New Frontiers and Advances in Discrete Integrable Systems. Integrable systems boast a long and venerable history, and have such famous members as the Kepler system, the Korteweg-de Vries equation, and the sine-Gordon equation. More recently, interest in integrable systems has expanded to include systems with discrete time, that is, ordinary difference equations (or maps) and integrable partial difference equations. These discrete integrable systems are arguably more fundamental than the continuo ....New Frontiers and Advances in Discrete Integrable Systems. Integrable systems boast a long and venerable history, and have such famous members as the Kepler system, the Korteweg-de Vries equation, and the sine-Gordon equation. More recently, interest in integrable systems has expanded to include systems with discrete time, that is, ordinary difference equations (or maps) and integrable partial difference equations. These discrete integrable systems are arguably more fundamental than the continuous-time ones. Based upon recent breakthroughs this study will combine analysis, geometry, and computer algebra to expand and systematise this new interdisciplinary field of discrete integrable systems.Read moreRead less
Energy efficient sensing, computing and communication. This research will study trade-offs in resource use: bandwidth, power, and computational capacity of systems of sensors such as cameras, radars, and distributed sensor networks based on a statistical mechanical theory of information processing, leading to practical algorithms to optimize resource use in the design of such systems.
Fractional dynamic models for MRI to probe tissue microstructure. This project aims to develop new mathematical tools for mapping tissue microstructural properties via the use of space-time fractional calculus methods. In magnetic resonance imaging, mathematical models and their parameters play a key role in associating information between images and biology, with the overall aim of producing spatially resolved maps of tissue property variations. However, models which can inform on changes in mi ....Fractional dynamic models for MRI to probe tissue microstructure. This project aims to develop new mathematical tools for mapping tissue microstructural properties via the use of space-time fractional calculus methods. In magnetic resonance imaging, mathematical models and their parameters play a key role in associating information between images and biology, with the overall aim of producing spatially resolved maps of tissue property variations. However, models which can inform on changes in microscale tissue properties are lacking. The tools developed by this project will be used to generate new magnetic resonance image based maps to convey information on tissue microstructure changes in the human brain. Additionally, the mathematical tools developed will be transferable to other applications where diffusion and transport in heterogeneous porous media play a role.Read moreRead less
Mathematical and statistical methods for modelling invivo pathogen dynamics. This project aims to develop mathematical models and Bayesian statistical methods that better capture how natural defence responses and drugs help control infection. When viruses (e.g. influenza) or parasites (e.g. malaria) invade the human body, they begin to replicate. To date, only simple mathematical models have been developed to capture these processes, and these models are not well formulated. This project will im ....Mathematical and statistical methods for modelling invivo pathogen dynamics. This project aims to develop mathematical models and Bayesian statistical methods that better capture how natural defence responses and drugs help control infection. When viruses (e.g. influenza) or parasites (e.g. malaria) invade the human body, they begin to replicate. To date, only simple mathematical models have been developed to capture these processes, and these models are not well formulated. This project will improve biomathematics and biostatistical algorithms for pathogen dynamics and is ultimately expected to benefit public health and clinical research aimed at alleviating the effect of infectious diseases on human health.Read moreRead less
How calcium makes the heart grow. This project aims to develop a mathematical model of calcium signalling in heart cells to understand how calcium makes the heart grow. Our hearts grow to adapt to long-term changes, such as during development and in pregnancy or heart disease. Biochemical reactions involving calcium control the growth of heart cells and heart cells also use calcium signalling to trigger contraction with each beat. How calcium controls the heartbeat and regulates cell growth is u ....How calcium makes the heart grow. This project aims to develop a mathematical model of calcium signalling in heart cells to understand how calcium makes the heart grow. Our hearts grow to adapt to long-term changes, such as during development and in pregnancy or heart disease. Biochemical reactions involving calcium control the growth of heart cells and heart cells also use calcium signalling to trigger contraction with each beat. How calcium controls the heartbeat and regulates cell growth is unknown. This project will develop a new mathematical model of calcium signalling in heart cells to understand important cellular adaption processes. This knowledge will lead to the ability to independently control cellular pathways mediated by calcium, opening new avenues in biotechnology and biomedicine.Read moreRead less
Stability and performance analysis for uncertain time-varying interconnected systems. This project will deliver new mathematical tools for the study of interconnected systems. This will broaden the scope for rigorously certifying engineered systems and understanding interactions in the natural world. Potential application areas include telecommunications, transport, water distribution, smart-grids, neuroscience and cell biology.
The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - t ....The Sakai scheme-Askey table correspondence, analogues of isomonodromy and determinantal point processes. The Australian mathematical sciences enjoys two research groups with active interests on Painleve equations in applied mathematics which are able to address difficult problems. Such a problem is to give a formulation of Sakai's 2001 classification of the Painleve equations in a form most suitable for applications. For this links will be made with a seemingly distinct area of mathematics - the Askey table from the theory of hypergeometric orthogonal polynomials. A number of tractable PhD projects are suggested by this proposal.Read moreRead less
Functional state observers for large-scale interconnected systems. This project will produce conceptual advances with new design rules to develop robust and efficient functional state observers for interconnected systems. The outcomes will advance the theory of functional observers and improve the operation, efficiency and performance of critical infrastructure such as power grids, water and traffic networks.
Optimising progress towards elimination of malaria. The project aims to advance mathematical knowledge by developing novel tools appropriate for modelling disease elimination. We will apply these new mathematical tools to the significant problem of malaria elimination in Vietnam. The expected outcomes are new tools for modelling disease elimination on a fine spatial resolution with heterogeneities in individual patient characteristics, calibrating models to household level data on disease transm ....Optimising progress towards elimination of malaria. The project aims to advance mathematical knowledge by developing novel tools appropriate for modelling disease elimination. We will apply these new mathematical tools to the significant problem of malaria elimination in Vietnam. The expected outcomes are new tools for modelling disease elimination on a fine spatial resolution with heterogeneities in individual patient characteristics, calibrating models to household level data on disease transmission and designing intervention strategies for maximum effect on disease transmission. The innovative combination of modelling, inference and optimisation ensures that the mathematical methods developed will be broadly applicable to modelling elimination strategies for other infectious diseases.
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Optimisation Geometry. Signal processing is an enabling technology for multi-billion dollar industries including broadband connections, medical imaging devices and automotive collision avoidance systems. A key component is real-time optimisation. While various aspects of real-time optimisation have been studied extensively, especially under the banner of adaptive signal processing, the general (non-adaptive) problem will be studied here from a geometric viewpoint. Morse theory and other tools fr ....Optimisation Geometry. Signal processing is an enabling technology for multi-billion dollar industries including broadband connections, medical imaging devices and automotive collision avoidance systems. A key component is real-time optimisation. While various aspects of real-time optimisation have been studied extensively, especially under the banner of adaptive signal processing, the general (non-adaptive) problem will be studied here from a geometric viewpoint. Morse theory and other tools from differential topology will be used to understand the inherent complexity of real-time optimisation problems and to develop novel numerical algorithms for classes of real-time optimisation problems having widespread applicability.Read moreRead less