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Associate Professor Sanjeeva Balasuriya
Senior Lecturer in Applied Mathematics


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Professor Robert Elliott
Adjunct Professor


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Professor Tony Roberts
Professor of Applied Mathematics


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Courses matching "Dynamical systems"

Chaotic Dynamical Systems

Many complex physical systems can be modelled accurately by a small number of deterministic coupled differential or difference equations. Examples include neuron dynamics, population modelling, chemical reactions, stirring and mixing, particle interactions, forced pendulums, weather modelling, and even dripping taps, to name just a few. Though these models are often simple to write down, their solutions often exhibit very complicated behaviours (which are observed in reality), including chaos, meaning extreme sensitivity to initial conditions. This course will provide an introduction to the theory of continuous and discrete dynamical systems, with a particular emphasis on bifurcations and routes to chaotic behaviour. The first few lectures will set the scene, showing how low-dimensional dynamical systems arise from various approximations to complex physical systems. We will observe that these systems can exhibit exotic changes in behaviour as governing parameters are varied smoothly, including chaotic dynamics. To understand this behaviour we will study in detail the stability and bifurcations of periodic structures in these systems, bringing together results from linear algebra, multivariable calculus, differential equations, topology and group theory.

Topics covered will include: an introduction to flows and maps; phase plane analysis of autonomous flows; periodic orbits and limit cycles; the Poincare-Bendixson theorem; saddle-node, transcritical, pitchfork and Hopf bifurcations; conservative versus dissipative systems; hysteresis; dimensional reduction and Poincare sections; analysis of interval maps; period doubling bifurcations and cascades; chaos; Sharkovskii's theorem; area preserving maps and homoclinic tangles; the Thurston-Nielsen theorem; mapping classes and braid groups (time permitting).

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Modelling and Simulation of Stochastic Systems

The course provides students with the skills to analyse and design systems using modelling and simulation techniques. Case studies will be undertaken involving hands-on use of simulation packages. The application of simulation in areas such as manufacturing, telecommunications and transport will be investigated. At the end of this course, students will be capable of identifying practical situations where simulation modelling can be helpful, reporting to management on how they would undertake such a project, collecting relevant data, building and validating a model, analysing the output and reporting their findings to management. Students complete a project in groups of two or three, write a concise summary of what they have done and report their findings to the class. The project report at the end of this course should be a substantial document that is a record of a student's practical ability in simulation modelling, which can also become part of a portfolio or CV. Topics covered are: Introduction to simulation, hand simulation, introduction to a simulation package, review of basic probabilty theory, introduction to random number generation, generation of random variates, anaylsis of simulation output, variance reduction techniques and basic analytic queeing models.

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Events matching "Dynamical systems"

Learning to Satisfy Actuator and Camera Networks
15:10 Fri 25 May, 2007 :: G08 Mathematics Building University of Adelaide :: Assistant Prof Mark Coates

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Wireless sensor and actuator networks (SANETs) represent an important extension of sensor networks, allowing nodes within the network to make autonomous decisions and perform actions (actuation) in response to sensor measurements and shared information. SANETS combine aspects of sensor networks and multi-robot systems, and the merger gives rise to an array of challenges absent from conventional sensor networks. SANETs are active systems that must use the sensed information to modify the environment in order to elicit a desired response. This involves the development of an actuation strategy, a set of decision rules that specify how the network responds to sensed conditions. In this talk, I will discuss the challenges involved in using distributed algorithms to learn suitable actuation strategies. I will draw connections with the class of learning satisfiability problems, which includes a range of learning tasks involving multiple constraints.
Add one part chaos, one part topology, and stir well...
13:10 Fri 19 Oct, 2007 :: Engineering North 132 :: Dr Matt Finn :: School of Mathematical Sciences

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Stirring and mixing of fluids occurs everywhere, from adding milk to a cup of coffee, right through to industrial-scale chemical blending. So why stir in the first place? Is it possible to do it badly? And how can you make sure you do it effectively? I will attempt to answer these questions using a few thought experiments, some dynamical systems theory and a little topology.
Betti's Reciprocal Theorem for Inclusion and Contact Problems
15:10 Fri 1 Aug, 2008 :: G03 Napier Building University of Adelaide :: Prof. Patrick Selvadurai :: Department of Civil Engineering and Applied Mechanics, McGill University

Enrico Betti (1823-1892) is recognized in the mathematics community for his pioneering contributions to topology. An equally important contribution is his formulation of the reciprocity theorem applicable to elastic bodies that satisfy the classical equations of linear elasticity. Although James Clerk Maxwell (1831-1879) proposed a law of reciprocal displacements and rotations in 1864, the contribution of Betti is acknowledged for its underlying formal mathematical basis and generality. The purpose of this lecture is to illustrate how Betti's reciprocal theorem can be used to full advantage to develop compact analytical results for certain contact and inclusion problems in the classical theory of elasticity. Inclusion problems are encountered in number of areas in applied mechanics ranging from composite materials to geomechanics. In composite materials, the inclusion represents an inhomogeneity that is introduced to increase either the strength or the deformability characteristics of resulting material. In geomechanics, the inclusion represents a constructed material region, such as a ground anchor, that is introduced to provide load transfer from structural systems. Similarly, contact problems have applications to the modelling of the behaviour of indentors used in materials testing to the study of foundations used to distribute loads transmitted from structures. In the study of conventional problems the inclusions and the contact regions are directly loaded and this makes their analysis quite straightforward. When the interaction is induced by loads that are placed exterior to the indentor or inclusion, the direct analysis of the problem becomes inordinately complicated both in terns of formulation of the integral equations and their numerical solution. It is shown by a set of selected examples that the application of Betti's reciprocal theorem leads to the development of exact closed form solutions to what would otherwise be approximate solutions achievable only through the numerical solution of a set of coupled integral equations.
Free surface Stokes flows with surface tension
15:10 Fri 5 Sep, 2008 :: G03 Napier Building University of Adelaide :: Prof. Darren Crowdy :: Imperial College London

In this talk, we will survey a number of different free boundary problems involving slow viscous (Stokes) flows in which surface tension is active on the free boundary. Both steady and unsteady flows will be considered. Motivating applications range from industrial processes such as viscous sintering (where end-products are formed as a result of the surface-tension-driven densification of a compact of smaller particles that are heated in order that they coalesce) to biological phenomena such as understanding how organisms swim (i.e. propel themselves) at low Reynolds numbers. Common to our approach to all these problems will be an analytical/theoretical treatment of model problems via complex variable methods -- techniques well-known at infinite Reynolds numbers but used much less often in the Stokes regime. These model problems can give helpful insights into the behaviour of the true physical systems.
The Mechanics of Nanoscale Devices
15:10 Fri 10 Oct, 2008 :: G03 Napier Building University of Adelaide :: Associate Prof. John Sader :: Department of Mathematics and Statistics, The University of Melbourne

Nanomechanical sensors are often used to measure environmental changes with extreme sensitivity. Controlling the effects of surfaces and fluid dissipation presents significant challenges to achieving the ultimate sensitivity in these devices. In this talk, I will give an overview of theoretical/experimental work we are undertaking to explore the underlying physical processes in these systems. The talk will be general and aimed at introducing some recent developments in the field of nanomechanical sensors.
Symmetry-breaking and the Origin of Species
15:10 Fri 24 Oct, 2008 :: G03 Napier Building University of Adelaide :: Toby Elmhirst :: ARC Centre of Excellence for Coral Reef Studies, James Cook University

The theory of partial differential equations can say much about generic bifurcations from spatially homogeneous steady states, but relatively little about generic bifurcations from unimodal steady states. In many applications, spatially homogeneous steady states correspond to low-energy physical states that are destabilized as energy is fed into the system, and in these cases standard PDE theory can yield some impressive and elegant results. However, for many macroscopic biological systems such results are less useful because low-energy states do not hold the same priviledged position as they do in physical and chemical systems. For example, speciation -- the evolutionary process by which new species are formed -- can be seen as the destabilization of a unimodal density distribution over phenotype space. Given the diversity of species and environments, generic results are clearly needed, but cannot be gained from PDE theory. Indeed, such questions cannot even be adequately formulated in terms of PDEs. In this talk I will introduce 'Pod Systems' which can provide an answer to the question; 'What happens, generically, when a unimodal steady state loses stability?' In the pod system formalization, the answer involves elements of equivariant bifurcation theory and suggests that new species can arise as the result of broken symmetries.
Oceanographic Research at the South Australian Research and Development Institute: opportunities for collaborative research
15:10 Fri 21 Nov, 2008 :: Napier G04 :: Associate Prof John Middleton :: South Australian Research and Development Institute

Increasing threats to S.A.'s fisheries and marine environment have underlined the increasing need for soundly based research into the ocean circulation and ecosystems (phyto/zooplankton) of the shelf and gulfs. With support of Marine Innovation SA, the Oceanography Program has within 2 years, grown to include 6 FTEs and a budget of over $4.8M. The program currently leads two major research projects, both of which involve numerical and applied mathematical modelling of oceanic flow and ecosystems as well as statistical techniques for the analysis of data. The first is the implementation of the Southern Australian Integrated Marine Observing System (SAIMOS) that is providing data to understand the dynamics of shelf boundary currents, monitor for climate change and understand the phyto/zooplankton ecosystems that under-pin SA's wild fisheries and aquaculture. SAIMOS involves the use of ship-based sampling, the deployment of underwater marine moorings, underwater gliders, HF Ocean RADAR, acoustic tracking of tagged fish and Autonomous Underwater vehicles.

The second major project involves measuring and modelling the ocean circulation and biological systems within Spencer Gulf and the impact on prawn larval dispersal and on the sustainability of existing and proposed aquaculture sites. The discussion will focus on opportunities for collaborative research with both faculty and students in this exciting growth area of S.A. science.

