Monash University > Science > Mathematical Sciences > Research

Research snippets for 2011

Prof. Paul Cally

Plucking the Sun

Alfvén waves are like the to-and-fro oscillations of a plucked harp string, except that instead of strings they live on magnetic field lines in ionized gases (plasmas) found everywhere in the universe. Our Sun in particular is a magnetic star: Alfvén waves are observed ubiquitously in its outer atmosphere, the corona, and detected by spacecraft in the solar wind as it rushes outward past the Earth at up to 750 km/s. These waves are believed to be crucial in accelerating the wind to these speeds, and may also contribute to heating the corona to over 1 million degrees. But where are Alfvén waves excited (where is the harpist)?

Recently, Professor Paul Cally and PhD student Shelley Hansen have shown that Alfvén waves can be made easily by a process called mode conversion from another sort of wave, the fast magneto-acoustic wave, that we know inhabits the lower solar atmosphere, the chromosphere. This has been confirmed using supercomputer simulations by Dr Elena Khomenko of the Instituto de Astrofísica de Canarias and Prof Cally. In their latest paper, Cally and Hansen have also shown that Alfvén waves made this way in the upper chromosphere are much better able to breach the huge hundred-fold temperature jump between chromosphere and corona that would normally reflect the waves back downward.

Reference: Click here. (to appear shortly in Astrophys J)

Schematic diagram showing how Alfv'en waves are created near the top of a fast wave's path before it reflects back into the Sun's interior. Alfv'en waves, like waves on harp strings, then must follow the magnetic field lines (shown in blue), either upwards or downwards depending on how the field is oriented. Here cs is the sound speed and vA is the Alfv'en speed, i.e., the speed at which the Alfv'en waves travel.

Dr Dietmar Dommenget

A climate model for everyone

The future climate change projections of the UN IPCC report are based on highly complex climate model simulations, which give a distinct global warming pattern, with the strongest warming in arctic sea in winter time (but not in summer), an equilibrium stronger warming over land than over oceans and an inter-hemispheric warming gradient. While these simulations are the most important tools of the IPCC predictions, the conceptual understanding of these predicted structures of climate change are very difficult to reach if only based on these highly complex simulations.

In this study presented here a very simple climate model based on strongly simplified physical processes is introduced, which is capable of simulating the main characteristics of global warming on a realistic global map. The model shall give a bridge between the theoretical 1-dimensional energy balance models and the fully complex state of the art climate models. Unlike an IPCC-class climate model that has about 1million lines of program code, that even experts in the field can hardly understand, the simple climate model introduced here has only 700 lines of code that to the largest part can be understood by high school students. The program runs on standard PC computer computing globally resolved climate simulation of an IPCC climate change scenario in 1 min. (100,000yrs simulation per day). For comparison an IPCC-class climate model runs only on supercomputers and would take about a month for the same IPCC climate change scenario (5yrs simulation per day).

The simple model allows you to take the climate system apart to understand how different elements in the climate system, such as atmospheric water vapor, winds, clouds, deep ocean heat up-take and topography, interact to result into the strong warming response to a doubling of the atmospheric CO2 concentrations. It is therefore a very good tool to teach the general public, but also the climate experts, about the fundamentals of climate change.

Reference: Dommenget, D. and J. Floeter, 2011: Conceptual Understanding of Climate Change with a Globally Resolved Energy Balance Mode. Climate Dynamics, 37, 11, 2143-2165.

Dr Alina Donea

Is My Halo Slipping? Angels on the Sun

Have you seen pictures of angels with halos? The angel's halo is a ring of light that surrounds the head in early religious art. Well, solar active regions on the sun's surface have such rings or halos, but these are made of sound waves and not light. We can see them using the modern science of local heliosesmology, invented in the 1980s. The reason for the existence of these high-frequency acoustic halos (5-7 mHz) has not yet been elucidated. We do not know why they are seen only around complex active regions and not around simple sunspots, why they do not appear in the lower-frequency acoustic regime, why they are spatially so extended, or why they present small-scale acoustic enhancements sometimes called "acoustic beads" or "glories"?. The presence of the acoustic halos has made us reassess the definition of a quiet sun region, especially around active regions. Active regions seem to enjoy noisy environments. They are surrounded by thousands of acoustic signals travelling through them, beneath and above, all tangled with strong magnetic field lines. There have been several models to explain this effect as a result of wave conversion and/or scattering in magnetic structures. More investigations of the acoustic excitation mechanisms are underway lead by Dr Alina Donea with her students Marie Newington and Chris Hanson using mainly state-of-the-art solar data from the Solar Dynamic Observatory launched into space in 2010.

