging from atmospheric CO2 capture to oceanic micro-plastic pollution.

** Wednesday ** August 29, 1pm (ONE HOUR EARLIER THAN USUAL!) in the AGR room

Dr. Robyn Araujo (Queensland Univ. of Tech.)

** Title:** Robust Perfect Adaptation in Complex Bionetworks

** Abstract: ** Robustness, and the ability to function and thrive amid changing and unfavourable environments,
is a fundamental requirement for all living systems. Moreover, it has been a long-standing mystery how the
extraordinarily complex communication networks inside living cells, comprising thousands of different interacting
molecules, are able to exhibit such remarkable robustness since complexity is generally associated with fragility.
In this talk I will give an overview of our recent research on robustness in cellular signalling networks, with
an emphasis on the robust functionality known as Robust Perfect Adaptation (RPA). This work is now published
in Nature Communications, and is available here:
https://rdcu.be/M46K. This work has suggested a resolution
to the complexity-robustness paradox through the discovery that robust adaptive signalling networks must be
decomposable into topological basis modules of just two possible types. This newly-discovered modularisation
of complex bionetworks has important implications for evolutionary biology, embryology and development, cancer
research and drug development.

** Wednesday ** September 5, 2pm in the AGR room

Dr. Justin Tzou (Macquarie University)

** Title:** Stability analysis of localised patterns in two and three spatial dimensions

** Abstract: ** We present a matched asymptotics framework for constructing and analysing the stability of localised patterns that arise in singularly perturbed activator-inhibitor reaction-diffusion systems. In two spatial dimensions, by way of analyses of nonlocal eigenvalue problems, we resolve two long-standing problems regarding 1) the stability of spot patterns to oscillatory instabilities, and 2) the stability of stripe patterns to break-up instabilities, the latter motivated by the persistence of striped vegetation patterns on steep hillsides. In three spatial dimensions, we calculate explicit stability thresholds for self-replication and annihilation of spots, and derive a gradient flow that governs their slow dynamics. Joint work with Theodore Kolokolnikov, Michael J. Ward, and Shuangquan Xie.

#### First semester

** Monday (!) ** March 26, 2pm in the AGR room

Prof. Gunther Uhlmann (University of Washington)

** Title:** Journey to the Center of the Earth

** Abstract: ** We will consider the inverse problem of determining the sound
speed or index of refraction of a medium by measuring the travel times of
waves going through the medium. This problem arises in global seismology
in an attempt to determine the inner structure of the Earth by measuring
travel times of earthquakes. It has also several applications in optics
and medical imaging among others.

The problem can be recast as a geometric problem: Can one determine the Riemannian metric of a Riemannian manifold with boundary by measuring the distance function between boundary points? This is the boundary rigidity problem. We will also consider the problem of determining the metric from the scattering relation, the so-called lens rigidity problem. The linearization of these problems involve the integration of a tensor along geodesics, similar to the X-ray transform.

We will also describe some recent results, join with Plamen Stefanov and Andras Vasy, on the partial data case, where you are making measurements on a subset of the boundary. No previous knowledge of Riemannian geometry will be assumed.

Wednesday February 21, 2pm in the AGR room

Prof. Herbert Huppert (University of Cambridge)

** Title:** How to frack into and out of trouble.

** Abstract: ** After a short introduction to the mechanism and politics of fracking, the talk will
concentrate on the fluid mechanics and elastodynamics of driving fluid into cracks and the quite different
response when the pressure is released and the fluid flows back out. Development of the governing equations
will be presented along with their numerical solution and asymptotic analysis in certain useful limits.
Videos of laboratory experiments will be shown and the results compared with the theoretical predictions.