Hunting Non-linear Mathematical Butterflies
15:10 Fri 23 Jan, 2009 :: Napier LG29 :: Prof Nalini Joshi :: University of Sydney

The utility of mathematical models relies on their ability to predict the future from a known set of initial states. But there are non-linear systems, like the weather, where future behaviours are unpredictable unless their initial state is known to infinite precision. This is the butterfly effect. I will show how to analyse functions to overcome this problem for the classical Painleve equations, differential equations that provide archetypical non-linear models of modern physics.
Dispersing and settling populations in biology
15:10 Tue 23 Jun, 2009 :: Napier G03 :: Prof Kerry Landman :: University of Melbourne

Partial differential equations are used to model populations (such as cells, animals or molecules) consisting of individuals that undergo two important processes: dispersal and settling. I will describe some general characteristics of these systems, as well as some of our recent projects.
Another proof of Gaboriau-Popa
13:10 Fri 3 Jul, 2009 :: School Board Room :: Prof Greg Hjorth :: University of Melbourne

Gaboriau and Popa showed that a non-abelian free group on finitely many generators has continuum many measure preserving, free, ergodic, actions on standard Borel probability spaces. The original proof used the notion of property (T). I will sketch how this can be replaced by an elementary, and apparently new, dynamical property.
Generalizations of the Stein-Tomas restriction theorem
13:10 Fri 7 Aug, 2009 :: School Board Room :: Prof Andrew Hassell :: Australian National University

The Stein-Tomas restriction theorem says that the Fourier transform of a function in L^p(R^n) restricts to an L^2 function on the unit sphere, for p in some range [1, 2(n+1)/(n+3)]. I will discuss geometric generalizations of this result, by interpreting it as a property of the spectral measure of the Laplace operator on R^n, and then generalizing to the Laplace-Beltrami operator on certain complete Riemannian manifolds. It turns out that dynamical properties of the geodesic flow play a crucial role in determining whether a restriction-type theorem holds for these manifolds.
Is the price really right?
12:10 Thu 22 Oct, 2009 :: Napier 210 :: Mr Sam Cohen :: University of Adelaide

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Making decisions when outcomes are uncertain is a common problem we all face. In this talk I will outline some recent developments on this question from the mathematics of finance-the theory of risk measures and nonlinear expectations. I will also talk about how decisions are currently made in the finance industry, and how some simple mathematics can show where these systems are open to abuse.
Manifold destiny: a talk on water, fire and life
15:10 Fri 6 Nov, 2009 :: MacBeth Lecture Theatre :: Dr Sanjeeva Balasuriya :: University of Adelaide

Manifolds are important entities in dynamical systems, and organise space into regions in which different motions occur. For example, intersections between stable and unstable manifolds in discrete systems result in chaotic motion. This talk will focus on manifolds and their locations in continuous dynamical systems, and in particular on Melnikov's method and its adaptations for determining the effect of perturbations on manifolds. The relevance of such adaptations to a surprising range of applications will be shown, in addition to recent theoretical developments inspired by such problems. The applications addressed in this talk include understanding the motion of fluid near oceanic eddies and currents, optimising mixing in nano-fluidic devices in order to improve reactions, computing the speed of a flame front, and finding the spreading rate of bacterial colonies.
Eigen-analysis of fluid-loaded compliant panels
15:10 Wed 9 Dec, 2009 :: Santos Lecture Theatre :: Prof Tony Lucey :: Curtin University of Technology

This presentation concerns the fluid-structure interaction (FSI) that occurs between a fluid flow and an arbitrarily deforming flexible boundary considered to be a flexible panel or a compliant coating that comprises the wetted surface of a marine vehicle. We develop and deploy an approach that is a hybrid of computational and theoretical techniques. The system studied is two-dimensional and linearised disturbances are assumed. Of particular novelty in the present work is the ability of our methods to extract a full set of fluid-structure eigenmodes for systems that have strong spatial inhomogeneity in the structure of the flexible wall.

We first present the approach and some results of the system in which an ideal, zero-pressure gradient, flow interacts with a flexible plate held at both its ends. We use a combination of boundary-element and finite-difference methods to express the FSI system as a single matrix equation in the interfacial variable. This is then couched in state-space form and standard methods used to extract the system eigenvalues. It is then shown how the incorporation of spatial inhomogeneity in the stiffness of the plate can be either stabilising or destabilising. We also show that adding a further restraint within the streamwise extent of a homogeneous panel can trigger an additional type of hydroelastic instability at low flow speeds. The mechanism for the fluid-to-structure energy transfer that underpins this instability can be explained in terms of the pressure-signal phase relative to that of the wall motion and the effect on this relationship of the added wall restraint.

We then show how the ideal-flow approach can be conceptually extended to include boundary-layer effects. The flow field is now modelled by the continuity equation and the linearised perturbation momentum equation written in velocity-velocity form. The near-wall flow field is spatially discretised into rectangular elements on an Eulerian grid and a variant of the discrete-vortex method is applied. The entire fluid-structure system can again be assembled as a linear system for a single set of unknowns - the flow-field vorticity and the wall displacements - that admits the extraction of eigenvalues. We then show how stability diagrams for the fully-coupled finite flow-structure system can be assembled, in doing so identifying classes of wall-based or fluid-based and spatio-temporal wave behaviour.

Integrable systems: noncommutative versus commutative
14:10 Thu 4 Mar, 2010 :: School Board Room :: Dr Cornelia Schiebold :: Mid Sweden University

After a general introduction to integrable systems, we will explain an approach to their solution theory, which is based on Banach space theory. The main point is first to shift attention to noncommutative integrable systems and then to extract information about the original setting via projection techniques. The resulting solution formulas turn out to be particularly well-suited to the qualitative study of certain solution classes. We will show how one can obtain a complete asymptotic description of the so called multiple pole solutions, a problem that was only treated for special cases before.
Exploratory experimentation and computation
15:10 Fri 16 Apr, 2010 :: Napier LG29 :: Prof Jonathan Borwein :: University of Newcastle

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The mathematical research community is facing a great challenge to re-evaluate the role of proof in light of the growing power of current computer systems, of modern mathematical computing packages, and of the growing capacity to data-mine on the Internet. Add to that the enormous complexity of many modern capstone results such as the Poincare conjecture, Fermat's last theorem, and the Classification of finite simple groups. As the need and prospects for inductive mathematics blossom, the requirement to ensure the role of proof is properly founded remains undiminished. I shall look at the philosophical context with examples and then offer some of five bench-marking examples of the opportunities and challenges we face.
A variance constraining ensemble Kalman filter: how to improve forecast using climatic data of unobserved variables
15:10 Fri 28 May, 2010 :: Santos Lecture Theatre :: A/Prof Georg Gottwald :: The University of Sydney

Data assimilation aims to solve one of the fundamental problems ofnumerical weather prediction - estimating the optimal state of the atmosphere given a numerical model of the dynamics, and sparse, noisy observations of the system. A standard tool in attacking this filtering problem is the Kalman filter.

We consider the problem when only partial observations are available. In particular we consider the situation where the observational space consists of variables which are directly observable with known observational error, and of variables of which only their climatic variance and mean are given. We derive the corresponding Kalman filter in a variational setting.

We analyze the variance constraining Kalman filter (VCKF) filter for a simple linear toy model and determine its range of optimal performance. We explore the variance constraining Kalman filter in an ensemble transform setting for the Lorenz-96 system, and show that incorporating the information on the variance on some un-observable variables can improve the skill and also increase the stability of the data assimilation procedure.

Using methods from dynamical systems theory we then systems where the un-observed variables evolve deterministically but chaotically on a fast time scale.

This is joint work with Lewis Mitchell and Sebastian Reich.

Topological chaos in two and three dimensions
15:10 Fri 18 Jun, 2010 :: Santos Lecture Theatre :: Dr Matt Finn :: School of Mathematical Sciences

Research into two-dimensional laminar fluid mixing has enjoyed a renaissance in the last decade since the realisation that the Thurston–Nielsen theory of surface homeomorphisms can assist in designing efficient "topologically chaotic" batch mixers. In this talk I will survey some tools used in topological fluid kinematics, including braid groups, train-tracks, dynamical systems and topological index formulae. I will then make some speculations about topological chaos in three dimensions.
Meteorological drivers of extreme bushfire events in southern Australia
15:10 Fri 2 Jul, 2010 :: Benham Lecture Theatre :: Prof Graham Mills :: Centre for Australian Weather and Climate Research, Melbourne

Bushfires occur regularly during summer in southern Australia, but only a few of these fires become iconic due to their effects, either in terms of loss of life or economic and social cost. Such events include Black Friday (1939), the Hobart fires (1967), Ash Wednesday (1983), the Canberra bushfires (2003), and most recently Black Saturday in February 2009. In most of these events the weather of the day was statistically extreme in terms of heat, (low) humidity, and wind speed, and in terms of antecedent drought. There are a number of reasons for conducting post-event analyses of the meteorology of these events. One is to identify any meteorological circulation systems or dynamic processes occurring on those days that might not be widely or hitherto recognised, to document these, and to develop new forecast or guidance products. The understanding and prediction of such features can be used in the short term to assist in effective management of fires and the safety of firefighters and in the medium range to assist preparedness for the onset of extreme conditions. The results of such studies can also be applied to simulations of future climates to assess the likely changes in frequency of the most extreme fire weather events, and their documentary records provide a resource that can be used for advanced training purposes. In addition, particularly for events further in the past, revisiting these events using reanalysis data sets and contemporary NWP models can also provide insights unavailable at the time of the events. Over the past few years the Bushfire CRC's Fire Weather and Fire Danger project in CAWCR has studied the mesoscale meteorology of a number of major fire events, including the days of Ash Wednesday 1983, the Dandenong Ranges fire in January 1997, the Canberra fires and the Alpine breakout fires in January 2003, the Lower Eyre Peninsula fires in January 2005 and the Boorabbin fire in December 2007-January 2008. Various aspects of these studies are described below, including the structures of dry cold frontal wind changes, the particular character of the cold fronts associated with the most damaging fires in southeastern Australia, and some aspects of how the vertical temperature and humidity structure of the atmosphere may affect the fire weather at the surface. These studies reveal much about these major events, but also suggest future research directions, and some of these will be discussed.
Explicit numerical simulation of multiphase and confined flows
15:10 Fri 8 Oct, 2010 :: Napier G04 :: Prof Mark Biggs :: University of Adelaide

Simulations in which the system of interest is essentially mimicked are termed explicit numerical simulations (ENS). Direct numerical simulation (DNS) of turbulence is a well known and long-standing example of ENS. Such simulations provide a basis for elucidating fundamentals in a way that is impossible experimentally and formulating and parameterizing engineering models with reduced experimentation. In this presentation, I will first outline the concept of ENS. I will then report a number of ENS-based studies of various multiphase fluid systems and flows in porous media. In the first of these studies, which is concerned with flow of suspensions in porous media accompanied by deposition, ENS is used to demonstrate the significant inadequacies of the classical trajectory models typically used for the study of such problems. In the second study, which is concerned with elucidating the change in binary droplet collision behaviour with Capillary number (Ca) and Reynolds number (Re), a range of collision scenarios are revealed as a function of Ca and Re and it appears that the boundaries between these scenarios in the Ca-Re space are not distinct but, rather, smeared. In the final study, it is shown that ENS an be used to predict ab initio the hydrodynamic properties of single phase flow through porous media from the Darcy to the turbulent regimes.
Comparison of Spectral and Wavelet Estimation of the Dynamic Linear System of a Wade Energy Device
12:10 Mon 2 May, 2011 :: 5.57 Ingkarni Wardli :: Mohd Aftar :: University of Adelaide

Renewable energy has been one of the main issues nowadays. The implications of fossil energy and nuclear energy along with its limited source have triggered researchers and industries to find another source of renewable energy for example hydro energy, wind energy and also wave energy. In this seminar, I will talk about the spectral estimation and wavelet estimation of a linear dynamical system of motion for a heaving buoy wave energy device. The spectral estimates was based on the Fourier transform, while the wavelet estimate was based on the wavelet transform. Comparisons between two spectral estimates with a wavelet estimate of the amplitude response operator(ARO) for the dynamical system of the wave energy device shows that the wavelet estimate ARO is much better for data with and without noise.
Priority queueing systems with random switchover times and generalisations of the Kendall-Takacs equation
16:00 Wed 1 Jun, 2011 :: 7.15 Ingkarni Wardli :: Dr Andrei Bejan :: The University of Cambridge

In this talk I will review existing analytical results for priority queueing systems with Poisson incoming flows, general service times and a single server which needs some (random) time to switch between requests of different priority. Specifically, I will discuss analytical results for the busy period and workload of such systems with a special structure of switchover times. The results related to the busy period can be seen as generalisations of the famous Kendall-Tak\'{a}cs functional equation for $M|G|1$: being formulated in terms of Laplace-Stieltjes transform, they represent systems of functional recurrent equations. I will present a methodology and algorithms of their numerical solution; the efficiency of these algorithms is achieved by acceleration of the numerical procedure of solving the classical Kendall-Tak\'{a}cs equation. At the end I will identify open problems with regard to such systems; these open problems are mainly related to the modelling of switchover times.
AustMS/AMSI Mahler Lecture: Chaos, quantum mechanics and number theory
18:00 Tue 9 Aug, 2011 :: Napier 102 :: Prof Peter Sarnak :: Institute for Advanced Study, Princeton