Seismic egression-power maps (a) of AR 8998 (May 18, 2000) in a 1 mHz band centered at 5 mHz, integrated over 24-hr interval (b) of AR 8179 (March 15, 1998). Reference arrow heads indicate the locations of the beading of small-scale elements of enhanced seismic emission which comprise the 5 mHz solar acoustic glories. The maps are normalised to the quiet sun areas.

Dr Tim Garoni

Chaotic traffic lights produce more reliable travel times

Perhaps the most obvious way of trying to increase the capacity of a road network is to simply build more roads. However, this approach involves significant financial and environmental costs, and is often simply not feasible in built up areas. By contrast, by designing novel algorithms which more efficiently control the traffic signals on the existing network, we can reduce traffic congestion essentially "for free". In a collaboration with traffic engineers at VicRoads which began in 2008, our group has developed a new way to model traffic flow on urban road networks, based on the theory of cellular automata. A cellular automaton is a model which is discrete in time, space, and state variables, whose dynamical rules are local. Our model was specifically designed to be able to quickly and easily test any conceivable algorithm for controlling traffic signals.

Using this model, we have recently investigated the benefits that can result from using certain highly-adaptive algorithms to control traffic signals, referred to as "self-organizing traffic lights". Such algorithms allow traffic signals to alter their behaviour in real time, based on current traffic conditions. Unlike traditional systems, they are not constrained to operate in a cyclic manner. In congested conditions, the resulting traffic signals tend to behave seemingly randomly. However, although the traffic lights become unpredictable, our modelling suggests that the variability in vehicle travel times is significantly reduced, compared with the networks using the type of predictable traffic signals currently in use. In effect, our results suggest that if drivers were willing to live with unpredictable traffic lights, then the benefit would be far more reliable travel times.

Reference: Jan de Gier, Timothy M. Garoni, and Omar Rojas, Traffic flow on realistic road networks with adaptive traffic lights, J. Stat. Mech. P04008 (2011).

Daniel Horsley

Smart compression methods

A digital camera taking a photograph, a mobile phone transmitting a voice, an MRI machine scanning a brain: these are all examples of signal sampling. In 2004 a revolutionary new approach to signal sampling and reconstruction, called compressed sensing, was introduced by several mathematicians including Fields Medallist Terry Tao. Compressed sensing exploits the "compressibility" of signals occurring in most applications to enable accurate reconstruction from dramatically fewer measurements than dictated by traditional thinking. It has generated an explosion of interest over the past few years, within both the mathematical and electrical engineering research communities.

With Charles Colbourn and Violet Syrotiuk of Arizona State University, I have been investigating a new approach to compressed sensing using combinatorial arrays called hash families. This work is still in its infancy, but we have already demonstrated that hash families can be used to create larger sensing matrices (vital objects in the sensing process) from smaller ones, and that we can exploit the structure that we build into the newly-created matrices in various interesting ways.

Dr Jules Kajtar and Prof. Joe Monaghan

Speed and efficiency studies of swimming fish

Why do marine animals swim with different styles? Have they evolved to swim with maximum efficiency? Can we use this information to improve the swimming techniques of Olympic athletes? We attempt to answer these and other related questions by solving the governing set of mathematical equations using computer programs. Our simulations are of basic swimming devices (see figure for an example), but nevertheless we can draw profound conclusions. Real fish swim by changing their shape: muscles move parts of their body and fins such that they push against the water and thus swim in a particular direction. We prescribe similar changes of shape for our model swimmer. The fluid is modelled using a numerical technique known as "Smoothed Particle Hydrodynamics", whereby the fluid is represented as a large set of interacting particles.