Wednesday March 7, 2pm in the AGR room

Prof. Martin Wechselberger (Applied Maths, University of Sydney)

** Title:** Two-stroke relaxation oscillators

** Abstract: ** In classic van der Pol-type oscillator theory, a relaxation cycle consists of two
slow and two fast orbit segments per period (slow-fast-slow-fast). A possible alternative relaxation
oscillator type consists of one slow and one fast segment only. In electrical circuit theory, Le Corbeiller
(published in IEEE 1960) termed this type a two-stroke oscillator (compared to the four-stroke vdP oscillator).
I will provide examples of two-stroke relaxation oscillators and discuss these problems from a geometric
singular perturbation theory point of view "beyond the standard form". It is worth mentioning that
Fenichel's seminal work on geometric singular perturbation theory (published in JDE 1979) discusses this
more general setting, but it has not received much attention in the literature.

Wednesday March 14, 2pm in the AGR room

Prof. Dmitry Pelinovsky (McMaster University, Canada)

** Title: ** Rogue periodic waves in the focusing MKDV and NLS equations

** Abstract:** Rogue periodic waves stand for gigantic waves on a periodic background. The nonlinear
Schrodinger equation in the focusing case admits two families of periodic wave solutions
expressed by the Jacobian elliptic functions dn and cn. Both periodic waves are
modulationally unstable with respect to long-wave perturbations. Exact solutions for the
rogue periodic waves are constructed by using the explicit expressions
for the periodic eigenfunctions of the Zakharov–Shabat spectral problem and
the Darboux transformations. These exact solutions generalize the classical rogue wave
(the so-called Peregrine’s breather). Computations of rogue periodic waves rely on
properties of the nonlinear Schrodinger equation due to its integrability.

### 2017

#### Second semester

Monday December 11, 2pm in the AGR room

Dr. Yulia Peet (Arizona State University)

** Title:** Overlapping and Moving Grid Approaches with Spectral-Element Methods: Concepts and Applications

** Abstract: ** In this talk, we present our recent development of overlapping and moving grid methodology
for high-fidelity computations of fluid flow problems with spectral element methods. Spectral element methods
belong to a class of high-order methods that combine exponential convergence of global spectral methods with
geometrical flexibility of finite-element methods. High-order methods possess low dissipation and low
dispersion errors and are well suited for high-accuracy simulations of turbulent flows. The current development
of overlapping and moving grid approaches enables the application of spectral element methods to a larger
class of problems that involve moving bodies and complex geometries. In this talk, the fundamental concepts
of both the spectral element methodology and the overlapping grid approach will be discussed, followed by a
description and analysis of the methods that we have developed, paying a special attention to the concepts
of stability and accuracy of the proposed methodology. We will proceed by discussing an application of the
developed method to Direct Numerical Simulations of airfoil dynamic stall in the presence of upstream
disturbances. We conclude by showing further potential extensions of the current methodology and list new
applications that can be successfully tackled with this method.

Wednesday July 19

Prof. Boris Khesin (Department of Mathematics, University of Toronto, Canada)

** Title:** Hamiltonian dynamics of vortex membranes

** Abstract: ** We show that an approximation of the hydrodynamical Euler equation
describes the skew-mean-curvature flow on vortex membranes in any
dimension. This generalizes the classical binormal, or vortex filament,
equation in 3D. We present a Hamiltonian framework for dynamics of
higher-dimensional vortex filaments and vortex sheets as singular
2-forms (Green currents) with support of codimensions 2 and 1,
respectively.

Wednesday July 26

Dr. Marianito Rodrigo (School of Mathematics and Applied Statistics University of Wollongong)

** Title:** On a fractional matrix exponential and an explicit method for its calculation

** Abstract: **The matrix exponential arises in many applications, particularly in the solution
of linear systems of ordinary differential equations. The nth derivative of the matrix exponential is equal
to the nth power of the matrix multiplied by the matrix exponential. What is the analogue of this when
the ordinary derivative is replaced by a fractional derivative? In this talk I will define a fractional
matrix exponential and then give an explicit method for calculating the fractional matrixexponential. An
overview of the fractional calculus will be given.