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The correspondence principle in quantum mechanics is concerned with the relation between a mechanical system and its quantization. When the mechanical system are relatively orderly ("integrable"), then this relation is well understood. However when the system is chaotic much less is understood. The key features already appear and are well illustrated in the simplest systems which we will review. For chaotic systems defined number-theoretically, much more is understood and the basic problems are connected with central questions in number theory. The Mahler lectures are a biennial activity organised by the Australian Mathematical Society with the assistance of the Australian Mathematical Sciences Institute.
Comparing Einstein to Newton via the post-Newtonian expansions
15:10 Fri 19 Aug, 2011 :: 7.15 Ingkarni Wardli :: Dr Todd Oliynyk :: Monash University

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Einstein's general relativity is presently the most accurate theory of gravity. To completely determine the gravitational field, the Einstein field equations must be solved. These equations are extremely complex and outside of a small set of idealized situations, they are impossible to solve directly. However, to make physical predictions or understand physical phenomena, it is often enough to find approximate solutions that are governed by a simpler set of equations. For example, Newtonian gravity approximates general relativity very well in regimes where the typical velocity of the gravitating matter is small compared to the speed of light. Indeed, Newtonian gravity successfully explains much of the behaviour of our solar system and is a simpler theory of gravity. However, for many situations of interest ranging from binary star systems to GPS satellites, the Newtonian approximation is not accurate enough; general relativistic effects must be included. This desire to include relativistic corrections to Newtonian gravity lead to the development of the post-Newtonian expansions.
Laplace's equation on multiply-connected domains
12:10 Mon 29 Aug, 2011 :: 5.57 Ingkarni Wardli :: Mr Hayden Tronnolone :: University of Adelaide

Various physical processes take place on multiply-connected domains (domains with some number of 'holes'), such as the stirring of a fluid with paddles or the extrusion of material from a die. These systems may be described by partial differential equations (PDEs). However, standard numerical methods for solving PDEs are not well-suited to such examples: finite difference methods are difficult to implement on multiply-connected domains, especially when the boundaries are irregular or moving, while finite element methods are computationally expensive. In this talk I will describe a fast and accurate numerical method for solving certain PDEs on two-dimensional multiply-connected domains, considering Laplace's equation as an example. This method takes advantage of complex variable techniques which allow the solution to be found with spectral accuracy provided the boundary data is smooth. Other advantages over traditional numerical methods will also be discussed.
Mixing, dynamics, and probability
15:10 Fri 2 Mar, 2012 :: B.21 Ingkarni Wardli :: A/Prof Gary Froyland :: University of New South Wales

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Many interesting natural phenomena are hard to predict. When modelled as a dynamical system, this unpredictability is often the result of rapid separation of nearby trajectories. Viewing the dynamics as acting on a probability measure, the mixing property states that two measurements (or random variables), evaluated at increasingly separated times, become independent in the time-separation limit. Thus, the later measurement becomes increasingly difficult to predict, given the outcome of the earlier measurement. If this approach to independence occurs exponentially quickly in time, one can profitably use linear operator tools to analyse the dynamics. I will give an overview of these techniques and show how they can be applied to answer mathematical questions, describe observed behaviour in fluid mixing, and analyse models of the ocean and atmosphere.
The entropy of an overlapping dynamical system
15:10 Fri 23 Mar, 2012 :: Napier G03 :: Prof Michael Barnsley :: Australian National University

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The term "overlapping" refers to a certain fairly simple type of piecewise continuous function from the unit interval to itself and also to a fairly simple type of iterated function system (IFS) on the unit interval. A correspondence between these two classes of objects is used to: 1. find a necessary and sufficient condition for a fractal transformation from the attractor of one overlapping IFS to the attractor of another overlapping IFS to be a homeomorphism and 2. find a formula for the topological entropy of the dynamical system associated with an overlapping function. These results suggest a new method for analysing clocks, weather systems and prime numbers.
Financial risk measures - the theory and applications of backward stochastic difference/differential equations with respect to the single jump process
12:10 Mon 26 Mar, 2012 :: 5.57 Ingkarni Wardli :: Mr Bin Shen :: University of Adelaide

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This is my PhD thesis submitted one month ago. Chapter 1 introduces the backgrounds of the research fields. Then each chapter is a published or an accepted paper. Chapter 2, to appear in Methodology and Computing in Applied Probability, establishes the theory of Backward Stochastic Difference Equations with respect to the single jump process in discrete time. Chapter 3, published in Stochastic Analysis and Applications, establishes the theory of Backward Stochastic Differential Equations with respect to the single jump process in continuous time. Chapter 2 and 3 consist of Part I Theory. Chapter 4, published in Expert Systems With Applications, gives some examples about how to measure financial risks by the theory established in Chapter 2. Chapter 5, accepted by Journal of Applied Probability, considers the question of an optimal transaction between two investors to minimize their risks. It's the applications of the theory established in Chapter 3. Chapter 4 and 5 consist of Part II Applications.
The mechanics of plant root growth
15:10 Fri 30 Mar, 2012 :: B.21 Ingkarni Wardli :: Dr Rosemary Dyson :: University of Birmingham

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Growing plant cells undergo rapid axial elongation with negligible radial expansion: high internal turgor pressure causes viscous stretching of the cell wall. We represent the cell wall as a thin fibre-reinforced viscous sheet, providing insight into the geometric and biomechanical parameters underlying bulk quantities such as wall extensibility and showing how either dynamical changes in material properties, achieved through changes in the cell-wall microstructure, 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.
Spatial-point data sets and the Polya distribution
15:10 Fri 27 Apr, 2012 :: B.21 Ingkarni Wardli :: Dr Benjamin Binder :: The University of Adelaide

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Spatial-point data sets, generated from a wide range of physical systems and mathematical models, can be analyzed by counting the number of objects in equally sized bins. We find that the bin counts are related to the Polya distribution. New indexes are developed which quantify whether or not a spatial data set is at its most evenly distributed state. Using three case studies (Lagrangian fluid particles in chaotic laminar flows, cellular automata agents in discrete models, and biological cells within colonies), we calculate the indexes and predict the spatial-state of the system.
Modelling protective anti-tumour immunity using a hybrid agent-based and delay differential equation approach
15:10 Fri 11 May, 2012 :: B.21 Ingkarni Wardli :: Dr Peter Kim :: University of Sydney

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Although cancers seem to consistently evade current medical treatments, the body's immune defences seem quite effective at controlling incipient tumours. Understanding how our immune systems provide such protection against early-stage tumours and how this protection could be lost will provide insight into designing next-generation immune therapies against cancer. To engage this problem, we formulate a mathematical model of the immune response against small, incipient tumours. The model considers the initial stimulation of the immune response in lymph nodes and the resulting immune attack on the tumour and is formulated as a hybrid agent-based and delay differential equation model.
The classification of Dynkin diagrams
12:10 Mon 21 May, 2012 :: 5.57 Ingkarni Wardli :: Mr Alexander Hanysz :: University of Adelaide

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The idea of continuous symmetry is often described in mathematics via Lie groups. These groups can be classified by their root systems: collections of vectors satisfying certain symmetry properties. The root systems are described in a concise way by Dynkin diagrams, and it turns out, roughly speaking, that there are only seven possible shapes for a Dynkin diagram. In this talk I'll describe some simple examples of Lie groups, explain what a root system is, and show how a Dynkin diagram encodes this information. Then I'll give a very brief sketch of the methods used to classify Dynkin diagrams.
Adventures with group theory: counting and constructing polynomial invariants for applications in quantum entanglement and molecular phylogenetics
15:10 Fri 8 Jun, 2012 :: B.21 Ingkarni Wardli :: Dr Peter Jarvis :: The University of Tasmania

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In many modelling problems in mathematics and physics, a standard challenge is dealing with several repeated instances of a system under study. If linear transformations are involved, then the machinery of tensor products steps in, and it is the job of group theory to control how the relevant symmetries lift from a single system, to having many copies. At the level of group characters, the construction which does this is called PLETHYSM. In this talk all this will be contextualised via two case studies: entanglement invariants for multipartite quantum systems, and Markov invariants for tree reconstruction in molecular phylogenetics. By the end of the talk, listeners will have understood why Alice, Bob and Charlie love Cayley's hyperdeterminant, and they will know why the three squangles -- polynomial beasts of degree 5 in 256 variables, with a modest 50,000 terms or so -- can tell us a lot about quartet trees!
Comparison of spectral and wavelet estimators of transfer function for linear systems
12:10 Mon 18 Jun, 2012 :: B.21 Ingkarni Wardli :: Mr Mohd Aftar Abu Bakar :: University of Adelaide

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We compare spectral and wavelet estimators of the response amplitude operator (RAO) of a linear system, with various input signals and added noise scenarios. The comparison is based on a model of a heaving buoy wave energy device (HBWED), which oscillates vertically as a single mode of vibration linear system. HBWEDs and other single degree of freedom wave energy devices such as the oscillating wave surge convertors (OWSC) are currently deployed in the ocean, making single degree of freedom wave energy devices important systems to both model and analyse in some detail. However, the results of the comparison relate to any linear system. It was found that the wavelet estimator of the RAO offers no advantage over the spectral estimators if both input and response time series data are noise free and long time series are available. If there is noise on only the response time series, only the wavelet estimator or the spectral estimator that uses the cross-spectrum of the input and response signals in the numerator should be used. For the case of noise on only the input time series, only the spectral estimator that uses the cross-spectrum in the denominator gives a sensible estimate of the RAO. If both the input and response signals are corrupted with noise, a modification to both the input and response spectrum estimates can provide a good estimator of the RAO. However, a combination of wavelet and spectral methods is introduced as an alternative RAO estimator. The conclusions apply for autoregressive emulators of sea surface elevation, impulse, and pseudorandom binary sequences (PRBS) inputs. However, a wavelet estimator is needed in the special case of a chirp input where the signal has a continuously varying frequency.
Geometry - algebraic to arithmetic to absolute
15:10 Fri 3 Aug, 2012 :: B.21 Ingkarni Wardli :: Dr James Borger :: Australian National University

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Classical algebraic geometry is about studying solutions to systems of polynomial equations with complex coefficients. In arithmetic algebraic geometry, one digs deeper and studies the arithmetic properties of the solutions when the coefficients are rational, or even integral. From the usual point of view, it's impossible to go deeper than this for the simple reason that no smaller rings are available - the integers have no proper subrings. In this talk, I will explain how an emerging subject, lambda-algebraic geometry, allows one to do just this and why one might care.
Infectious diseases modelling: from biology to public health policy
15:10 Fri 24 Aug, 2012 :: B.20 Ingkarni Wardli :: Dr James McCaw :: The University of Melbourne

