All talks will be held in the School of Mathematics and Statistics in room 4082 of the Red Centre at University of New South Wales, Sydney.

Registration in the Foyer of the School of Mathematics and Statistics.

Program

Abstracts

Speaker: Asma Alharbi, University of Wollongong

Title: Mathematical Analysis of a model for the activated sludge process
Abstract: The activated Sludge process is widely used in wastewater treatment plants to reduce levels of pollutants in contaminated wastewaters originating from both the municipal and industrial sectors. The activated sludge process can be modelled by a simple reactor configuration consisting of a settling unit attached to a continuously stirred flow bioreactor. The standard model for this process assumes that the volatile suspended solids only contain one component. We examine a more complicated model which has three components in order to better characterise both the wastewater and the activated sludge. We investigate how engineering control parameters affect the performance of the activated sludge process. We find the steady state solutions of the recycle model and determine their stability as a function of the residence time.

Speaker: Hakim Al Garalleh, University of Wollongong

Title: Modelling interaction of antiviral compound binding with fullerene C60 and HIV proteases
Abstract: Fullerenes have gained a great deal of interest in different scientific fields since their discovery in recent years. This discovery leads the scientists to paying more attention because of their unique structure and distinct properties. In this paper, we highlight the underlying of the unique structure and physical and chemical properties of fullerenes. We study the mechanism of treatment of human immunodeficiency virus and to obtain mathematical model which describes the interaction energy resulting from the connection between the antiviral compounds binding and conjugated with fullerenes as inhibitors of the human immunodeficiency virus. The current paper aims to calculate the interaction and force energies arising from antiviral compounds which contain different molecules and shapes inside the on the external surface of the single-walled carbon nanotube with radii r. These interaction are calculated on the basis of certain shape, either atoms or molecules. The present work investigate the single atoms of oxygen and nitrogen as discrete points and by considering OH, CO and NH molecules as lines with the two atoms uniformly distributed which make an angle ? with the z-axis and hydrogen atoms as a spherical shell interacting inside and on the external surface of single-walled carbon nanotubes.

Speaker: Sangeeta Bhatia, University of Western Sydney

Title: Using groups to infer mitochondrial DNA evolutionary history
Abstract: Mitochondrial DNA resides outside the nucleus in mitochondria. It has several interesting features which make it particularly useful for inferring phylogenies.Rearrangement events are believed to be major forces in its evolution. These events include inversion, deletion, and translocation of a fragment of DNA. Duplication of part of the DNA followed by random loss is also considered an important rearrangement event. My work concerns the modeling of these rearrangement events using a group-theoretic framework. In this talk, I'll lay out the problem statement and our approach to modeling it.

Speaker: Dr Wenting Chen, University of Wollongong

Title: Analytically pricing European-style options under the modified Black-Scholes equation with a spatial-fractional derivative
Abstract: This paper investigates the option pricing under the FMLS (finite moment log stable) model, which can effectively capture the leptokurtic feature observed in many financial markets. However, under the FMLS model, the option price is governed by a modified Black-Scholes equation with a spatial-fractional derivative. In comparison with standard derivatives of integer order, the fractional-order derivatives are characterized by their “globalness”, i.e., the rate of change of a function near a point is affected by the property of the function defined in the entire domain of definition rather than just near the point itself. This has added additional degree of difficulty not only when a purely numerical solution is sought but also when an analytical method is attempted. Despite this difficulty, we have managed to find a closed-form analytical solution for European style options after successfully solving the FPDE (fractional partial differential equation) derived from the FMLS model. After the validity of the put-call parity under the FMLS model is verified both financially and mathematically, we have also proposed an efficient numerical evaluation technique to facilitate the implementation of our formula so that it can be easily used in trading practice. A numerical example is provided to illustrate the efficiency of the proposed numerical technique.