A snapshot of the model swimmer in motion. When it moves near a free surface, it generates large waves. Somewhat surprisingly, we have found that its motion is more efficient near the surface rather than deep down under water.

We have investigated various swimming styles with which the swimmer can move forwards, backwards, or turn left or right. We have found favourable agreement between the swimming speed of our model swimmer and a real swimming leech observed in the laboratory. Our most recent work studied swimming speeds and energy usage near free surfaces. We found that as the swimmer moves closer to the surface, its speed is reduced. This can be related to the fact that near the surface, larger waves are produced, and the swimmer pushes against the fluid less effectively. However, more surprisingly, we found that the energy used by the swimmer near the surface is also reduced. So although the swimmer does not swim quite as fast near the surface, it expends less energy in travelling the same distance. This may be part of the reason why marine animals, such as dolphins, prefer to swim near the surface.

Reference: Click here.

Dr. Jonathan Keith

Eradicating the Red Fire Ant

The red imported fire ant is considered one of the world's 100 worst invasive species. Fire ants were first discovered in Brisbane, Australia in 2001 and a massive eradication program began immediately. Eradication in Brisbane is thought possible, although nests are still being discovered. To design an effective eradication strategy, it is imperative to understand the spatio-temporal patterns of nest spread.

In recent work, I developed a new Bayesian method that estimates parameters of invasion dynamics including local and long-distance dispersal, reproduction, and genetic divergence. The method was applied to data collected during the past ten years of the Brisbane eradication programme, and provides a detailed picture of interacting processes of nest reproduction, spatial dispersion, genetic divergence, search and eradication, at fine spatial and temporal scales. A major intended application of the model is to provide information for simulating alternative search and eradication strategies. The method is applicable to other invasive species.

Prof. John Lattanzio

Computer Crash helps Understand Death of Stars

We know that computers tend to have a mind of their own. But what do you do when a trusted program suddenly stops working for no apparent reason! This happened to the SINS group, working on the late evolution of red-giants. During the calculations the code would get to a certain point and then refuse to go further. It told us the gas pressure was negative and it did not know what to do. That is fair enough - we did not know either! But we did know that a negative gas pressure was not allowed.

So we went back to pen and paper to see if this problem was something in the computer code, or perhaps a numerical instability, or something real that might happen in real stars. Postdoctoral research Herbert Lau found that the energy generation was so high that it was driving a radiative bubble in the gas, where the local luminosity was so high (ie above the Eddington limit, for those who have done ASP3012) that there was no longer a hydrostatic solution. No wonder the computer could not find one!

So with PhD student Carolyn Doherty and Spanish collaborator Dr Pilar Gil Pons we have been trying to determine what the real star does! It seems that it will result in the envelope being ejected in one big burp! And it turns out that there are some stars that look just like these predictions...

Note that this research was initiated by the computer failing to work suddenly! You just never know where research will lead you...

Dr Maria Lugaro

Ancient stars tell us about the origin of gold

The origin of the elements heavier than iron, such as silver, gold, lead, and uranium, is one of the mysteries of contemporary science. Up to today we have divided these elements in two main groups based on the nuclear reaction processes that produce them and the astrophysical sites where these processes can occur. The first group, including lead, is produced by nuclei capturing free neutrons in giant stars over times of tens of thousands of years (the "slow" neutron-capture process). The second group, including gold, is produced by nuclei capturing free neutrons during supernova explosions and the merging of two neutron stars over times of less than a second (the "rapid" neutron-capture process).

However, this scenario may be too simplified. We demonstrate that the observed chemical composition of the oldest stars in the Milky Way, born at the dawn of the Universe, cannot be explained by either of the two processes above. These ancient stars are telling us that there must be a different process during which the two groups of elements are produced together by a "slow/rapid" neutron-capture process, where the two neutron-capture processes can intermingle with each other. Now, we need to discover in which possible ways and astrophysical objects this can happen.

Reference: Lugaro, M., Karakas, A.I, Stancliffe, R.J., Rijs, C. 2011 The Astrophysical Journal, accepted.