Wednesday August 2

A/Prof Zhi-An Wang (Department of Applied Mathematics, Hong Kong Polytechnic University, Hong Kong)

** Title:** Boundary layers arising from chemotaxis models

** Abstract:** The original well-known Keller-Segel system describing the chemotactic wave propagation
remains poorly understood in many aspects due to the logarithmic singularity. As the chemical assumption rate
is linear, the singular Keller-Segel model can be converted, via a Cole-Hopf type transformation, into a
system of viscous conservation laws without singularity. In this talk, we first consider the dynamics of
the transformed Keller-Segel system in a bounded interval with time-dependent Dirichlet boundary conditions.
By imposing some conditions on the boundary data, we show that boundary layer profiles are present as
chemical diffusion tends to zero and large-time profile of solutions will be determined by the boundary
data (i.e. boundary stabilization). We employ the refined (weighted) energy estimates with the "effective viscous
flux" technique to show the emergence of boundary layer profiles. For asymptotic dynamics of solutions, we
develop a new idea by exploring the convexity of an entropy expansion to get the basic $L^1$-estimate, on
which our results are built up by the method of energy estimates. Finally we gain the results for the original
singular Keller-Segel system by reversing the Cole-Hopf transformation. Numerical simulations are performed
to interpret our analytical results and their implications.

Wednesday August 9

Prof Kenji Kajiwara (Institute of Mathematics for Industry, Kyushu University, Japan)

** Title:** Construction and simulation of discrete integrable model for soil water infiltration problem

** Abstract: ** In this talk, we propose an integrable model and its discretization describing
one-dimensional soil water infiltration problem. The model is formulated as the nonlinear boundary value
problem for a nonlinear diffusion-convection equation, which is transformed
to the Burgers equation by a certain independent variable transformation incorporating the dependent
variable, called the hodograph transformation or the reciprocal transformation. We construct
the discrete model preserving the underlying integrability nature and formulate it as the
self-adaptive moving mesh scheme. If we require the numerical stability and high-precision
coincidence with the special case where the exact solution is obtained, we need some investigation
and modification of the discrete model from the point of view different from integrability.
We discuss this point and show some numerical results.

This talk is based on the paper arXiv:1705.03129 by D. Triadis (Kyushu/La Trobe) , P. Broadbridge (La Trobe), K. Maruno (Waseda) and myself.

Wednesday September 6

Dr. Sophie Calabretto (Department of Mathematics,Macquarie University)

** Title:** Flow external to a rotating torus (or a sphere)

** Abstract: ** The unsteady flow generated due to the impulsive motion of a torus or sphere is a
paradigm for the study of many temporally developing boundary layers. The boundary layer is known to
exhibit a finite-time singularity at the equator. We present results of a study that focuses upon the
behaviour of the flow after the onset of this singularity. Our computational results demonstrate that the
singularity in the boundary layer manifests as the ejection of a radial jet. This radial jet is preceded
by a toroidal starting vortex pair, which detaches and propagates away from the sphere. The radial jet
subsequently develops an absolute instability, which propagates upstream towards the sphere surface.

Wednesday September 13

Dr. Ananta K. Majee (Mathematisches Institut, University of Tuebingen, Germany)

** Title:** On stochastic optimal control in ferromagnetism

** Abstract: ** In this presentation, we study an optimal control problem for the stochastic
Landau-Lifshitz-Gilbert equation on a bounded domain in R^d (d = 1, 2, 3). We first establish existence
of a relaxed optimal control for relaxed version of the problem. As the control acts in the equation
linearly, we then establish existence of an optimal control for the underlying problem. Furthermore,
convergence of a structure presrving finite element approximation for d = 1 and physically relevant
computational studies will be discussed.