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The mathematical study of human-to-human transmissible pathogens has established itself as a complementary methodology to the traditional epidemiological approach. The classic susceptible--infectious--recovered model paradigm has been used to great effect to gain insight into the epidemiology of endemic diseases such as influenza and pertussis, and the emergence of novel pathogens such as SARS and pandemic influenza. The modelling paradigm has also been taken within the host and used to explain the within-host dynamics of viral (or bacterial or parasite) infections, with implications for our understanding of infection, emergence of drug resistance and optimal drug-interventions. In this presentation I will provide an overview of the mathematical paradigm used to investigate both biological and epidemiological infectious diseases systems, drawing on case studies from influenza, malaria and pertussis research. I will conclude with a summary of how infectious diseases modelling has assisted the Australian government in developing its pandemic preparedness and response strategies.
Wave propagation in disordered media
15:10 Fri 31 Aug, 2012 :: B.21 Ingkarni Wardli :: Dr Luke Bennetts :: The University of Adelaide

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Problems involving wave propagation through systems composed of arrays of scattering sources embedded in some background medium will be considered. For example, in a fluids setting, the background medium is the open ocean surface and the scatterers are floating bodies, such as wave energy devices. Waves propagate in very different ways if the system is structured or disordered. If the disorder is random the problem is to determine the `effective' wave propagation properties by considering the ensemble average over all possible realisations of the system. I will talk about semi-analytical (i.e. low numerical cost) approaches to determining the effective properties.
Krylov Subspace Methods or: How I Learned to Stop Worrying and Love GMRes
12:10 Mon 17 Sep, 2012 :: B.21 Ingkarni Wardli :: Mr David Wilke :: University of Adelaide

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Many problems within applied mathematics require the solution of a linear system of equations. For instance, models of arterial umbilical blood flow are obtained through a finite element approximation, resulting in a linear, n x n system. For small systems the solution is (almost) trivial, but what happens when n is large? Say, n ~ 10^6? In this case matrix inversion is expensive (read: completely impractical) and we seek approximate solutions in a reasonable time. In this talk I will discuss the basic theory underlying Krylov subspace methods; a class of non-stationary iterative methods which are currently the methods-of-choice for large, sparse, linear systems. In particular I will focus on the method of Generalised Minimum RESiduals (GMRes), which is of the most popular for nonsymmetric systems. It is hoped that through this presentation I will convince you that a) solving linear systems is not necessarily trivial, and that b) my lack of any tangible results is not (entirely) a result of my own incompetence.
Probability, what can it tell us about health?
13:10 Tue 9 Oct, 2012 :: 7.15 Ingkarni Wardli :: Prof Nigel Bean :: School of Mathematical Sciences

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Clinical trials are the way in which modern medical systems test whether individual treatments are worthwhile. Sophisticated statistics is used to try and make the conclusions from clinical trials as meaningful as possible. What can a very simple probability model then tell us about the worth of multiple treatments? What might the implications of this be for the whole health system?

This talk is based on research currently being conducted with a physician at a major Adelaide hospital. It requires no health knowledge and was not tested on animals. All you need is an enquiring and open mind.
Fair and Loathing in State Parliament
12:10 Mon 29 Oct, 2012 :: B.21 Ingkarni Wardli :: Mr Casey Briggs :: University of Adelaide

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The South Australian electoral system has a history of bias, malapportionment and perceived unfairness. These days, it is typical of most systems across Australia, except with one major difference - a specific legislated criterion designed to force the system to be 'fair'. In reality, fairness is a hard concept to define, and an even harder concept to enforce. In this talk I will briefly take you through the history of South Australian electoral reform, the current state of affairs and my proposed research. There will be very little in the way of rigorous mathematics. No knowledge of politics is assumed, but an understanding of the process of voting would be useful.
A multiscale approach to reaction-diffusion processes in domains with microstructure
15:10 Fri 15 Mar, 2013 :: B.18 Ingkarni Wardli :: Prof Malte Peter :: University of Augsburg

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Reaction-diffusion processes occur in many materials with microstructure such as biological cells, steel or concrete. The main difficulty in modelling and simulating accurately such processes is to account for the fine microstructure of the material. One method of upscaling multi-scale problems, which has proven reliable for obtaining feasible macroscopic models, is the method of periodic homogenisation. The talk will give an introduction to multi-scale modelling of chemical mechanisms in domains with microstructure as well as to the method of periodic homogenisation. Moreover, a few aspects of solving the resulting systems of equations numerically will also be discussed.
Neuronal excitability and canards
15:10 Fri 10 May, 2013 :: B.18 Ingkarni Wardli :: A/Prof Martin Wechselberger :: University of Sydney

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The notion of excitability was first introduced in an attempt to understand firing properties of neurons. It was Alan Hodgkin who identified three basic types (classes) of excitable axons (integrator, resonator and differentiator) distinguished by their different responses to injected steps of currents of various amplitudes. Pioneered by Rinzel and Ermentrout, bifurcation theory explains repetitive (tonic) firing patterns for adequate steady inputs in integrator (type I) and resonator (type II) neuronal models. In contrast, the dynamic behavior of differentiator (type III) neurons cannot be explained by standard dynamical systems theory. This third type of excitable neuron encodes a dynamic change in the input and leads naturally to a transient response of the neuron. In this talk, I will show that "canards" - peculiar mathematical creatures - are well suited to explain the nature of transient responses of neurons due to dynamic (smooth) inputs. I will apply this geometric theory to a simple driven FitzHugh-Nagumo/Morris-Lecar type neural model and to a more complicated neural model that describes paradoxical excitation due to propofol anesthesia.
K-theory and solid state physics
12:10 Fri 13 Sep, 2013 :: Ingkarni Wardli B19 :: Dr Keith Hannabuss :: Balliol College, Oxford

More than 50 years ago Dyson showed that there is a nine-fold classification of random matrix models, the classes of which are each associated with Riemannian symmetric spaces. More recently it was realised that a related argument enables one to classify the insulating properties of fermionic systems (with the addition of an extra class to give 10 in all), and can be described using K-theory. In this talk I shall give a survey of the ideas, and a brief outline of work with Guo Chuan Thiang.
Symmetry gaps for geometric structures
15:10 Fri 20 Sep, 2013 :: B.18 Ingkarni Wardli :: Dr Dennis The :: Australian National University

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Klein's Erlangen program classified geometries based on their (transitive) groups of symmetries, e.g. Euclidean geometry is the quotient of the rigid motion group by the subgroup of rotations. While this perspective is homogeneous, Riemann's generalization of Euclidean geometry is in general very "lumpy" - i.e. there exist Riemannian manifolds that have no symmetries at all. A common generalization where a group still plays a dominant role is Cartan geometry, which first arose in Cartan's solution to the equivalence problem for geometric structures, and which articulates what a "curved version" of a flat (homogeneous) model means. Parabolic geometries are Cartan geometries modelled on (generalized) flag varieties (e.g. projective space, isotropic Grassmannians) which are well-known objects from the representation theory of semisimple Lie groups. These curved versions encompass a zoo of interesting geometries, including conformal, projective, CR, systems of 2nd order ODE, etc. This interaction between differential geometry and representation theory has proved extremely fruitful in recent years. My talk will be an example-based tour of various types of parabolic geometries, which I'll use to outline some of the main aspects of the theory (suppressing technical details). The main thread throughout the talk will be the symmetry gap problem: For a given type of Cartan geometry, the maximal symmetry dimension is realized by the flat model, but what is the next possible ("submaximal") symmetry dimension? I'll sketch a recent solution (in joint work with Boris Kruglikov) for a wide class of parabolic geometries which gives a combinatorial recipe for reading the submaximal symmetry dimension from a Dynkin diagram.
Dynamics and the geometry of numbers
14:10 Fri 27 Sep, 2013 :: Horace Lamb Lecture Theatre :: Prof Akshay Venkatesh :: Stanford University

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It was understood by Minkowski that one could prove interesting results in number theory by considering the geometry of lattices in R^n. (A lattice is simply a grid of points.) This technique is called the "geometry of numbers." We now understand much more about analysis and dynamics on the space of all lattices, and this has led to a deeper understanding of classical questions. I will review some of these ideas, with emphasis on the dynamical aspects.
Braids, conformal module and entropy
12:10 Fri 8 Nov, 2013 :: Ingkarni Wardli B19 :: Prof Burglind Joricke :: Australian National University

I will discuss two invariants of conjugacy classes of braids. The first invariant is the conformal module which implicitly occurred already in a paper of Gorin and Lin in connection with their interest in Hilbert's 13th problem. The second is a popular dynamical invariant, the entropy. It appeared in connection with Thurston's theory of surface homeomorphisms. It turns out that these invariants are related: They are inversely proportional. In a preparatory talk (at 10:10 am) I will give a brief introduction to some aspects of braid theory and to entropy.
All at sea with spectral analysis
11:10 Tue 19 Nov, 2013 :: Ingkarni Wardli Level 5 Room 5.56 :: A/Prof Andrew Metcalfe :: The University of Adelaide

The steady state response of a single degree of freedom damped linear stystem to a sinusoidal input is a sinusoidal function at the same frequency, but generally with a different amplitude and a phase shift. The analogous result for a random stationary input can be described in terms of input and response spectra and a transfer function description of the linear system. The practical use of this result is that the parameters of a linear system can be estimated from the input and response spectra, and the response spectrum can be predicted if the transfer function and input spectrum are known. I shall demonstrate these results with data from a small ship in the North Sea. The results from the sea trial raise the issue of non-linearity, and second order amplitude response functons are obtained using auto-regressive estimators. The possibility of using wavelets rather than spectra is consedred in the context of single degree of freedom linear systems. Everybody welcome to attend. Please not a change of venue - we will be in room 5.56
The structuring role of chaotic stirring on pelagic ecosystems
11:10 Fri 28 Feb, 2014 :: B19 Ingkarni Wardli :: Dr Francesco d'Ovidio :: Universite Pierre et Marie Curie (Paris VI)

The open ocean upper layer is characterized by a complex transport dynamics occuring over different spatiotemporal scales. At the scale of 10-100 km - which covers the so called mesoscale and part of the submesoscale - in situ and remote sensing observations detect strong variability in physical and biogeochemical fields like sea surface temperature, salinity, and chlorophyll concentration. The calculation of Lyapunov exponent and other nonlinear diagnostics applied to the surface currents have allowed to show that an important part of this tracer variability is due to chaotic stirring. Here I will extend this analysis to marine ecosystems. For primary producers, I will show that stable and unstable manifolds of hyperbolic points embedded in the surface velocity field are able to structure the phytoplanktonic community in fluid dynamical niches of dominant types, where competition can locally occur during bloom events. By using data from tagged whales, frigatebirds, and elephant seals, I will also show that chaotic stirring affects the behaviour of higher trophic levels. In perspective, these relations between transport structures and marine ecosystems can be the base for a biodiversity index constructued from satellite information, and therefore able to monitor key aspects of the marine biodiversity and its temporal variability at the global scale.
The effects of pre-existing immunity
15:10 Fri 7 Mar, 2014 :: B.18 Ingkarni Wardli :: Associate Professor Jane Heffernan :: York University, Canada