Speaker: Christopher Angstmann, Isaac Donnelly, Bruce Henry, University of New South Wales

Title: A Continuous Time Random Walk Approach to Pattern Formation on Networks with Time Dependent Reaction Kinetics
Abstract: We present the derivation from an underlying stochastic process, the continuous time random walk (CTRW) for the generalised master equation for reaction-diffusion on networks. In this derivation, we allow the reaction and diffusion kinetics to vary with time. We show that the CTRW defines the Laplacian operator on the network in a non ad-hoc manner and that the structure of this Laplacian is a key pattern formation mechanism. We have investigated different types of pattern formation across the vertices on a range of random and deterministic networks whereby the CTRWs have exponential waiting times and are defined to be one of two cases; firstly where the rate parameter is constant for all vertices and secondly where the rate parameter is proportional to the vertex degree. These different rates result in non-symmetric and symmetric CTRW Laplacians respectively. In the case of non-symmetric Laplacians, pattern formation may be possible with or without a Turing instability. However in symmetric Laplacians, pattern formation follows from a Turing instability.

Speaker: Lucia-Marie Billie Ganendran, University of New South Wales, Canberra

Title: Is seabird survival affected by wind? A mark-recapture analysis of adult Little Penguins Eudyptula minor in south-eastern Australia using Matlab.
Abstract: We present a framework for using computational techniques to analyse mark-recapture data arising from studies of any wildlife species. Here we conduct a detailed statistical analysis of a valuable and unique 42-year life-history data set for Little Penguins Eudyptula minor from Phillip Island, south-eastern Australia. We illustrate our use of the computer package Matlab to investigate possible relationships between strong wind events and the survival of adult penguins. Our parameters of interest, the survival and recapture probabilities, are modelled via a series of biologically-sensible age structures and covariate dependencies.  Using the life history data, we form likelihood contributions for each bird and estimate the parameters using maximum likelihood methods. 

We use the statistical software package R to summarize the raw data from each record corresponding to a mark-recapture occasion. Matlab was used to summarise the wind data, to create the covariates for survival, to form the likelihood function under various models and to estimate the parameters using the function “fmincon” from Matlab’s Optimisation Toolbox.
Our earlier study indicated that wind from each of the four cardinal compass points affects the survival of first-year and adult Little Penguins. Southerly winds in the winter prior to a chick's birth are associated with an increased annual survival probability in its first year of life and easterly winds in the summer of hatching/fledging are linked with a decreased survival. Furthermore, adult survival is positively associated with increasing northerly winds in the autumn following moult, and negatively associated with easterly winds in the preceding summer. The lagged nature of these effects highlights the complexity of the mechanisms affecting survival.

In the current study, at the first stage of modelling, results show that adult survival is negatively associated with strong overall winds in the autumn following moult.  When a second covariate is added, we find that strong autumnal winds still have a negative association with adult survival, but strong winter winds following the breeding season have a positive association with adult survival.  The mechanisms by which strong wind events affect adult penguin survival are not yet fully understood but are possibly linked to the movement of nutrient-rich waters into or out of the penguins’ foraging areas, or to the mixing of the water column, making prey acquisition more difficult.

Speaker: Dr Emmanuel Hanert, Université Catholique de Louvain, Belgium

Title: Front dynamics in fractional-order multi-species models: Applications in epidemiology and ecology
Abstract: A number of recent studies have shown that mobility patterns for both humans and biological species can be quite complex and exhibit a scale-free dynamics, characteristic of Lévy flights. Lévy-flight patterns have been observed in the dispersion of bank notes, in human mobility patterns derived from mobile phone data as well as in the foraging patterns of a numbers of animal species. However, current reaction-diffusion models used to describe the spread of humans and other biological species do not account for the superdiffusive effect due to Lévy flights. This could result in higher spreading speeds than predicted by classical models.

We have considered two-species reaction-diffusion models driven by Lévy flights. That family of models is based on the Lotka-Volterra equations and has been obtained by replacing the second-order diffusion operator by a fractional-order one. Depending on the parameter values, it can be used to represent the interaction between susceptible and infective populations in an epidemics model or the interactions between ecological species competing for the same resources.

