A new asymptotic toolbox for nonlinear discrete systems and particle chains. This project aims to pioneer a mathematical toolbox of new asymptotic techniques for discrete systems driven by vanishingly small influences. The purpose of these techniques is to permit the asymptotic study of discrete problems in which significant effects originate due to subtle causes that are invisible to existing asymptotic methods. Discrete systems play a significant role in modern applied mathematics, and it is v ....A new asymptotic toolbox for nonlinear discrete systems and particle chains. This project aims to pioneer a mathematical toolbox of new asymptotic techniques for discrete systems driven by vanishingly small influences. The purpose of these techniques is to permit the asymptotic study of discrete problems in which significant effects originate due to subtle causes that are invisible to existing asymptotic methods. Discrete systems play a significant role in modern applied mathematics, and it is vital that mathematical tools be designed in order to explore their behaviour. The aim of this project is to open new pathways for resolving open scientific problems, providing benefits such as understanding the energy dissipation of particle chains and granular lattices contained in small-scale technological components.Read moreRead less
High Dimensional Computation and Uncertainty. This project aims to establish powerful computational methods for high-dimensional problems - methods that are rigorous, and carefully tailored to specific applications, from physics, environment, manufacturing and finance, and often driven by uncertainty. The project will generate new knowledge in the area of high-dimensional computation, and develop technological innovations in key areas of science and industry. Expected outcomes include improved c ....High Dimensional Computation and Uncertainty. This project aims to establish powerful computational methods for high-dimensional problems - methods that are rigorous, and carefully tailored to specific applications, from physics, environment, manufacturing and finance, and often driven by uncertainty. The project will generate new knowledge in the area of high-dimensional computation, and develop technological innovations in key areas of science and industry. Expected outcomes include improved control of uncertainty in industry, enhanced international and interdisciplinary collaborations, and significant publications and presentations in international forums. The technological advancements will help boost Australia's position as a world leader in innovation.Read moreRead less
An optimisation-based framework for non-classical Chebyshev approximation. This project aims to solve open mathematical problems in multivariate and piecewise polynomial approximations, two directions that correspond to fundamental obstacles to extending classical approximation results. Through an innovative combination of optimisation and algebraic technique, the project intends to develop foundations for new results in approximation theory, and new insights into other areas of mathematics, mos ....An optimisation-based framework for non-classical Chebyshev approximation. This project aims to solve open mathematical problems in multivariate and piecewise polynomial approximations, two directions that correspond to fundamental obstacles to extending classical approximation results. Through an innovative combination of optimisation and algebraic technique, the project intends to develop foundations for new results in approximation theory, and new insights into other areas of mathematics, most notably optimisation. The techniques and methods developed should also have significant benefits in the many disciplines where approximation problems appear, such as engineering, physics or data mining. The research outputs resulting from this project will be used in a wide range of fields to help implement programs, policies and improve decision making.Read moreRead less
Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elus ....Geometric analysis of nonlinear systems. Modern science derives its power from mathematics. The project aims to capture, identify and describe pivotal, transcendental solutions of nonlinear systems that are universal in science, in the sense that they always arise as mathematical models under certain physical limits. The project expects to produce new mathematical methods to describe such functions by using a newly discovered geometric framework. Expected outcomes include the description of elusive solutions of discrete and higher-dimensional nonlinear systems. This should provide significant benefits, such as new mathematical knowledge, innovative techniques, enhanced scientific capacity in Australia.Read moreRead less