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Immune system memory, also called immunity, is gained as a result of primary infection or vaccination, and can be boosted after vaccination or secondary infections. Immunity is developed so that the immune system is primed to react and fight a pathogen earlier and more effectively in secondary infections. The effects of memory, however, on pathogen propagation in an individual host (in-host) and a population (epidemiology) are not well understood. Mathematical models of infectious diseases, employing dynamical systems, computer simulation and bifurcation analysis, can provide projections of pathogen propagation, show outcomes of infection and help inform public health interventions. In the Modelling Infection and Immunity (MI^2) lab, we develop and study biologically informed mathematical models of infectious diseases at both levels of infection, and combine these models into comprehensive multi-scale models so that the effects of individual immunity in a population can be determined. In this talk we will discuss some of the interesting mathematical phenomenon that arise in our models, and show how our results are directly applicable to what is known about the persistence of infectious diseases.
Dynamical systems approach to fluid-plasma turbulence
15:10 Fri 14 Mar, 2014 :: 5.58 Ingkarni Wardli :: Professor Abraham Chian

Sun-Earth system is a complex, electrodynamically coupled system dominated by multiscale interactions. The complex behavior of the space environment is indicative of a state driven far from equilibrium whereby instabilities, nonlinear waves, and turbulence play key roles in the system dynamics. First, we review the fundamental concepts of nonlinear dynamics in fluids and plasmas and discuss their relevance to the study of the Sun-Earth relation. Next, we show how Lagrangian coherent structures identify the transport barriers of plasma turbulence modeled by 3-D solar convective dynamo. Finally, we show how Lagrangian coherent structures can be detected in the solar photospheric turbulence using satellite observations.
Semiclassical restriction estimates
12:10 Fri 4 Apr, 2014 :: Ingkarni Wardli B20 :: Melissa Tacy :: University of Adelaide

Eigenfunctions of Hamiltonians arise naturally in the theory of quantum mechanics as stationary states of quantum systems. Their eigenvalues have an interpretation as the square root of E, where E is the energy of the system. We wish to better understand the high energy limit which defines the boundary between quantum and classical mechanics. In this talk I will focus on results regarding the restriction of eigenfunctions to lower dimensional subspaces, in particular to hypersurfaces. A convenient way to study such problems is to reframe them as problems in semiclassical analysis.
Ergodicity and loss of capacity: a stochastic horseshoe?
15:10 Fri 9 May, 2014 :: B.21 Ingkarni Wardli :: Professor Ami Radunskaya :: Pomona College, the United States of America

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Random fluctuations of an environment are common in ecological and economical settings. The resulting processes can be described by a stochastic dynamical system, where a family of maps parametrized by an independent, identically distributed random variable forms the basis for a Markov chain on a continuous state space. Random dynamical systems are a beautiful combination of deterministic and random processes, and they have received considerable interest since von Neuman and Ulam's seminal work in the 1940's. Key questions in the study of a stochastic dynamical system are: does the system have a well-defined average, i.e. is it ergodic? How does this long-term behavior compare to that of the state variable in a constant environment with the averaged parameter? In this talk we answer these questions for a family of maps on the unit interval that model self-limiting growth. The techniques used can be extended to study other families of concave maps, and so we conjecture the existence of a "stochastic horseshoe".
Group meeting
15:10 Fri 6 Jun, 2014 :: 5.58 Ingkarni Wardli :: Meng Cao and Trent Mattner :: University of Adelaide

Meng Cao:: Multiscale modelling couples patches of nonlinear wave-like simulations :: Abstract: The multiscale gap-tooth scheme is built from given microscale simulations of complicated physical processes to empower macroscale simulations. By coupling small patches of simulations over unsimulated physical gaps, large savings in computational time are possible. So far the gap-tooth scheme has been developed for dissipative systems, but wave systems are also of great interest. This article develops the gap-tooth scheme to the case of nonlinear microscale simulations of wave-like systems. Classic macroscale interpolation provides a generic coupling between patches that achieves arbitrarily high order consistency between the multiscale scheme and the underlying microscale dynamics. Eigen-analysis indicates that the resultant gap-tooth scheme empowers feasible computation of large scale simulations of wave-like dynamics with complicated underlying physics. As an pilot study, we implement numerical simulations of dam-breaking waves by the gap-tooth scheme. Comparison between a gap-tooth simulation, a microscale simulation over the whole domain, and some published experimental data on dam breaking, demonstrates that the gap-tooth scheme feasibly computes large scale wave-like dynamics with computational savings. Trent Mattner :: Coupled atmosphere-fire simulations of the Canberra 2003 bushfires using WRF-Sfire :: Abstract: The Canberra fires of January 18, 2003 are notorious for the extreme fire behaviour and fire-atmosphere-topography interactions that occurred, including lee-slope fire channelling, pyrocumulonimbus development and tornado formation. In this talk, I will discuss coupled fire-weather simulations of the Canberra fires using WRF-SFire. In these simulations, a fire-behaviour model is used to dynamically predict the evolution of the fire front according to local atmospheric and topographic conditions, as well as the associated heat and moisture fluxes to the atmosphere. It is found that the predicted fire front and heat flux is not too bad, bearing in mind the complexity of the problem and the severe modelling assumptions made. However, the predicted moisture flux is too low, which has some impact on atmospheric dynamics.
Hydrodynamics and rheology of self-propelled colloids
15:10 Fri 8 Aug, 2014 :: B17 Ingkarni Wardli :: Dr Sarthok Sircar :: University of Adelaide

The sub-cellular world has many components in common with soft condensed matter systems (polymers, colloids and liquid crystals). But it has novel properties, not present in traditional complex fluids, arising from a rich spectrum of non-equilibrium behavior: flocking, chemotaxis and bioconvection. The talk is divided into two parts. In the first half, we will (get an idea on how to) derive a hydrodynamic model for self-propelled particles of an arbitrary shape from first principles, in a sufficiently dilute suspension limit, moving in a 3-dimensional space inside a viscous solvent. The model is then restricted to particles with ellipsoidal geometry to quantify the interplay of the long-range excluded volume and the short-range self-propulsion effects. The expression for the constitutive stresses, relating the kinetic theory with the momentum transport equations, are derived using a combination of the virtual work principle (for extra elastic stresses) and symmetry arguments (for active stresses). The second half of the talk will highlight on my current numerical expertise. In particular we will exploit a specific class of spectral basis functions together with RK4 time-stepping to determine the dynamical phases/structures as well as phase-transitions of these ellipsoidal clusters. We will also discuss on how to define the order (or orientation) of these clusters and understand the other rheological quantities.
Mathematics: a castle in the sky?
14:10 Mon 25 Aug, 2014 :: Ingkarni Wardli 715 Conference Room :: Dr. David Roberts :: School of Mathematical Sciences

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At university you are exposed to more rigorous mathematics than at school, exemplified by definitions such as those of real numbers individually or as a whole. However, what does mathematics ultimately rest on? Definitions depend on things defined earlier, and this process must stop at some point. Mathematicians expended a lot of energy in the late 19th and early 20th centuries trying to pin down the absolutely fundamental ideas of mathematics, with unexpected results. The results of these efforts are called foundations and are still an area of active research today. This talk will explain what foundations are, some of the historical setting in which they arose, and several of the various systems on which mathematics can be built -- and why most of the mathematics you will do only uses a tiny portion of it!
Exploration vs. Exploitation with Partially Observable Gaussian Autoregressive Arms
15:00 Mon 29 Sep, 2014 :: Engineering North N132 :: Julia Kuhn :: The University of Queensland & The University of Amsterdam

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We consider a restless bandit problem with Gaussian autoregressive arms, where the state of an arm is only observed when it is played and the state-dependent reward is collected. Since arms are only partially observable, a good decision policy needs to account for the fact that information about the state of an arm becomes more and more obsolete while the arm is not being played. Thus, the decision maker faces a tradeoff between exploiting those arms that are believed to be currently the most rewarding (i.e. those with the largest conditional mean), and exploring arms with a high conditional variance. Moreover, one would like the decision policy to remain tractable despite the infinite state space and also in systems with many arms. A policy that gives some priority to exploration is the Whittle index policy, for which we establish structural properties. These motivate a parametric index policy that is computationally much simpler than the Whittle index but can still outperform the myopic policy. Furthermore, we examine the many-arm behavior of the system under the parametric policy, identifying equations describing its asymptotic dynamics. Based on these insights we provide a simple heuristic algorithm to evaluate the performance of index policies; the latter is used to optimize the parametric index.
Micro Magnetofluidics - Wireless Manipulation for Microfluidics
15:10 Fri 24 Oct, 2014 :: N.132 Engineering North :: Professor Nam-Trung Nguyen :: Griffith University

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Microfluidics is rich in multi-physics phenomena, which offer fundamentally new capabilities in the manipulation and detection of biological particles. Most current microfluidic applications are based on hydrodynamic, electrokinetic, acoustic and optic actuation. Implementing these concepts requires bulky external pumping/valving systems and energy supplies. The required wires and connectors make their fabrication and handling difficult. Most of the conventional approaches induce heat that may affect sensitive bio particles such as cells. There is a need for a technology for fluid handling in microfluidic devices that is of low-cost, simple, wireless, free of induced heat and independent of pH level or ion concentration. The use of magnetism would provide a wireless solution for this need. Micro magnetofluidics is a newly established research field that links magnetism and microfluidics to gain new capabilities. Magnetism provides a convenient and wireless way for control and manipulation of fluid flow in the microscale. Investigation of magnetism-induced phenomena in a microfluidic device has the advantage of well-defined experimental condition such as temperature and magnetic field because of the system size. This talk presents recent interesting phenomena in both continuous-flow and digital micro magnetofluidics.
Multiscale modelling of multicellular biological systems: mechanics, development and disease
03:10 Fri 6 Mar, 2015 :: Lower Napier LG24 :: Dr James Osborne :: University of Melbourne

When investigating the development and function of multicellular biological systems it is not enough to only consider the behaviour of individual cells in isolation. For example when studying tissue development, how individual cells interact, both mechanically and biochemically, influences the resulting tissues form and function. In this talk we present a multiscale modelling framework for simulating the development and function of multicellular biological systems (in particular tissues). Utilising the natural structural unit of the cell, the framework consists of three main scales: the tissue level (macro-scale); the cell level (meso-scale); and the sub-cellular level (micro-scale), with multiple interactions occurring between all scales. The cell level is central to the framework and cells are modelled as discrete interacting entities using one of a number of possible modelling paradigms, including lattice based models (cellular automata and cellular Potts) and off-lattice based models (cell centre and vertex based representations). The sub-cellular level concerns numerous metabolic and biochemical processes represented by interaction networks rendered stochastically or into ODEs. The outputs from such systems influence the behaviour of the cell level affecting properties such as adhesion and also influencing cell mitosis and apoptosis. At the tissue level we consider factors or restraints that influence the cells, for example the distribution of a nutrient or messenger molecule, which is represented by field equations, on a growing domain, with individual cells functioning as sinks and/or sources. The modular approach taken within the framework enables more realistic behaviour to be considered at each scale. This framework is implemented within the Open Source Chaste library (Cancer Heart and Soft Tissue Environment, (http://www.cs.ox.ac.uk/chaste/) and has been used to model biochemical and biomechanical interactions in various biological systems. In this talk we present the key ideas of the framework along with applications within the fields of development and disease.
On the analyticity of CR-diffeomorphisms
12:10 Fri 13 Mar, 2015 :: Engineering North N132 :: Ilya Kossivskiy :: University of Vienna