Theoretical developments and numerical simulations show that fractional-order diffusion leads to an exponential acceleration of the population fronts and a powerlaw decay of the fronts’ leading tail. Depending on the skewness of the fractional derivative, we can derive catch-up conditions for different types of fronts. Our results confirm that second-order reaction-diffusion models are not well-suited to simulate the spatial spread of modern epidemics and biological species that follow a Lévy random walk as they are inclined to underestimate the propagation speeds at which they spread.

References:
Hanert E. (2012) Front dynamics in a two-species competition model driven by Lévy flights, Journal of Theoretical Biology 300, 134-142.
Hanert E., E. Schumacher and E. Deleersnijder (2011) Front dynamics in fractional-order epidemic models. Journal of Theoretical Biology 279, 9-16.

Speaker: Dr Roslyn Hickson, University of Newcastle

Title: Old flu, young flu, more flu, less flu: Who to vaccinate?
Abstract: An important concern in public health is what population group should be prioritised for vaccination. To this end, we present a model capable of using arbitrary distributions for population susceptibility, and corresponding infectivity distributions. We consider three scenarios: first, a population with heterogeneous susceptibility with a uniform distribution, but homogeneous infectivity. Second, a uniform heterogeneously susceptible population with linear heterogeneous infectivity functions, namely that either the most susceptible are either the most or least infectious. Finally, we consider the effects of pre‐epidemic immunity, ostensibly through vaccination, on the epidemic dynamics. For a seasonal influenza‐like infectious disease, we find the smallest final size and overall deaths for the epidemic if the most susceptible are vaccinated, corresponding to children.

Speaker: Natalya Levenkova, University of New South Wales

Title: Modelling a contact network using random geometric graphs and random motion
Abstract: We propose a model of a random network which combines ideas from random geometric graphs and random walks. The model produces a spatially-embedded, degree-inhomogeneous, time-evolving random network. Motivation for this work comes from epidemiology, since the structure of the underlying contact network has a large effect on the spread of an epidemic over time. In this talk we consider the one-dimensional discrete case of the model, in which vertices are uniformly distributed on a circle and each vertex is assigned a radius, drawn from a binomial distribution. We will look at several properties of simulated networks such as the number of connected components, clustering coefficient, degree distribution, and other measures which are of interest for epidemiologists.

Speaker: Timothy Ling, University of Technology

Title: Study of some functionals of standard and fractional Brownian motions with applications in statistics and finance
Abstract: This talk contains results on two important problems arising in Statistics and
Quantitative Finance. The first problem is on study of simulation algorithms for a fractional Brownian motion (fBM) which is considered in modern Mathematical finance as a good alternative to model processes with long memory. We have reviewed the known simulation algorithms and have implemented the fastest of them (“circulant embedding”) on a modern multicore desktop PC. We applied the algorithm to two longstanding open problems in statistics, namely, to study distributions of exponential functionals of fBm and the maximum of fBm. The results of our simulations exhibit new and striking properties of these distributions. The second problem is about option pricing with a volume weighted average price (VWAP) as an underlying process. The VWAP is a very important quantity in finance; it appears in Australian taxation law to specify the price of share-buybacks for publicly-listed companies and it is a benchmark price for market participants.

Pricing options on VWAP is a challenging problem from a mathematical point of view because it involves two sources of randomness: the price of the asset and its traded volume. To solve the problem we have applied the moment-matching approach to a range of “stock and volume” models and as a result obtained an accurate approximation for prices of “call” options. All results have been verified by intensive Monte Carlo simulation.

