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Time scales calculus is a relatively new field of analysis with the goal of unifying discreet and continuum forms of calculus. It considers functions whose domain is a closed subset of the real numbers, and a “derivative” with the property that if the domain was all of the real numbers it is consistent with the normal derivative. But when the domain is the integers it is consistent with the difference operator seen in discrete analysis. Fractional calculus is the study of non-integer orders of integration and differentiation. In recent work the study of non-integer orders of the difference operators has gained momentum. This talk will investigate the open problem of defining fractional calculus on time scales, such that it unifies the definitions in the literature. No prior knowledge of time scales calculus or fractional calculus will be assumed in the talk.
For More Information:contact Guoqi Qian, email: g.qian@ms.unimelb.edu.au
This talk gives a brief statement of the general approach through posterior likelihoods, set out in detail in my 2010 book Statistical Inference: an Integrated Bayesian/Likelihood Approach (Chapman and Hall/CRC Press). The idea is illustrated by a very simple example of a court case involving a possibly increased risk of encephalitis following MMR vaccination. A complex example of mixture modelling of the recession velocities of galaxies illustrates the idea of stochastic ordering of likelihood or deviance distributions, and resolves the confusing inconsistencies of previous Bayesian analyses.
For More Information:contact Guoqi Qian, g.qian@ms.unimelb.edu.au
Configuration spaces of points in \(R^n\) give a family of interesting geometric objects. They and their variants have numerous applications in geometry, topology, representation theory, and number theory. In this talk, we will review several of these manifestations (for instance, as moduli spaces, function spaces, and the like), and use them to address certain conjectures in number theory regarding distributions of number fields.
Many problems in group theory are known to be undecidable; they are well-posed but cannot be answered by any deterministic computer program. But what happens when some extra information is known about the group before attempting such problems? Are they still undecidable? Or does this extra information make the problems computable? I have considered many such problems throughout my PhD, and will present several cases where additional information is useful, as well as cases where seemingly useful information is not sufficient to compute such problems.
For More Information:Contact Guoqi Qian, g.qian@ms.unimelb.edu.au
One of the major industries where mathematical optimisation has been adopted is the airline industry. There are a wide variety of airline problems which have been studied in the literature including the assignment of aircraft and crews to flights. This scheduling problem is typically separated into four different subproblems. A major push in the current research is to solve as many of these problems as possible in an integrated fashion. Guided by an new integrated formulation of the airline scheduling problem, this talk consider a variety of network optimisation problems which form important subproblems. Specifically; solution methods are proposed, refined and investigated computationally for the resource constrained shortest path with replenishment with either one or many resources. The use of a multiple path variant as a subproblem is also discussed. No prior knowledge of mathematical optimisation techniques is expected.
For More Information:Contact g.qian@ms.unimelb.edu.au
A random matrix is, broadly speaking, a matrix with entries randomly chosen from some distribution. In the non-random case eigenvalues can occur anywhere in the complex plane, but, remarkably, random elements imply predictable behaviour, albeit in a probabilistic sense. Correlation functions are one measure of a probabilistic characterisation and we discuss a 5-part scheme, based upon orthogonal polynomials, to calculate the eigenvalue correlation functions. We apply this scheme to three ensembles of random matrices, each of which can be identified with one of the surfaces of constant Gaussian curvature: the plane, the sphere and the anti- or pseudo-sphere. We will be using real random matrices, which possess the added complication of having a finite probability of real eigenvalues. This talk aims to be accessible, and to that end we will start with a general overview of random matrices and then discuss the 5-step method, hopefully keeping technicalities to a minimum, and with plenty of pictures.
Problems in statistical mechanics often take the following form: Given a graph, G, consider a probability distribution defined on a space of "configurations" constructed from G, e.g. the set of all k-vertex colourings. In particular, we are interested in the behaviour of such models as the size (number of vertices) of G tends to infinity, because in this limit the probability distribution on the configurations can display "phase transitions". Since exact solutions of such models are rare, much of our understanding of them relies on numerical methods, such as Markov-chain Monte Carlo. Unfortunately, the non-analyticities that make these phase transitions of significant physical interest also manifest themselves in the size of the autocorrelations of the Markov chains employed in their study, often severely affecting their computational efficiency. In this talk I'll discuss some of the current paradigms used to construct Markov-chain Monte Carlo algorithms which avoid, or at least significantly reduce, these efficiency pitfalls near phase transitions.
For More Information:Contact g.qian@ms.unimelb.edu.au for more information
Take hot dry air, then sequentially expand it, spray cool it and re-compress it to the inlet pressure whilst allowing further evaporative cooling. That thermodynamic cycle defines a heat engine with advantages for certain applications. The thermodynamic cycle will be analysed and two applications will be described. For a continuous-flow version, the result is a 20% boost in gas turbine output with no extra fuel consumption or emissions. For a piston-in-cylinder version, there is the prospect of unconventional but cheap solar power, including the possibility of thermal storage so as to give 24/7 despatchability.
An introduction to the key ideas of enumerative combinatorics, and the presentation of some novel results relating to sets of paths on the integer lattice. Three main results of particular interest will be discussed: (i) an involution for enumerating osculating lattice paths; (ii) a combinatorial proof of a lemma which enables a purely combinatorial method for enumerating non-intersecting paths; and (iii) a combinatorial proof of another recurrence which, in addition to the previous lemma, enables combinatorial interpretations of product forms. The result in (i) is the first result for osculating paths which allows an arbitrary number of paths. Previous methods have been found for two or three paths, but none previously known were able to be generalised to the full problem. The results in (ii) and (iii) are the first known wholly combinatorial proofs for so-called `product forms'. These represent the first step in a combinatorial approach to solving the remainder of this very large class of problems.
The Gromov-Witten theory of the two dimensional sphere contains lots of interesting geometry and can be difficult to calculate. I will give an intuitive introduction to Gromov-Witten invariants and map counting, and describe how polynomial expressions can be derived using the methods of Eynard and Orantin. This places the invariants amongst a larger class of examples, and general properties will be presented that can be used to provide insight into the structure of the Gromov-Witten theory.