Wednesday October 4

Dr. Lewis Mitchell (School of Mathematical Sciences, University of Adelaide)

** Title:** Information flows in online social networks

** Abstract: ** The flow of information online is a significant factor in social contagion, rumour
and “fake news” propagation, and protest organisation. Further, online social platforms provide
a unique opportunity for computational social scientists to observe individuals’ spontaneous
interactions over social ties, often through structural or temporal proxies for information. However,
such approaches do not leverage the full extent of information available, namely the time-ordered textual
content of messages. Here we apply information-theoretic techniques to social media data to identify the
extent to which predictive information is encoded in social ties, and that in principle one can profile
an individual from their contacts even if the individual is ``hidden'' within the network. Analysis of
Twitter users shows that 95% of the potential predictive accuracy attainable for an individual is embedded
within their social ties, and numerical simulations on a paradigmatic model of information flow shows that
these techniques are robust.

Wednesday October 11

Dr. David A. Smith (Science (Mathematics), Yale-NUS, Singapore)

** Title:** Nonlocal Problems for Linear Evolution Equations

** Abstract: ** Linear evolution equations, such as the heat and linearized KdV equations, are commonly
studied on finite spatial domains via initial-boundary value problems. Typically, the boundary conditions
specify information about the solution and its derivatives at the edges of the spatial domain. Alternatively,
in place of the boundary conditions, consider "multipoint conditions", where one specifies some linear
combination of the solution and its derivatives evaluated at internal points of the spatial domain. A further
generalization is the "nonlocal" specification of the integral over space of the solution against some
continuous weight. We describe a general framework for studying such problems, and provide solution
representations for 2nd and 3rd order examples.

#### First semester

Wednesday January 25

Dr. Paul Griffiths (Oxford Brookes University, UK)

** Title:** Shear-thinning: A stabilising effect? Yes, no, maybe?

** Abstract: ** In this talk we will investigate how viscosity effects the stability of a fluid flow. By
assuming a shear-thinning viscosity relationship, where an increase in shear-rate results in a decrease in
fluid viscosity, we show that flows can be both stabilised or destabilised, depending on (i) the fluid model
in question and (ii) the ‘amount’ of shear-thinning the fluid exhibits. Using a two-dimensional
boundary-layer flow as our ‘toy model’ we are able to show equivalence between different
shear-thinning models. The effect shear-thinning has on important parameters such as the critical Reynolds
number, and the maximum frequency of the disturbances will be discussed and interpreted in the wider context.

Wednesday February 22

Dr. Maria Vlassiou (Eindhoven University of Technology, Netherlands)

** Title:** Heavy-traffic limits for layered queueing networks

** Abstract: ** Heavy-traffic limits for queueing networks are a topic of continuing interest. Presently,
the class of networks for which these limits have been rigorously derived is restricted. An important
ingredient in such work is the demonstration of state space collapse (SSC), which loosely speaking shows
that in diffusion scale the queuing process for the stochastic model can be approximately recovered as a
continuous lifting of the workload process. This often results in a reduction of the dimensions of the
original system in the limit, leading to improved tractability. In this talk, we discuss diffusion
approximations of layered queuing networks through two examples.

In the first example, we establish a heavy-traffic limit through SSC for a computer network model. For this model, SSC is related to an intriguing separation of time scales in heavy traffic. The main source of randomness occurs at the top layer; the interactions at the other layer are shown to converge to a fixed point at a faster time scale.

The second example focuses on a network of parallel single-server queues, where the speeds of the servers are varying over time and governed by a single continuous-time Markov chain. We obtain heavy-traffic limits for the distributions of the joint workload, waiting-time and queue length processes. We do so by using a functional central limit theorem approach, which requires the interchange of steady-state and heavy-traffic limits. For this model, we show that the SSC property does not hold.

Wednesday March 1

Dr. Daniel Lecoanet (Princeton University, Princeton, USA)

** Title:** Measuring Core Stellar Magnetic Field using Wave Conversion

** Abstract: **
By studying oscillation modes at the surface of stars, astrophysicists are able to infer characteristics of
their deep interior structure. This was first applied to observations of the Sun, but recently space-based
telescopes have measured oscillations in many other stars, leading to many new mysteries in stellar structure
and evolution. Recent work has suggested that low dipole oscillation amplitudes in evolved red giant branch
stars may indicate strong core magnetic fields. Here we present both numerical simulations and analytic
calculations of the interactions of waves with a strong magnetic field. We can solve the problem very accurately
by using the WKB approximation to reduce the 2D PDE into a series of ODEs for different heights. We find that
magnetic fields convert the buoyancy-driven waves observable at the surface of the star to magnetic waves, which
are not present at the surface, in agreement with observations.