Speaker: Stephen Maher, University of New South Wales

Title: Solving the integrated recovery problem with column-and-row generation
Abstract: The airline recovery problem is a very large and complex problem generally solved as a series of sequential stages by the airline operations control centre. There has been much research into the development of automated approaches for each of these stages to improve operational performance. To further improve the solution quality of the airline recovery problem, an integrated approach has been proposed. In this talk, an integrated schedule, aircraft and crew recovery problem is presented. To reduce the size of the integrated recovery problem and improve runtimes the solution method of column-and-row generation has been employed. The inclusion of row generation significantly reduces the problem size by limiting the available flight delay options and iteratively improving the solution quality. The results demonstrate the benefits of column-and-row generation over a standard column generation approach in both solution quality and runtime.

Speaker: Dr Mike Meylan, University of Newcastle

Title: Wave Scattering in the Marginal Ice Zone
Abstract: The theory of linear water waves is a classical area of fluid mechanics which was extensively studied in the last century. In some sense the field is very mature and the methodology developed has a wide range of applications in marine engineering. However, there are still many open questions and I will discuss some of these. I will also present my own research into developing a connection between the solution in the time and frequency domains.

Speaker: Chao Qian, University of New South Wales, Canberra

Title: Numerical method for solving combustion waves arising from sequential exothermic and endothermic reactions
Abstract: We consider flame propagation in a two-stage reaction of a solid fuel under
adiabatic conditions in one-dimension. Specifically we consider the following
sequential reaction:

Here both reactions are assumed to proceed with temperature-dependent rates. The first reaction transforms the fuel A into the intermediate product B and heat Q1, while the second reaction transforms the intermediate species B into the final product P and heat Q2. In the exothermic-endothermic scheme considered here, it is assumed that Q1>0 (exothermic) and Q2<0 (endothermic). The above reaction scheme has direct relevance to blast furnace chemistry. For example, under certain condition, the iron ore reduction process can be modelled as a two-stage sequential exo-endo reaction scheme. In our study, the properties of the travelling combustion waves were investigated numerically by solving the system of nonlinear differential equations subject to appropriate boundary conditions. We use a standard shooting algorithm with a fourth-order Runge-Kutta integration scheme. First, we applied forward shooting method. According to our combustion system, adjusting shooting direction in a 2-D plane with 2 free parameters is hard to find an accurate solution, especially on boundary conditions. Then we tried to apply backward shooting method by changing the original combustion system to its negative system. After that, we adjust shooting direction in an 1-D line with only 1 free parameter and get a fairly good solution. Finally we use the solution from backward shooting to calculate shooting direction in forward shooting and get the same results, which verified our backward shooting solution.

Speaker: Amirah Rahman, University of New South Wales

Title: Scheduling Line and Yard Operations in a Freight Network
Abstract: In freight railway planning, the area of demand fulfilment is one that combines both line and yard operations. The problem is to schedule train and product movement between unloading and multiple loading yards without violating line capacity. The schedule must fulfil as much of the demand as possible within requested time windows while keeping stockpile levels at all yards below their maximum capacities. In this talk I will briefly discuss the problem model and formulation.

Speaker: Michael Rose, University of Newcastle

Title: Expectations on Fractal Sets
Abstract: Motivated by laboratory studies on the distribution of brain synapses, the classical theory of box integrals - being expectations on unit hypercubes - is extended to a new class of fractal “string-generated Cantor sets” that facilitate fine-tuning of their fractal dimension through a suitable choice of generating string. Closed forms for certain statistical moments on these fractal sets will be presented, together with a precision algorithm for higher embedding dimensions. This is based on joint work with Laur. Prof. Jon Borwein, Prof. David Bailey and Dr. Richard Crandall.

