# 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 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.

## Previous seminars

### 2017

#### 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 spectral 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:

1. N. Ay, S.I. Amari. A Novel Approach to Canonical Divergences within Information Geometry. Entropy (2015) 17: 8111-8129.

2. N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Information geometry and sufficient statistics. Probability Theory and Related Fields (2015) 162: 327-364.

3. N. Ay, J. Jost, H. V. Le, L. Schwachhöfer. Parametrized measure models. Bernoulli (2016) accepted. arXiv:1510.07305.

4. 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.

5. 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. {\bf 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 {\bf 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.