Dr Gregory Markowsky

Using the motion of particles to sum series

A light particle in a fluid undergoes a random motion due to being continuous bombarded in all directions by surrounding molecules. This motion is commonly referred to as Brownian motion. This type of motion has interested mathematicians for a number of decades now and has found application in many places, ranging from models of the motion of black holes to models of the stock market.

A conformal map is a function which maps a region in the xy-plane into a different region in the plane and which preserves angles(there are a few other technical requirements). That is, a pair of curves which meet at a point in the initial region are mapped to a new pair of curves which meet at the same angle in the image region.

There are a number of connections between these conformal maps in the plane and Brownian motion, and recently I have been studying one of them. We can start a Brownian motion in the plane and measure the average amount of time it takes to leave a region. It turns out that this average time can also be measured with respect to a particular series which measures the "energy" of a conformal map onto the region. If we equate these two quantities we obtain a value of the sum of the series. In this way, the motion of particles in fluid can be use to obtain the sum of a series. An example of the type of series that this method works well with is something like

1 + 1/4 + 1/9 + 1/16 + 1/25 + ...

That is, the sum of the reciprocal of the squares. This was famously found by Euler several hundred years ago to be equal to the square of pi over 6. I have been able to sum this and a few more difficult series using this method, and hope to find many more which this method can be applied to.

A/Prof. Burkard Polster

What is the best way to draw a cube?

Is there still something new and beautiful to be discovered about as common an object as a cube? Absolutely!

Together with my colleague Marty Ross I recently pondered the different ways to draw a cube. We managed to show that among all the different orthographic drawings* of a wireframe cube the one shown in the picture is best in the sense that in it the eight corners are as widely spaced as possible.

We also showed that this drawing is characterised among the orthographic drawings by the fact that the three orange distances highlighted on the right are of equal length.

*The different orthographic projections/drawings of a cubical wireframe are simply its possible shadows when the sun is directly overhead. Orthographic drawings are used widely in technical drawing.

Dr Daniel Price

Jets from a newborn star

Understanding star formation is key to understanding the Universe, since stars are the basic building blocks from which galaxies and star clusters are made. Also, the birth of stars is inseparable from the problem of how planets are formed. Our best insights into the star formation process arise from peering into nearby molecular clouds -- the "maternity wards" of the Milky Way -- with instruments like the Spitzer, Herschel and Hubble Space Telescopes, where newborn stars can be seen still embedded in their dusty placental cocoons. Observations of this stage reveal a plethora of rich phenomena associated with starbirth, one of the most beautiful of which is the production of narrow jets emanating from swirling discs of gas surrounding the forming star.

We have been performing computer simulations of the star formation process, including the effects of magnetic fields in the gas as it collapses to form a star (magnetic fields are thought to be key to producing jets). With recent improvements to our simulation methods, we indeed find narrow, fast jets, very similar to their observed counterparts in real star forming clouds. These are produced in our simulations without any "faking it", arising naturally from solving the equations governing the motion of the gas including the effects of magnetism. As in observations, simulated jets are a beautiful phenomena (see image, showing rendering of gas column density from one of our simulations, the newborn star is in the centre) and even richer for our understanding of the physical mechanism behind their production. From Price, Tricco and Bate (2012), paper submitted to Monthly Notices of the Royal Astronomical Society.

Prof. Michael Reeder

Very High Resolution Simulations of the Weather on Black Saturday

Although bushfires are a common and integral part of our natural environment, from time to time they burn tragically into our history. As underscored by Black Saturday, they are especially dangerous in those areas where Australia's large urban communities and rural townships merge with the surrounding bush. Therefore, predicting the behaviour and spread of bushfires, especially in complex terrain and in extreme and under rapidly changing weather conditions, is of great practical benefit. Moreover, it is clear that the interplay between the atmosphere and the fire plays a fundamental role in determining the behaviour and spread of bushfires and understanding and predicting this interplay is critical.

In a study with two colleagues from University of Melbourne, the extreme weather conditions on Black Saturday were simulated at very high resolution with a numerical model, and the features of relevance to the behaviour of the fires investigated. These features include enhanced downslope winds, organized vortical rolls in the boundary layer and, of course, the passage of an extreme cold front. In addition, two nocturnal bores were generated by the cold front, one of which was found rapidly invigorate the Beechworth fire in the evening of Black Saturday.