Wednesday March 22

Sheehan Olver (School of Mathematics and Statistics, University of Sydney)

** Title:** Solving PDEs on triangles using multivariate orthogonal polynomials

** Abstract: ** Univariate orthogonal polynomials have long history in applied and computational mathematics,
playing a fundamental role in quadrature, spectral theory and solving differential equations with spectra
methods. Unfortunately, while numerous theoretical results concerning multivariate orthogonal polynomials
exist, they have an unfair reputation of being unwieldy on non-tensor product domains. In reality, many of
the powerful computational aspects of univariate orthogonal polynomials translate naturally to multivariate
orthogonal polynomials, including the existence of Jacobi operators and the ability to construct sparse partial
differential operators, a la the ultrapsherical spectral method [Olver & Townsend 2012]. We demonstrate
these computational aspects using multivariate orthogonal polynomials on a triangle, including the fast
solution of general partial differential equations.

Wednesday April 5

Professor Shige Peng (Shandong University, Jinan, China)

** Title:** Backward Stochastic Differential Equations Driven by G-Brownian Motion in Finance

** Abstract: ** We present some recent developments in the theory of Backward Stochastic Differential
Equations (BSDEs) driven by a new type of a Brownian motion under a nonlinear expectation space and we
discuss applications of this new class of BSDEs to financial models in which
the uncertainty of volatility is taken into account.

Wednesday April 12

Professor Holger Dullin (School of Mathematics, University of Sydney)

** Title:** A new twisting somersault - 513XD

** Abstract: ** Abstract: Modelling an athlete as a system of coupled rigid body we derive a time-dependent reduced Euler
equation for the dynamics of shape changing bodies. Reconstruction allows to recover the full dynamics
in SO(3), and the number of somersaults is decomposed into a geometric phase and a dynamics phase.
A kick model is used to approximate the dynamics, and using the insight gained from this we propose
a new 10 meter platform twisting somersault dive (FINA code 513XD) that incorporates 5 full twists.

Wednesday April 19 ** Different Location! Carslaw room 535 **

Prof. Nihat Ay (Max-Planck-Institute for the Mathematics in the Sciences, Leipzig, Germany)

** Title:**Information Geometry and its Application to Complexity Theory

** Abstract: ** In the first part of my talk, I will review information-geometric structures and
highlight the important role of divergences. I will present a novel approach to canonical divergences
which extends the classical definition and recovers, in particular, the well-known Kullback-Leibler
divergence and its relation to the Fisher-Rao metric and the Amari-Chentsov tensor.

Divergences also play an important role within a geometric approach to complexity. This approach is based on the general understanding that the complexity of a system can be quantified as the extent to which it is more than the sum of its parts. In the second part of my talk, I will motivate this approach and review corresponding work.

References:

- N. Ay, S.I. Amari. A Novel Approach to Canonical Divergences within Information Geometry. Entropy (2015) 17: 8111-8129.
- N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Information geometry and sufficient statistics. Probability Theory and Related Fields (2015) 162: 327-364.
- N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Parametrized measure models. Bernoulli (2016) accepted. arXiv:1510.07305.
- N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Information geometry. Ergebnisse der Mathematik und Ihrer Grenzgebiete/A Series of Modern Surveys in Mathematics, Springer 2017, forthcoming book.
- N. Ay. Information Geometry on Complexity and Stochastic Interaction. Entropy (2015) 17(4): 2432-2458.