Speaker: Hamizah Mohd Safuan, H. S. Sidhu, Z. Jovanoski, I. N. Towers, University of New South Wales, Canberra

Title: A two population model with varying carrying capacity
Abstract:  Classical population growth models assume that the environmental carrying capacity is a fixed quantity which is not often realistic. As an alternative, time-dependent functional forms for carrying capacities have been widely applied. Inter-dependency between population and the carrying capacity as a nonlinear system was considered by Safuan et al.1 for a single species population. This study proposes a two population model with varying carrying capacity by incorporating intra- and inter-specific interaction terms in each equation. Stability, bifurcation and numerical analyses are presented to illustrate the system's dynamical behaviour. Bistability occurs in certain parameter regions that could describe the transition between a safe and detrimental environment. For some parameter ranges, Hopf bifurcation is found to exist in the system which generates solutions that posses limit-cycle behaviour between the populations.
1Safuan, H. M., Towers, I. N., Jovanoski, Z. and Sidhu, H. Coupled logistic carrying capacity model. , 53, C142-C154, 2012.

Speaker: Dr Boris Savkovic, University of New South Wales

Title: Mathematical modelling in anti-HIV gene therapy: estimating clinically relevant parameters and predicting likely clinical outcomes
Abstract: Gene therapy represents a promising modality to treat HIV/AIDS. The concept involves the introduction of an anti-HIV gene into a patient’s immune cells, thereby conferring protection against the HIV virus. While in-vitro and humanized mice studies, by our group and others, have demonstrated strong anti-viral effects of the therapy, it is the translation of this therapy into clinical practice that represents current challenges. In this talk, I will outline how mathematical modelling is providing a clearer understanding of the quantitative factors involved in the delivery of such gene therapy in-vivo, with the aim of maximizing therapeutic effect against HIV. This work is funded by an ARC Linkage Grant and is in collaboration with the industry partner Calimmune Australia Pty Ltd.

Speaker: Luke Sciberras, University of Wollongong

Title: Vortices in bounded soft media
Abstract: Background information into and reasons for using polarised light beams (lasers) to form solitons (or optical waves) in a liquid crystal will be initially discussed in this seminar. Extending from this, there will be an examination of a specific type of optical wave called an optical vortex and a brief discussion of its formation. Using this background knowledge along with variational techniques and Lagrangian methods in a nonlinear system of PDE's, conversations will be directed towards a study on the evolution of an optical vortex in a finite nematic liquid crystal cell. Indeed, this study requires linearised stability analysis about the steady state for the given system to determined a relationship between the instability of an optical vortex and the minimum distance of approach to the boundary. Results from the mathematical study, show that the simple asymptotic approximations capture the amplitude of the optical vortex and its path towards its final steady state within a finite cell. The variational analysis results are compared to the full numerical solution for the non linear system. Good agreement is shown with all results.

Speaker: Yang Sun, University of New South Wales, Canberra

Title: Soliton dynamics in a frequency-modulated potential
Abstract: Currently, experimental and theoretical studies of solitons have been conducted in the context of several areas of science, from applied mathematics and physics to chemistry and biology. A significant amount of soliton research is concentrated on nonlinear optics (light waves) and Bose‐Einstein condensates (BECs matter waves) in optical lattices, which are often described by the Gross‐Pitaevskii equation or the nonlinear Schrödinger with a periodic potential. Although many theoretical results have been published, intensive studies of solitons dynamics in nonlinear lattices started only recently. The study of the dynamics in nonlinear lattices requires the understanding of new problems. Thus, soliton dynamics in nonlinear lattices is considered primary in theoretical research.

Speaker: Dr Shuaian Wang, University of Wollongong

Title: Bunker Consumption Optimization Methods in Shipping: A Critical Review and Some Extensions
Abstract: It is crucial nowadays for shipping companies to reduce bunker consumption while maintain a certain level of shipping service in view of the high bunker price and concerned shipping emissions. After introducing the three bunker consumption optimization contexts: minimize total operating cost, minimize emission and collaborative mechanisms between port operators and shipping companies, this presentation gives a critical and timely literature review on the bunker consumption optimization methods. Several novel bunker consumption methods are subsequently proposed. The applicability, optimality, and efficiency of the existing and newly proposed methods are also analyzed. This presentation would provide some technical guidelines and insights for researchers and practitioners dealing with the bunker consumption issues.

