A new generation of fractals: theory, computation, and applications particularly to digital imaging. The project develops the mathematical and algorithmic foundations of superfractals and applies these results to a number of different areas, including in particular, digital imaging. For example, the ``third generation'' of mobile communications (3G), combines wireless mobile technology with high data transmission capacities. Currently the requirement for extensive bandwidth is a problem for e ....A new generation of fractals: theory, computation, and applications particularly to digital imaging. The project develops the mathematical and algorithmic foundations of superfractals and applies these results to a number of different areas, including in particular, digital imaging. For example, the ``third generation'' of mobile communications (3G), combines wireless mobile technology with high data transmission capacities. Currently the requirement for extensive bandwidth is a problem for efficient use. Superfractals and the associated colouring algorithm could be used to develop a new system to produce synthetic content for wireless devices that would require only low bandwidth.Read moreRead less
Adaptiveness of self-organised decision making. Complex systems are an important international research focus in many disciplines, and their engineering applications are plentiful. The new mathematical approach developed by this project will enable different disciplines for the first time to communicate using a common formal framework. This will open the path to a generalized understanding of self-organized systems in dynamic environments. Creating the tools for a unified interdisciplinary a ....Adaptiveness of self-organised decision making. Complex systems are an important international research focus in many disciplines, and their engineering applications are plentiful. The new mathematical approach developed by this project will enable different disciplines for the first time to communicate using a common formal framework. This will open the path to a generalized understanding of self-organized systems in dynamic environments. Creating the tools for a unified interdisciplinary approach will allow Australia to gain a stronger position in biomimetic engineering and to take a lead in international research on collective behavior.
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Developing Sophisticated e-Business Automation. This project will provide theoretical foundations and a practical platform for developing advanced e-business automation technology. As such, it will significantly enhance Australia's leading role in the cutting edge research on e-business automation. By applying the new methodology and technology, Australian IT industries will be able to develop highly efficient e-market application systems, which will be financially beneficial to most organizatio ....Developing Sophisticated e-Business Automation. This project will provide theoretical foundations and a practical platform for developing advanced e-business automation technology. As such, it will significantly enhance Australia's leading role in the cutting edge research on e-business automation. By applying the new methodology and technology, Australian IT industries will be able to develop highly efficient e-market application systems, which will be financially beneficial to most organizations as Australia business and institutions are moving towards a more electronically oriented future. With a strong research team and collaborative research training environment, this project will further promote Australia's international reputation as a leader in Computing and IT research.Read moreRead less
3D Image segmentation and shape characterisation driven by topological persistence. Tomographic imaging is emerging as a new tool to help tackle a remarkable array of scientific challenges. What distinguishes healthy bone from that of osteoporosis sufferers? How does groundwater contamination spread? Why is a macadamia nut so hard to crack? What causes the iridescence in a butterfly wing? These are just a few of the questions being answered at tomographic facilities in Australia alone. By co ....3D Image segmentation and shape characterisation driven by topological persistence. Tomographic imaging is emerging as a new tool to help tackle a remarkable array of scientific challenges. What distinguishes healthy bone from that of osteoporosis sufferers? How does groundwater contamination spread? Why is a macadamia nut so hard to crack? What causes the iridescence in a butterfly wing? These are just a few of the questions being answered at tomographic facilities in Australia alone. By combining sophisticated mathematics with cutting edge image-processing algorithms, this project will yield a new class of topology driven image analysis techniques that will improve the accuracy and reliability of predictions made from tomographic images.Read moreRead less