Wednesday April 26

Professor Robert Dewar (Research School of Physics & Eng., Australian National Univ., Canberra)

** Title:**Variational constructions of almost-invariant tori for 1 1/2-D Hamiltonian systems

** Abstract: ** Action-angle variables are normally defined only for integrable systems, but in order to
describe 3D magnetic field systems a generalization of this concept was proposed recently [1,2] that
unified the concepts of ghost surfaces and quadratic-flux-minimizing (QFMin) surfaces (two strategies for
minimizing action gradient). This was based on a simple canonical transformation generated by a change of
variable, $\theta = \theta(\Theta ,\zeta)$, where $\theta$ and $\zeta$ (a time-like variable) are poloidal
and toroidal angles, respectively, with $\Theta$ a new poloidal angle chosen to give pseudo-orbits that are
(a) straight when plotted in the $\zeta,\Theta$ plane and (b) QFMin pseudo-orbits in the transformed
coordinate. These two requirements ensure that the pseudo-orbits are also (c) ghost pseudo-orbits, but they
do not uniquely specify the transformation owing to a relabelling symmetry. Variational methods of solution
that remove this lack of uniqueness are discussed.

[1] R.L. Dewar and S.R. Hudson and A.M. Gibson, Commun. Nonlinear Sci. Numer. Simulat.

**17**, 2062 (2012) http://dx.doi.org/10.1016/j.cnsns.2011.04.022

[2] R.L. Dewar and S.R. Hudson and A.M. Gibson, Plasma Phys. Control. Fusion

**55**, 014004 (2013) http://dx.doi.org/10.1088/0741-3335/55/1/014004

Wednesday May 3

Prof. Michael Small (The University of Western Australia)

** Title:** Communities Within Networks

** Abstract: ** Many complex systems are naturally represented as networks which lack an underlying geodesic
space. That is, elements of the network are naturally represented by their interconnection and not by their
position in any real space. A favourite problem in complex systems is then how best to infer sensible communities
from the network adjacency matrix. To be able to better frame this question, we first need to more precisely
say something about what we mean by "sensible" communities. The usual way to do this is to define a statistical
measure that quantifies the relative number of inter- to intra- community links - which we call "modularity".
With this in mind, there are several methods one can apply to choose suitable sets of communities which achieve
local optimality of this measure. I will describe some standard methods and some of our own approaches to this
problem. Most recently we have developed methods that embed the network in a suitable geodesic space and then
borrow ideas from computational clustering algorithms to detect communities (joint work with Arif Mahmood,
formerly of UWA now with Qatar University). If I get time, I hope to finish by spending a few minutes talking
about generative algorithms for networks with communities - the problem here is that while we have algorithms
to generate networks with specific "nice" properties (preferential attachment, for example), and we have
algorithms to generate communities, the algorithms to generate "nice" networks with communities are rather clunky.

Wednesday May 17

Dr. Milena Radnovic (The University of Sydney)

** Title:**Geometry, billiards, integrability.

** Abstract: **Starting from the celebrated Poncelet porism, we will present classical and modern results
concerning integrable billiards.

Wednesday May 24

Dr. David Galloway (The University of Sydney)

** Title:**Slow-burning instabilities of Dufort-Frankel finite differencing.

** Abstract: **Du Fort-Frankel is a tactic to stabilise Richardson's unstable 3-level leapfrog time-stepping
scheme. By including the next time level in the right hand side evaluation, it is implicit, but it can be
re-arranged to give an explicit updating formula, thus apparently giving the best of both worlds. Textbooks
prove unconditional stability for the heat equation, and extensive use on a variety of advection-diffusion
equations has produced many useful results. Nonetheless, for some problems the scheme can fail in an interesting
and surprising way, leading to instability at very long times. An analysis for a simple teaching problem
involving a pair of evolution equations that describe the spread of a rabies epidemic gives insight into how
this occurs. An even simpler modified diffusion equation suffers from the same instability. Attempts to fix the
rabies problem by additional averaging are described. One method works for a limited parameter range but beyond
that, instability can take a very long time to appear and its analysis displays interesting subtleties.
This is joint work with David Ivers.