Speaker: Yu Guang Wang, University of New South Wales

Title: Local Behaviour of Fourier, Cesaro and Filtered Operators on the Sphere
Abstract: Summation methods on the sphere have been discussed for decades. The classical summation methods such as Fourier-Laplace, Cesaro summations and filtered Fourier operators were studied by many authors. Recently, a filtered Fourier approximation on the sphere, was constructed via cubature rules and its convergence behaviours were discussed by Sloan and Womersley. The approximation properties discussed were over the whole sphere, and most of them inherit analogous behaviours of those over circles.

However, it is well-known that the same summation method may behave in a distinctly different way over arcs from over the entire circle. In fact, as Zygmund pointed out, the integrals of Dirichlet kernels over arcs containing the origin converges more slowly than over those without origin. On the other hand, the local approximation behaviours of summation methods on sphere were demonstrated numerically. This talk shows local approximate phenomena of Fourier, Cesaro and filtered Fourier kernels on sphere and provides their theoretical explanation.

Speaker: Wilson Wee, University of New South Wales, Canberra

Title: Properties of reaction fronts in a two-stage exothermic-endothermic competitive reaction scheme
Abstract: Studies of exothermic reaction fronts are of interest to science and industry; they form the basis of a combustion or reaction wave and find relevant applications in the synthesis of advanced materials, the design of explosives and power generation. Most investigations of the properties of reaction fronts have only utilised the simplest models with a one-step exothermic chemistry, while models with two-stage reaction kinetics have been less comprehensively studied, especially in a competitive scheme where two thermally and chemically coupled reactions feed on the same reactant. In this presentation, an analysis of a competitive reaction scheme will be presented, focusing on the propagating wave speed and the regions of stability within the parameter space.

Speaker: Assoc. Prof. Annette Worthy, University of Wollongong

Title: Generation and Control of Solitons using Various Nematic Geometries and Regimes
Abstract: A mathematical treatise of recent experimental work on the generation and control of optical solitons in the nonlinear nonlocal nematic media will be discussed. It will be shown that having an additional localized voltage to form various suitable regimes causes the director or nematic to have different orientations with the cell. Some geometries that will be studied are the rectangular cross section, circular and elliptical regions.

Using asymptotic methods along with the nematicon being is largely independent of functional form of its profile, it is shown how the beam evolves. It will be revealed that the nematicon sheds radiation whereby the velocity and position decoupled from its width and amplitude oscillations. Further, due to the additional geometrics caused by an external applied voltage to the nematic upon twisting the molecules, the resulting polarized self propagating beams distorts and refracts.

The resulting mathematical analysis is quick and efficient and is shown to give excellent agreement to both experimental work and numerical simulations.

Speaker: Yan Xu, University of New South Wales, Canberra

Title: Instability dynamics of a damped-driven oscillator chain with non-autonomous and parametric driving
Abstract: Damped‐driven oscillator models are at the heart of theoretical investigations in a broad range of physical systems, from models of friction, to mechanical systems as diverse as coupled pendulum and the operation of a crane. In this work we consider how the type of driving may affect the oscillator dynamics by investigating the interplay of the usual non‐autonomous driving and an additional parametric driving. Our results may be applied to oscillator systems more generally; however, motivated by unexplained observations in recent experiments, we consider in particular a chain of pendulums coupled by torsional springs with a force which depends on both the frequency of the driving, and the angular displacement of each pendulum. We examine firstly the dynamics of a single pendulum and find that the mixed driving supports bistability in a critical frequency range, and that the development of chaos also depends sensitively on the driving frequency. After presenting in detail the key features of the phase space we consider the instability dynamics of a chain of oscillators with mixed driving. We uncover the conditions for the development of modulational instability (MI), and ultimately the formation of localised oscillations in the form of discrete breathers. We conclude with an outline of the conditions required for the experimental confirmation of our predictions.