One of the fundamental objects in several complex variables is CR-mappings. CR-mappings naturally occur in complex analysis as boundary values of mappings between domains, and as restrictions of holomorphic mappings onto real submanifolds. It was already observed by Cartan that smooth CR-diffeomorphisms between CR-submanifolds in C^N tend to be very regular, i.e., they are restrictions of holomorphic maps. However, in general smooth CR-mappings form a more restrictive class of mappings. Thus, since the inception of CR-geometry, the following general question has been of fundamental importance for the field: Are CR-equivalent real-analytic CR-structures also equivalent holomorphically? In joint work with Lamel, we answer this question in the negative, in any positive CR-dimension and CR-codimension. Our construction is based on a recent dynamical technique in CR-geometry, developed in my earlier work with Shafikov.
Singular Pfaffian systems in dimension 6
12:10 Fri 20 Mar, 2015 :: Napier 144 :: Pawel Nurowski :: Center for Theoretical Physics, Polish Academy of Sciences

We consider a pair of rank 3 distributions in dimension 6 with some remarkable properties. They define an analog of the celebrated nearly-Kahler structure on the 6 sphere, with the exceptional simple Lie group G2 as a group of symmetries. In our case the metric associated with the structure is pseudo-Riemannian, of split signature. The 6 manifold has a 5-dimensional boundary with interesting induced geometry. This structure on the boundary has no analog in the Riemannian case.
Group Meeting
15:10 Fri 24 Apr, 2015 :: N218 Engineering North :: Dr Ben Binder :: University of Adelaide

Talk (Dr Ben Binder): How do we quantify the filamentous growth in a yeast colony? Abstract: In this talk we will develop a systematic method to measure the spatial patterning of yeast colony morphology. The methods are applicable to other physical systems with circular spatial domains, for example, batch mixing fluid devices. A hybrid modelling approach of the yeast growth process will also be discussed. After the seminar, Ben will start a group discussion by sharing some information and experiences on attracting honours/PhD students to the group.
Complex Systems, Chaotic Dynamics and Infectious Diseases
15:10 Fri 5 Jun, 2015 :: Engineering North N132 :: Prof Michael Small :: UWA

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In complex systems, the interconnection between the components of the system determine the dynamics. The system is described by a very large and random mathematical graph and it is the topological structure of that graph which is important for understanding of the dynamical behaviour of the system. I will talk about two specific examples - (1) spread of infectious disease (where the connection between the agents in a population, rather than epidemic parameters, determine the endemic state); and, (2) a transformation to represent a dynamical system as a graph (such that the "statistical mechanics" of the graph characterise the dynamics).
Mathematical Modeling and Analysis of Active Suspensions
14:10 Mon 3 Aug, 2015 :: Napier 209 :: Professor Michael Shelley :: Courant Institute of Mathematical Sciences, New York University

Complex fluids that have a 'bio-active' microstructure, like suspensions of swimming bacteria or assemblies of immersed biopolymers and motor-proteins, are important examples of so-called active matter. These internally driven fluids can have strange mechanical properties, and show persistent activity-driven flows and self-organization. I will show how first-principles PDE models are derived through reciprocal coupling of the 'active stresses' generated by collective microscopic activity to the fluid's macroscopic flows. These PDEs have an interesting analytic structures and dynamics that agree qualitatively with experimental observations: they predict the transitions to flow instability and persistent mixing observed in bacterial suspensions, and for microtubule assemblies show the generation, propagation, and annihilation of disclination defects. I'll discuss how these models might be used to study yet more complex biophysical systems.
Vanishing lattices and moduli spaces
12:10 Fri 28 Aug, 2015 :: Ingkarni Wardli B17 :: David Baraglia :: The University of Adelaide

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Vanishing lattices are symplectic analogues of root systems. As with roots systems, they admit a classification in terms of certain Dynkin diagrams (not the usual ones from Lie theory). In this talk I will discuss this classification and if there is time I will outline my work (in progress) showing that the monodromy of the SL(n,C) Hitchin fibration is essentially a vanishing lattice.
Pattern Formation in Nature
12:10 Mon 31 Aug, 2015 :: Benham Labs G10 :: Saber Dini :: University of Adelaide

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Pattern formation is a ubiquitous process in nature: embryo development, animals skin pigmentation, etc. I will talk about how Alan Turing (the British genius known for the Turing Machine) explained pattern formation by linear stability analysis of reaction-diffusion systems.
The Calderon Problem: From the Past to the Present
15:10 Fri 11 Sep, 2015 :: Ingkarni Wardli B21 :: Dr Leo Tzou :: University of Sydney

The problem of determining the electrical conductivity of a body by making voltage and current measurements on the object's surface has various applications in fields such as oil exploration and early detection of malignant breast tumour. This classical problem posed by Calderon remained open until the late '80s when it was finally solved in a breakthrough paper by Sylvester-Uhlmann.

In the recent years, geometry has played an important role in this problem. The unexpected connection of this subject to fields such as dynamical systems, symplectic geometry, and Riemannian geometry has led to some interesting progress. This talk will be an overview of some of the recent results and an outline of the techniques used to treat this problem.

Expanding maps
12:10 Fri 18 Mar, 2016 :: Eng & Maths EM205 :: Andy Hammerlindl :: Monash University

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Consider a function from the circle to itself such that the derivative is greater than one at every point. Examples are maps of the form f(x) = mx for integers m > 1. In some sense, these are the only possible examples. This fact and the corresponding question for maps on higher dimensional manifolds was a major motivation for Gromov to develop pioneering results in the field of geometric group theory. In this talk, I'll give an overview of this and other results relating dynamical systems to the geometry of the manifolds on which they act and (time permitting) talk about my own work in the area.
Chaos in dimensions 2 and 3
15:10 Fri 18 Mar, 2016 :: Engineering South S112 :: Dr Andy Hammerlindl :: Monash University

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I will talk about known models of chaotic dynamical systems in dimensions two and three, and results which classify the types of chaotic dynamics that are robust under perturbation. I will also talk about my own work towards understanding chaotic dynamics for discrete-time systems in dimension three. This is joint work with C. Bonatti, A. Gogolev, and R. Potrie.
Harmonic Analysis in Rough Contexts
15:10 Fri 13 May, 2016 :: Engineering South S112 :: Dr Pierre Portal :: Australian National University

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In recent years, perspectives on what constitutes the ``natural" framework within which to conduct various forms of mathematical analysis have shifted substantially. The common theme of these shifts can be described as a move towards roughness, i.e. the elimination of smoothness assumptions that had previously been considered fundamental. Examples include partial differential equations on domains with a boundary that is merely Lipschitz continuous, geometric analysis on metric measure spaces that do not have a smooth structure, and stochastic analysis of dynamical systems that have nowhere differentiable trajectories. In this talk, aimed at a general mathematical audience, I describe some of these shifts towards roughness, placing an emphasis on harmonic analysis, and on my own contributions. This includes the development of heat kernel methods in situations where such a kernel is merely a distribution, and applications to deterministic and stochastic partial differential equations.
Etale ideas in topological and algebraic dynamical systems
12:10 Fri 5 Aug, 2016 :: Ingkarni Wardli B18 :: Tuyen Truong :: University of Adelaide

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In etale topology, instead of considering open subsets of a space, we consider etale neighbourhoods lying over these open subsets. In this talk, I define an etale analog of dynamical systems: to understand a dynamical system f:(X,\Omega )->(X,\Omega ), we consider other dynamical systems lying over it. I then propose to use this to resolve the following two questions: Question 1: What should be the topological entropy of a dynamical system (f,X,\Omega ) when (X,\Omega ) is not a compact space? Question 2: What is the relation between topological entropy of a rational map or correspondence (over a field of arbitrary characteristic) to the pullback on cohomology groups and algebraic cycles?
Mathematical modelling of social spreading processes
15:10 Fri 19 Aug, 2016 :: Napier G03 :: Prof Hans De Sterck :: Monash University

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Social spreading processes are intriguing manifestations of how humans interact and shape each others' lives. There is great interest in improving our understanding of these processes, and the increasing availability of empirical information in the era of big data and online social networks, combined with mathematical and computational modelling techniques, offer compelling new ways to study these processes. I will first discuss mathematical models for the spread of political revolutions on social networks. The influence of online social networks and social media on the dynamics of the Arab Spring revolutions of 2011 are of particular interest in our work. I will describe a hierarchy of models, starting from agent-based models realized on empirical social networks, and ending up with population-level models that summarize the dynamical behaviour of the spreading process. We seek to understand quantitatively how political revolutions may be facilitated by the modern online social networks of social media. The second part of the talk will describe a population-level model for the social dynamics that cause cigarette smoking to spread in a population. Our model predicts that more individualistic societies will show faster adoption and cessation of smoking. Evidence from a newly composed century-long composite data set on smoking prevalence in 25 countries supports the model, with potential implications for public health interventions around the world. Throughout the talk, I will argue that important aspects of social spreading processes can be revealed and understood via quantitative mathematical and computational models matched to empirical data. This talk describes joint work with John Lang and Danny Abrams.
Collective and aneural foraging in biological systems
15:10 Fri 3 Mar, 2017 :: Lower Napier LG14 :: Dr Jerome Buhl and Dr David Vogel :: The University of Adelaide

The field of collective behaviour uses concepts originally adapted from statistical physics to study how complex collective phenomena such as mass movement or swarm intelligence emerge from relatively simple interactions between individuals. Here we will focus on two applications of this framework. First we will have look at new insights into the evolution of sociality brought by combining models of nutrition and social interactions to explore phenomena such as collective foraging decisions, emergence of social organisation and social immunity. Second, we will look at the networks built by slime molds under exploration and foraging context.
Lagrangian transport in deterministic flows: from theory to experiment
16:10 Tue 16 May, 2017 :: Engineering North N132 :: Dr Michel Speetjens :: Eindhoven University of Technology

Transport of scalar quantities (e.g. chemical species, nutrients, heat) in deterministic flows is key to a wide range of phenomena and processes in industry and Nature. This encompasses length scales ranging from microns to hundreds of kilometres, and includes systems as diverse as viscous flows in the processing industry, micro-fluidic flows in labs-on-a-chip and porous media, large-scale geophysical and environmental flows, physiological and biological flows and even continuum descriptions of granular flows. Essential to the net transport of a scalar quantity is its advection by the fluid motion. The Lagrangian perspective (arguably) is the most natural way to investigate advection and leans on the fact that fluid trajectories are organized into coherent structures that geometrically determine the advective transport properties. Lagrangian transport is typically investigated via theoretical and computational studies and often concerns idealized flow situations that are difficult (or even impossible) to create in laboratory experiments. However, bridging the gap from theoretical and computational results to realistic flows is essential for their physical meaningfulness and practical relevance. This presentation highlights a number of fundamental Lagrangian transport phenomena and properties in both two-dimensional and three-dimensional flows and demonstrates their physical validity by way of representative and experimentally realizable flows.
Stokes' Phenomenon in Translating Bubbles
15:10 Fri 2 Jun, 2017 :: Ingkarni Wardli 5.57 :: Dr Chris Lustri :: Macquarie University