A/Prof. Steven Siems

Precipitation in wintertime storms across Southeast Australia, Tasmania and the Southern Ocean

New satellite technologies have noted that much of the cloud cover over the Southern Ocean is composed of supercooled liquid water, i.e. cloud droplets at temperatures below freezing but remain as liquid instead of ice. These observations have implications not only for the climate over the Southern Ocean, but also for the rate at which these clouds form precipitation. These clouds also are a major component of the wintertime storms over Tasmania, the Victorian Alps and the Snowy Mountains.

Using a combination of satellite observations, field observations and numerical simulations we are working to better understand the development of precipitation in these systems with the intent of being better able to simulate and predict them. Better predictions will lead to better water management throughout the region.

Prof. Kate Smith-Miles

Understanding why some optimization problems are harder than others

Optimization problems like university timetabling or graph coloring can be easy or hard for various algorithms to solve depending on certain statistical properties of the problem instances. For example, if we have lots of lecture theatres, lots of time slots, and lots of lecturers, then it is easy to find a clash-free timetable for all students. But an instance of the problem becomes harder when there is less slack or degrees of freedom. How various algorithms cope with harder instances is still very unclear, and we need to develop new methods to understand and visualize the strengths and weaknesses of optimization algorithms - otherwise we are left with trial and error algorithm choice, or adopting an algorithm that may not be well suited to the properties of a given instance.

We have developed new methods to visualize instances of optimization problems in a high-dimensional space that summarizes instances based on features indicative of difficulty. The performance of optimization algorithms can then be viewed in this space, and the boundaries between strong and weak performance can be measured. This tells us which types of instances are ideally suited to be solved by certain algorithms. We have applied this approach to optimization problems like timetabling, graph coloring, travelling salesman problem, job scheduling, and logistics problems.

References: Smith-Miles, K. and van Hemert, J. "Discovering the Suitability of Optimisation Algorithms by Learning from Evolved Instances", Annals of Mathematics and Artificial Intelligence, DOI 10.1007/s10472-011-9230-5, published online 19th April 2011.

A/Prof. Tianhai Tian

Mathematical modeling of complex biological networks

A fundamental challenge facing researchers and clinicians is that cancers are inherently robust biological systems, able to survive, adapt and proliferate despite the perturbations resulting from anticancer drugs. Without a thorough understanding of the principles of tumor robustness, strategies to overcome therapy resistance are unlikely to be found. Degeneracy describes the ability of structurally distinct system components (e.g. proteins, pathways, cells, organisms) to be conditionally interchangeable in their contribution to system traits and it has been broadly implicated in the robustness and evolvability of complex biological systems. We focused on one of the most important mechanisms underpinning tumor robustness and degeneracy, the cellular heterogeneity that is the hallmark of most solid tumors. Based on a combination of computational, experimental and clinical studies we argued that stochastic noise was an underlying cause of tumor heterogeneity and particularly degeneracy. Drawing from a number of recent data sets, we proposed an integrative model for the evolution of therapy resistance, and discussed recent computational studies that propose new therapeutic strategies aimed at defeating the adaptable cancer phenotype.

It has been estimated that between 350 and 400 million people around the world are infected chronically with hepatitis B virus (HBV). Primarily because of the HBV's significance as a global public health threat, the virus and its associated disease have attracted considerable attentions from a wide range of researchers including mathematicians and theoretical biologists. In the last decade, mathematical model has become an effective approach to provide insights on the long-term dynamics of the HBV in liver cells because chronic infection may last decades. To study the effect of different treatment schemes, we proposed a mathematical model to study the transmission dynamics of HBV treated with impulsive releasing immune factor. Aimed at investigating the effect of the periodic applications of immune factors in treating HBV disease, we investigate the existence and sufficient conditions for the permanence of HBV. Analysis results indicated that a short releasing period of the immune factor or a proper pulse releasing quantity leads to the eradication of the HBV.

Reference: Click here.