Automated Determination of the Pose of a Human from Visual Information - Markerless 3D Pose Recovery of Humans from Videos. The development of 3D human pose recovery has been sought by computer vision researchers for many years. Our results will, firstly, have benefit for Australia's standing in the international computer vision community. Over time, the research outcomes will be developed into a software product for rehabilitation analysis by recognizing discrepancies between the walking pat ....Automated Determination of the Pose of a Human from Visual Information - Markerless 3D Pose Recovery of Humans from Videos. The development of 3D human pose recovery has been sought by computer vision researchers for many years. Our results will, firstly, have benefit for Australia's standing in the international computer vision community. Over time, the research outcomes will be developed into a software product for rehabilitation analysis by recognizing discrepancies between the walking patterns of healthy individuals and those with abnormalities as a result of accidents or diseases. The Australian economy will benefit by the reduction in the lifetime cost of injuries. This software will also provide benefits to the movie animation, computer games industry, and the training of athletes.Read moreRead less
Group orbits in garmonic analysis and ergodic theory. Researchers from many areas need a type of mathematical analysis which involves the behaviour of a system - which may be a set of data points - under repeated application of some operation or group of operations. The structures arising from this kind of process are known as group orbits. The project gives information about their nature. Two major types of orbits are considered, coming from actions of discrete groups on measure spaces, and fro ....Group orbits in garmonic analysis and ergodic theory. Researchers from many areas need a type of mathematical analysis which involves the behaviour of a system - which may be a set of data points - under repeated application of some operation or group of operations. The structures arising from this kind of process are known as group orbits. The project gives information about their nature. Two major types of orbits are considered, coming from actions of discrete groups on measure spaces, and from smooth actions of Lie groups on manifolds, where powerful geometric methods are available. The project will yield new understandings of entropy, and new approaches to Fourier analysis.Read moreRead less
Ergodic theory and number theory. Recent advances in the theory of measured dynamical systems investigated by the proponents include new versions of entropy, and the study of spectral theory for non-singular systems. These will be further developed in this joint project with the French CNRS. The results are expected to have interesting applications in physics and number theory.
Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expe ....Operator Integrals and Derivatives. The project is a contribution to the study of non-commutative differential and integral calculus. The novelty of the present project lies in the study of smoothness properties of functions whose domains and ranges are spaces of unbounded, non-commuting operators on some Hilbert space. Our general approach will be based on a detailed investigation of properties of double operator integrals, which permit smoothness estimates of operator-functions. It can be expected that the new techniques generated will find further application in areas of mathematical physics and non-commutative geometry related to quantized calculus.
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Entropy and maximal entropy in Markov systems. Entropy is a measure of how well-ordered a system is: chaotic systems have high entropy. Two approaches to entropy are available, via the limiting behaviour of the orbits of points, which yields topological entropy, and via the behaviour of the distributions of measures of partitions, yielding measure-theoretic entropy. The topological entropy is the least upper bound of entropies of all possible measures. We study when there is a measure which real ....Entropy and maximal entropy in Markov systems. Entropy is a measure of how well-ordered a system is: chaotic systems have high entropy. Two approaches to entropy are available, via the limiting behaviour of the orbits of points, which yields topological entropy, and via the behaviour of the distributions of measures of partitions, yielding measure-theoretic entropy. The topological entropy is the least upper bound of entropies of all possible measures. We study when there is a measure which realises this bound, describing the structure of such systems via Markov and Bratteli diagrams. Our methods will be applied to new versions of entropy for non-singular systems. This will assist in the description of chaotic behaviour.Read moreRead less
Symmetries in analysis. Technical research is like an iceberg. The 10% you see in applications is supported by 90% hidden, long-term, sometimes abstruse or theoretical-sounding work. The area of mathematical analysis has, for over 200 years, proved its worth as part of the unseen 90%, giving us such important tools as Fourier analysis, statistical mechanics and quantum mechanics. Australia is known as a world leader in mathematical analysis, and it is important for the country to maintain that e ....Symmetries in analysis. Technical research is like an iceberg. The 10% you see in applications is supported by 90% hidden, long-term, sometimes abstruse or theoretical-sounding work. The area of mathematical analysis has, for over 200 years, proved its worth as part of the unseen 90%, giving us such important tools as Fourier analysis, statistical mechanics and quantum mechanics. Australia is known as a world leader in mathematical analysis, and it is important for the country to maintain that edge in a number of key disciplines, so we can continue to participate in global technological advance. The project has an international focus which will enable that to happen. It will also provide training for the next generation of mathematicians. Read moreRead less