This study of translating air bubbles in a Hele-Shaw cell containing viscous fluid reveals the critical role played by surface tension in these systems. The standard zero-surface-tension model of Hele-Shaw flow predicts that a continuum of bubble solutions exists for arbitrary flow translation velocity. The inclusion of small surface tension, however, eliminates this continuum of solutions, instead producing a discrete, countably infinite family of solutions, each with distinct translation speeds. We are interested in determining this discrete family of solutions, and understanding why only these solutions are permitted. Studying this problem in the asymptotic limit of small surface tension does not seem to give any particular reason why only these solutions should be selected. It is only by using exponential asymptotic methods to study the Stokes’ structure hidden in the problem that we are able to obtain a complete picture of the bubble behaviour, and hence understand the selection mechanism that only permits certain solutions to exist. In the first half of my talk, I will explain the powerful ideas that underpin exponential asymptotic techniques, such as analytic continuation and optimal truncation. I will show how they are able to capture behaviour known as Stokes' Phenomenon, which is typically invisible to classical asymptotic series methods. In the second half of the talk, I will introduce the problem of a translating air bubble in a Hele-Shaw cell, and show that the behaviour can be fully understood by examining the Stokes' structure concealed within the problem. Finally, I will briefly showcase other important physical applications of exponential asymptotic methods, including submarine waves and particle chains.
Weil's Riemann hypothesis (RH) and dynamical systems
12:10 Fri 11 Aug, 2017 :: Engineering Sth S111 :: Tuyen Truong :: University of Adelaide

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Weil proposed an analogue of the RH in finite fields, aiming at counting asymptotically the number of solutions to a given system of polynomial equations (with coefficients in a finite field) in finite field extensions of the base field. This conjecture influenced the development of Algebraic Geometry since the 1950’s, most important achievements include: Grothendieck et al.’s etale cohomology, and Bombieri and Grothendieck’s standard conjectures on algebraic cycles (inspired by a Kahlerian analogue of a generalisation of Weil’s RH by Serre). Weil’s RH was solved by Deligne in the 70’s, but the finite field analogue of Serre’s result is still open (even in dimension 2). This talk presents my recent work proposing a generalisation of Weil’s RH by relating it to standard conjectures and a relatively new notion in complex dynamical systems called dynamical degrees. In the course of the talk, I will present the proof of a question proposed by Esnault and Srinivas (which is related to a result by Gromov and Yomdin on entropy of complex dynamical systems), which gives support to the finite field analogue of Serre’s result.
How oligomerisation impacts steady state gradient in a morphogen-receptor system
15:10 Fri 20 Oct, 2017 :: Ingkarni Wardli 5.57 :: Mr Phillip Brown :: University of Adelaide

In developmental biology an important process is cell fate determination, where cells start to differentiate their form and function. This is an element of the broader concept of morphogenesis. It has long been held that cell differentiation can occur by a chemical signal providing positional information to 'undecided' cells. This chemical produces a gradient of concentration that indicates to a cell what path it should develop along. More recently it has been shown that in a particular system of this type, the chemical (protein) does not exist purely as individual molecules, but can exist in multi-protein complexes known as oligomers. Mathematical modelling has been performed on systems of oligomers to determine if this concept can produce useful gradients of concentration. However, there are wide range of possibilities when it comes to how oligomer systems can be modelled and most of them have not been explored. In this talk I will introduce a new monomer system and analyse it, before extending this model to include oligomers. A number of oligomer models are proposed based on the assumption that proteins are only produced in their oligomer form and can only break apart once they have left the producing cell. It will be shown that when oligomers are present under these conditions, but only monomers are permitted to bind with receptors, then the system can produce robust, biologically useful gradients for a significantly larger range of model parameters (for instance, degradation, production and binding rates) compared to the monomer system. We will also show that when oligomers are permitted to bind with receptors there is negligible difference compared to the monomer system.
Stochastic Modelling of Urban Structure
11:10 Mon 20 Nov, 2017 :: Engineering Nth N132 :: Mark Girolami :: Imperial College London, and The Alan Turing Institute

Media...
Urban systems are complex in nature and comprise of a large number of individuals that act according to utility, a measure of net benefit pertaining to preferences. The actions of individuals give rise to an emergent behaviour, creating the so-called urban structure that we observe. In this talk, I develop a stochastic model of urban structure to formally account for uncertainty arising from the complex behaviour. We further use this stochastic model to infer the components of a utility function from observed urban structure. This is a more powerful modelling framework in comparison to the ubiquitous discrete choice models that are of limited use for complex systems, in which the overall preferences of individuals are difficult to ascertain. We model urban structure as a realization of a Boltzmann distribution that is the invariant distribution of a related stochastic differential equation (SDE) that describes the dynamics of the urban system. Our specification of Boltzmann distribution assigns higher probability to stable configurations, in the sense that consumer surplus (demand) is balanced with running costs (supply), as characterized by a potential function. We specify a Bayesian hierarchical model to infer the components of a utility function from observed structure. Our model is doubly-intractable and poses significant computational challenges that we overcome using recent advances in Markov chain Monte Carlo (MCMC) methods. We demonstrate our methodology with case studies on the London retail system and airports in England.
Chaos in higher-dimensional complex dynamics
13:10 Fri 20 Apr, 2018 :: Barr Smith South Polygon Lecture theatre :: Finnur Larusson :: University of Adelaide

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I will report on new joint work with Leandro Arosio (University of Rome, Tor Vergata). Complex manifolds can be thought of as laid out across a spectrum characterised by rigidity at one end and flexibility at the other. On the rigid side, Kobayashi-hyperbolic manifolds have at most a finite-dimensional group of symmetries. On the flexible side, there are manifolds with an extremely large group of holomorphic automorphisms, the prototypes being the affine spaces $\mathbb C^n$ for $n \geq 2$. From a dynamical point of view, hyperbolicity does not permit chaos. An endomorphism of a Kobayashi-hyperbolic manifold is non-expansive with respect to the Kobayashi distance, so every family of endomorphisms is equicontinuous. We show that not only does flexibility allow chaos: under a strong anti-hyperbolicity assumption, chaotic automorphisms are generic. A special case of our main result is that if $G$ is a connected complex linear algebraic group of dimension at least 2, not semisimple, then chaotic automorphisms are generic among all holomorphic automorphisms of $G$ that preserve a left- or right-invariant Haar form. For $G=\mathbb C^n$, this result was proved (although not explicitly stated) some 20 years ago by Fornaess and Sibony. Our generalisation follows their approach. I will give plenty of context and background, as well as some details of the proof of the main result.
Stability Through a Geometric Lens
15:10 Fri 18 May, 2018 :: Horace Lamb 1022 :: Dr Robby Marangell :: University of Sydney

Focussing on the example of the Fisher/KPP equation, I will show how geometric information can be used to establish (in)stability results in some partial differential equations (PDEs). Viewing standing and travelling waves as fixed points of a flow in an infinite dimensional system, leads to a reduction of the linearised stability problem to a boundary value problem in a linear non-autonomous ordinary differential equation (ODE). Next, by exploiting the linearity of the system, one can use geometric ideas to reveal additional structure underlying the determination of stability. I will show how the Riccati equation can be used to produce a reasonably computable detector of eigenvalues and how such a detector is related to another, well-known eigenvalue detector, the Evans function. If there is time, I will try to expand on how to generalise these ideas to systems of PDEs.
Modelling phagocytosis
15:10 Fri 25 May, 2018 :: Horace Lamb 1022 :: Prof Ngamta (Natalie) Thamwattana :: University of Wollongong

Phagocytosis refers to a process in which one cell type fully encloses and consumes unwanted cells, debris or particulate matter. It plays an important role in immune systems through the destruction of pathogens and the inhibiting of cancerous cells. In this study, we combine models on cell-cell adhesion and on predator-prey modelling to generate a new model for phagocytosis that is capable of relating the interaction between cells in both space and time. Numerical results are presented, demonstrating the behaviours of cells during the process of phagocytosis.
Tales of Multiple Regression: Informative Missingness, Recommender Systems, and R2-D2
15:10 Fri 17 Aug, 2018 :: Napier 208 :: Prof Howard Bondell :: University of Melbourne

In this talk, we briefly discuss two projects tangentially related under the umbrella of high-dimensional regression. The first part of the talk investigates informative missingness in the framework of recommender systems. In this setting, we envision a potential rating for every object-user pair. The goal of a recommender system is to predict the unobserved ratings in order to recommend an object that the user is likely to rate highly. A typically overlooked piece is that the combinations are not missing at random. For example, in movie ratings, a relationship between the user ratings and their viewing history is expected, as human nature dictates the user would seek out movies that they anticipate enjoying. We model this informative missingness, and place the recommender system in a shared-variable regression framework which can aid in prediction quality. The second part of the talk deals with a new class of prior distributions for shrinkage regularization in sparse linear regression, particularly the high dimensional case. Instead of placing a prior on the coefficients themselves, we place a prior on the regression R-squared. This is then distributed to the coefficients by decomposing it via a Dirichlet Distribution. We call the new prior R2-D2 in light of its R-Squared Dirichlet Decomposition. Compared to existing shrinkage priors, we show that the R2-D2 prior can simultaneously achieve both high prior concentration at zero, as well as heavier tails. These two properties combine to provide a higher degree of shrinkage on the irrelevant coefficients, along with less bias in estimation of the larger signals.
Interactive theorem proving for mathematicians
15:10 Fri 5 Oct, 2018 :: Napier 208 :: A/Prof Scott Morrison :: Australian National University

Mathematicians use computers to write their proofs (LaTeX), and to do their calculations (Sage, Mathematica, Maple, Matlab, etc, as well as custom code for simulations or searches). However today we rarely use computers to help us to construct and understand proofs. There is a long tradition in computer science of interactive and automatic theorem proving; particularly today these are important tools in engineering correct software, as well as in optimisation and compilation. There have been some notable examples of formalisation of modern mathematics (e.g. the odd order theorem, the Kepler conjecture, and the four-colour theorem). Even in these cases, huge engineering efforts were required to translate the mathematics to a form a computer could understand. Moreover, in most areas of research there is a huge gap between the interests of human mathematicians and the abilities of computer provers. Nevertheless, I think it's time for mathematicians to start getting interested in interactive theorem provers! It's now possible to write proofs, and write tools that help write proofs, in languages which are expressive enough to encompass most of modern mathematics, and ergonomic enough to use for general purpose programming. I'll give an informal introduction to dependent type theory (the logical foundation of many modern theorem provers), some examples of doing mathematics in such a system, and my experiences working with mathematics students in these systems.
Some advances in the formulation of analytical methods for linear and nonlinear dynamics
15:10 Tue 20 Nov, 2018 :: EMG07 :: Dr Vladislav Sorokin :: University of Auckland

In the modern engineering, it is often necessary to solve problems involving strong parametric excitation and (or) strong nonlinearity. Dynamics of micro- and nanoscale electro-mechanical systems, wave propagation in structures made of corrugated composite materials are just examples of those. Numerical methods, although able to predict systems behavior for specific sets of parameters, fail to provide an insight into underlying physics. On the other hand, conventional analytical methods impose severe restrictions on the problem parameters space and (or) on types of the solutions. Thus, the quest for advanced tools to deal with linear and nonlinear structural dynamics still continues, and the lecture is concerned with an advanced formulation of an analytical method. The principal novelty aspect is that the presence of a small parameter in governing equations is not requested, so that dynamic problems involving strong parametric excitation and (or) strong nonlinearity can be considered. Another advantage of the method is that it is free from conventional restrictions on the excitation frequency spectrum and applicable for problems involving combined multiple parametric and (or) direct excitations with incommensurate frequencies, essential for some applications. A use of the method will be illustrated in several examples, including analysis of the effects of corrugation shapes on dispersion relation and frequency band-gaps of structures and dynamics of nonlinear parametric amplifiers.

News matching "Dynamical systems"

ARC Grant successes
Congratulations to Tony Roberts, Charles Pearce, Robert Elliot, Andrew Metcalfe and all their collaborators on their success in the current round of ARC grants. The projects are "Development of innovative technologies for oil production based on the advanced theory of suspension flows in porous media" (Tony Roberts et al.), "Perturbation and approximation methods for linear operators with applications to train control, water resource management and evolution of physical systems" (Charles Pearce et al.), "Risk Measures and Management in Finance and Actuarial Science Under Regime-Switching Models" (Robert Elliott et al.) and "A new flood design methodology for a variable and changing climate" (Andrew Metcalfe et al.) Posted Mon 26 Oct 09.
ARC Grant successes
The School of Mathematical Sciences has again had outstanding success in the ARC Discovery and Linkage Projects schemes. Congratulations to the following staff for their success in the Discovery Project scheme: Prof Nigel Bean, Dr Josh Ross, Prof Phil Pollett, Prof Peter Taylor, New methods for improving active adaptive management in biological systems, $255,000 over 3 years; Dr Josh Ross, New methods for integrating population structure and stochasticity into models of disease dynamics, $248,000 over three years; A/Prof Matt Roughan, Dr Walter Willinger, Internet traffic-matrix synthesis, $290,000 over three years; Prof Patricia Solomon, A/Prof John Moran, Statistical methods for the analysis of critical care data, with application to the Australian and New Zealand Intensive Care Database, $310,000 over 3 years; Prof Mathai Varghese, Prof Peter Bouwknegt, Supersymmetric quantum field theory, topology and duality, $375,000 over 3 years; Prof Peter Taylor, Prof Nigel Bean, Dr Sophie Hautphenne, Dr Mark Fackrell, Dr Malgorzata O'Reilly, Prof Guy Latouche, Advanced matrix-analytic methods with applications, $600,000 over 3 years. Congratulations to the following staff for their success in the Linkage Project scheme: Prof Simon Beecham, Prof Lee White, A/Prof John Boland, Prof Phil Howlett, Dr Yvonne Stokes, Mr John Wells, Paving the way: an experimental approach to the mathematical modelling and design of permeable pavements, $370,000 over 3 years; Dr Amie Albrecht, Prof Phil Howlett, Dr Andrew Metcalfe, Dr Peter Pudney, Prof Roderick Smith, Saving energy on trains - demonstration, evaluation, integration, $540,000 over 3 years Posted Fri 29 Oct 10.
ARC Grant Success
Congratulations to the following staff who were successful in securing funding from the Australian Research Council Discovery Projects Scheme. Associate Professor Finnur Larusson awarded $270,000 for his project Flexibility and symmetry in complex geometry; Dr Thomas Leistner, awarded $303,464 for his project Holonomy groups in Lorentzian geometry, Professor Michael Murray Murray and Dr Daniel Stevenson (Glasgow), awarded $270,000 for their project Bundle gerbes: generalisations and applications; Professor Mathai Varghese, awarded $105,000 for his project Advances in index theory and Prof Anthony Roberts and Professor Ioannis Kevrekidis (Princeton) awarded $330,000 for their project Accurate modelling of large multiscale dynamical systems for engineering and scientific simulation and analysis Posted Tue 8 Nov 11.
AMSI-ANZIAM Lecture Tour - Public Lecture

AMSI-ANZIAM Lecture Tour, Public Lecture

The Role of Embedded Optimisation in Smart Systems and Products
September 23, 6:00pm Horace Lamb Lecture Theatre

Professor Stephen Boyd, Stanford University, Samsung Professor of Engineering, Professor of Electrical Engineering

Further details here

Posted Mon 23 Sep 13.

Publications matching "Dynamical systems"

Publications
A self tuning model for risk estimation
Elliott, Robert; Filinkov, Alexei, Expert Systems with Applications 34 (1692–1697) 2008
Normal form transforms separate slow and fast modes in stochastic dynamical systems
Roberts, Anthony John, Physics Letters A 387 (12–38) 2008
A preliminary approach to structural identifiability of uncontrolled linear switching systems
Whyte, Jason, MABE'07, Gold Coast 17/01/07
Perturbing singular systems and the correlating of uncorrelated random sequences
Pearce, Charles; Allison, Andrew; Abbott, Derek, International Conference on Numerical Analysis and Applied Mathematics, Corfu, Greence 16/09/07
On deterministic identifiability of uncontrolled linear switching systems
Whyte, Jason, WSEAS Transactions on Systems 5 (1028–1035) 2007
Nonlinear dynamics on centre manifolds describing turbulent floods: k-omega model
Georgiev, D; Roberts, Anthony John; Strunin, D, Discrete and Continuous Dynamical Systems Supplement (419–428) 2007
The solution of a free boundary problem related to environmental management systems
Elliott, Robert; Filinkov, Alexei, Stochastic Analysis and Applications 25 (1189–1202) 2007
Guide expansions for the recursive parametric solution of polynomial dynamical systems
Duff, G; Leipnik, R; Pearce, Charles, The ANZIAM Journal 47 (387–396) 2006
Prolongations of geometric overdetermined systems
Branson, T; Cap, A; Eastwood, Michael; Gover, A, International Journal of Mathematics 17 (641–664) 2006
Accurately model the Kuramoto-Sivashinsky dynamics with holistic discretization
MacKenzie, T; Roberts, Anthony John, SIAM Journal on Applied Dynamical Systems 5 (365–402) 2006
An algorithmic estimation scheme for hybrid stochastic systems
Malcolm, William; Elliott, Robert; Dufour, F; Arulampalam, M, The 44th IEEE Conference on Decision and Control and European Control Conference ECC 2005, Seville, Spain 12/12/05
Best approximation of nonlinear systems: Optimal causal feedback models
Torokhti, Anatoli; Howlett, P; Pearce, Charles, The 9th world multi-conference on systemics, cybernetics and informatics, Orlando, Florida, USA 10/07/05
Exact smoothers for discrete-time hybrid stochastic systems
Elliott, Robert; Malcolm, William; Dufour, F, The 44th IEEE Conference on Decision and Control and the European Control Conference, Seville, Spain 12/12/05
New Gaussian mixture state estimation schemes for discrete time hybrid Gauss-Markov systems
Elliott, Robert; Dufour, F; Malcolm, William, The 2005 American Control Conference, Portland, OR, USA 08/06/05
Best causal mathematical models for a nonlinear system
Torokhti, Anatoli; Howlett, P; Pearce, Charles, IEEE Transactions on Circuits and Systems I - regular papers 52 (1013–1020) 2005
Prolongations of linear overdetermined systems on affine and riemannian manifolds
Eastwood, Michael, Circolo Matmeatico di Palermo. Rendiconti 75 (89–108) 2005
Representations via overdetermined systems
Eastwood, Michael, Contemporary Mathematics 368 (201–210) 2005
State and mode estimation for discrete-time jump Markov systems
Elliott, Robert; Dufour, F; Malcolm, William, Siam Journal on Control and Optimization 44 (1081–1104) 2005
Finite-dimensional filtering and control for continuous-time nonlinear systems
Elliott, Robert; Aggoun, L; Benmerzouga, A, Stochastic Analysis and Applications 22 (499–505) 2005
A fundamental solution for linear second-order elliptic systems with variable coefficients
Clements, David, Journal of Engineering Mathematics 49 (209–216) 2004
A sufficient condition for the uniform exponential stability of time-varying systems with noise
Grammel, G; Maizurna, Isna, Nonlinear Analysis-Theory Methods & Applications 56 (951–960) 2004
Macrophage-tumour interactions: In vivo dynamics
Byrne, H; Cox, Stephen; Kelly, C, Discrete and Continuous Dynamical Systems-Series B 4 (81–98) 2004
Robust M-ary detection filters and smoothers for continuous-time jump Markov systems
Elliott, Robert; Malcolm, William, IEEE Transactions on Automatic Control 49 (1046–1055) 2004
Shear dispersion along circular pipes is affected by bends, but the torsion of the pipe is negligible
Roberts, Anthony John, SIAM Journal on Applied Dynamical Systems 3 (433–462) 2004
A philosophy for the modelling of realistic nonlinear systems
Howlett, P; Torokhti, Anatoli; Pearce, Charles, Proceedings of the American Mathematical Society 132 (353–363) 2003
Method of hybrid approximations for modelling of multidimensional nonlinear systems
Torokhti, Anatoli; Howlett, P; Pearce, Charles, Multidimensional Systems and Signal Processing 14 (397–410) 2003
Optimal mathematical models for nonlinear dynamical systems
Torokhti, Anatoli; Howlett, P; Pearce, Charles, Mathematical and Computer Modelling of Dynamical Systems 9 (327–343) 2003
Low-dimensional modelling of dynamical systems applied to some dissipative fluid mechanics
Roberts, Anthony John, chapter in Nonlinear dynamics: from lasers to butterflies (World Scientific Publishing) 257–313, 2003
Martingale methods for analysing single-server queues
Roughan, Matthew; Pearce, Charles, Queueing Systems 41 (205–239) 2002
On-line almost-sure parameter estimation for partially observed discrete-time linear systems with known noise characteristics
Elliott, Robert; Ford, J; Moore, J, International Journal of Adaptive Control and Signal Processing 16 (435–453) 2002
Best approximation of operators in the modeling of nonlinear systems
Torokhti, Anatoli; Howlett, P, IEEE Transactions on Circuits and Systems I - regular papers 49 (1792–1798) 2002
Robust M-ary detection filters for continuous-time jump Markov systems
Elliott, Robert; Malcolm, William, The 40th IEEE Conference on Decision and Control (CDC), Orlando, Florida 04/12/01
Coupled Painlev systems and quartic potentials
Hone, Andrew, Journal of Physics A: Mathematical and Theoretical (Print Edition) 34 (2235–2245) 2001
Hilbert C*-systems for actions of the circle group
Baumgaertel, H; Carey, Alan, Reports on Mathematical Physics 47 (349–361) 2001
On the best quadratic approximation of nonlinear systems
Torokhti, Anatoli; Howlett, P, IEEE Transactions on Circuits and Systems I - regular papers 48 (595–602) 2001
Optimal fixed rank transform of the second degree
Torokhti, Anatoli; Howlett, P, IEEE Transactions on Circuits and Systems II - express briefs 48 (309–315) 2001
State-dependent coupling of quasireversible nodes
Henderson, William; Taylor, Peter, Queueing Systems 37 (163–197) 2001
The modelling and numerical simulation of causal non-linear systems
Howlett, P; Torokhti, Anatoli; Pearce, Charles, Nonlinear Analysis-Theory Methods & Applications 47 (5559–5572) 2001
Bearing-only tracking from a stationary platform
Sworder, D; Boyd, J; Hutchins, R; Elliott, Robert, Asilomar Conference on Signals, Systems and Computers. Conference Record 2 (1428–1432) 2001
OBITUARY Associate Professor W. (Bill) Henderson (1943?2001)
Pearce, Charles; Taylor, Peter; Tuxk, E, Queueing Systems 39 (335–336) 2001
A new perspective on the normalization of invariant measures for loss networks and other product form systems
Bean, Nigel; Stewart, Mark, Mathematical and Computer Modelling 31 (47–54) 2000
Algorithms for second moments in batch-movement queueing systems
Hunt, Emma, Mathematical and Computer Modelling 31 (299–305) 2000
m-systems of polar spaces and maximal arcs in projective planes
Hamilton, N; Quinn, Catherine, Bulletin of the Belgian Mathematical Society-Simon Stevin 7 (237–248) 2000

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