Science, Technology and Engineering

Department of Mathematics and Statistics, Bendigo

Research Projects

The principal research projects that are being undertaken within the Department of Mathematics and Statistics at La Trobe University, Bendigo are as follows. Please follow the links for more detailed information.

Current Research Projects

Axiomatic Foundations of Space-time – Assoc Prof John Schutz

John's current active interests in this area include the axiomatic foundations of Euclidean and non-Euclidean geometries and the space-time structure of Einstein's theory of general relativity.

One of the aims of mathematics is to express theories in the simplest possible form, both in terms of the intuitive acceptability of the primitive concepts and in the assumed properties which are stated as axioms. Recently John has proposed an axiomatic system for hyperbolic geometry that appears to be simpler than other such axiomatic systems, although full proofs of the mutual independence of the axioms are yet to be established. The axiomatic system uses a recently developed characterisation of ellipsoids in projective space. This work will appear shortly in two papers in the Journal of Geometry.

John's earlier research on the foundations of relativity has been published in several journal articles and in the monograph Independent Axioms for Minkowski Space-Time.

Graph Theory – Dr Christopher Lenard and Prof Terry Mills

During 2008, we will conduct a regular seminar on the book R. Diestel "Graph Theory", 3rd edition, New York: Springer-Verlag.

Graph theory is full of interesting unsolved problems. One such problem is the "3 longest paths problem". It is well known that every pair of longest paths in a graph must share a common vertex. Is the same true for every set of three longest paths? Christopher Lenard is investigating this question.

Christopher Lenard and Terry Mills are collaborating with Graeme Byrne and Angela Pezic to develop models from graph theory to describe aspects of internal migration in regional Australia. This project will demonstrate a new tool to demographers and other social scientists, and lead to an interesting application of graph theory in an Australian context.

Hilbert's Sixth Problem: Foundations of Physics – Assoc Prof John Schutz

John is interested in the general problem of the axiomatisation of the theories of physics, which is one of the still-outstanding Hilbert problems.

A project of current interest is the investigation of the foundations of the two major theories of the electromagnetic field in order to ascertain how and why they differ, especially in relation to their descriptions of the vacuum state.

History of Mathematics – Dr Christopher Lenard, Prof Terry Mills and Ms Lex Milne

This research program explores the history of mathematics. Current projects are described below.

Marx and Mathematics: Charles Fahey (Faculty of Humanities and Social Sciences, LTU), Christopher Lenard, Terry Mills and Lex Milne are studying the mathematical manuscripts of Karl Marx.

Fibonacci and Fractions: Diane Itter (Faculty of Education, LTU), Christopher Lenard and Terry Mills are investigating Fibonacci's approach to fractions in his 13th century work "Liber Abaci".

Mathematical Models in Health Care – Dr Robert Champion, Dr Christopher Lenard and Prof Terry Mills

The aim of this research program is to apply ideas from mathematics, statistics and operations research to problems that arise in health care. This program involves collaboration between La Trobe University (LTU) and Bendigo Health (BH). A few current projects are described below.

Learning about patient satisfaction: Assessing patient satisfaction is one aspect of the increased emphasis on consumer participation in health care. Chris Matthews (LTU), Terry Mills (LTU) and Michael Oerlemans (BH) are applying methods from multivariate statistics and statistical learning theory to analyse patient satisfaction data from Bendigo Health. They will be assisted by Rachael Hamilton-Keene (LTU) who was supported by an ICE-EM Vacation Scholarship for 2005-6.

Applications of compartmental models in health care: Congestion in hospitals is a common phenomenon in the Australian health care system. This project explores ways in which compartmental modelling can be used to describe how patients flow through a hospital. The research team for this project is Glenis Beaumont (BH), Robert Champion (LTU), Christopher Lenard (LTU), Terry Mills (LTU) and Mark Mackay (TRACsa).

Streaming: An innovative model for emergency care in a regional hospital: This project applies operations research techniques to describe a new approach to managing the accident and emergency department of a hospital. The research team for this project is led by Leigh Kinsman (LTU) with support from June Dyson (BH), Paulett Thomas (BH), Geraldine Lee (LTU), Robert Champion (LTU) and Terry Mills (LTU). The project is supported by an Industry Collaborative Grant 2006, Faculty of Health Sciences, LTU.

Models for delivery of cancer services: In collaboration with researchers at Loddon Mallee Integrated Cancer Service, Terry Mills is involved in a number of research projects around the theme of improving the delivery of cancer services in Victoria, with a special emphasis on regional Victoria.

A set of of case studies that illustrate mathematical modelling in health care is being prepared by Jon Karnon (University of Adelaide), Mark Mackay (TRACsa) and Terry Mills. This work will be presented in a plenary lecture by Terry at MODSIM09.

Modelling Internal Migration in a Regional Context – Ms Angela Pezic (PhD student), Dr Graeme Byrne and Prof Terry Mills

Over the five-year period between 1991 and 1996, more than 6.5 million Australians – 43 percent of the total population – changed their place of usual residence. This represents one of the highest population mobility rates in the world. A principal goal of this project is to construct mathematical and statistical models of this internal migratory behaviour in a regional context. The model(s) and other data will be used to predict the demographic profiles of Bendigo and Warrnambool over the next ten to twenty years. These predictions will be used to forecast the infrastructure needs within these two cities with an emphasis on the aged population.

Initially, a comprehensive survey of the available literature is being conducted to assess the current "state of the art" in modelling internal migration. Much of the work carried out in this field has a national focus and it is anticipated that specialized models will need to be developed to accurately predict population structures and needs within regional centres. The data for this project will be gathered from a number of sources including the Australian Bureau of Statistics (ABS), the Victorian Departments of Infrastructure (DOI) and Human Services (DHS) and from surveys carried out as part of this project.

Numerical Methods in Statistics – Dr Robert Champion, Dr Christopher Lenard and Prof Terry Mills

A team consisting of Christopher Lenard, Robert Champion, Terry Mills and Chris Matthews (Computer Science & Computer Engineering, La Trobe) is studying applications of cross-validation methods to a variety of problems in applied mathematics.

Oscillation of a Mass Suspended on a Spring – Dr Robert Champion

The oscillatory motion of a mass suspended on a spiral spring is a standard application of second order linear differential equations in undergraduate mathematics subjects, and a commonly used illustration of simple harmonic motion and damped oscillations in physics texts. This topic is often a significant milestone for mathematics students in particular, in that it opens the door to the world of modelling with differential equations. The so-called spring-mass system is conceptually simple, and the elementary theory provides at least a qualitative understanding of observation. However, if one seeks answers to the question "How well does elementary theory actually compare with measurement?", a wealth of hidden, but fruitful, theoretical and practical problems are revealed.

Polynomial Interpolation – Dr Graeme Byrne, Prof Terry Mills and Assoc Prof Simon Smith

The major emphasis of the work in polynomial interpolation at Bendigo has been on the classical Lagrange and Hermite-Fejér interpolation methods, and on their generalization to so-called (0,1,.....,m) Hermite-Fejér interpolation. In particular, the work has concentrated on these interpolation techniques for special node systems, such as equally-spaced points in the interval [-1, 1], and the zeros of the Chebyshev polynomials of the first kind. Problems that have been studied include issues relating to the convergence of the interpolation polynomials and to the behaviour and determination of Lebesgue functions and Lebesgue constants.

A more detailed account of the work in polynomial interpolation at Bendigo is available, as is a list of publications.

The Bendigo group in polynomial interpolation has supervised one successful PhD candidate and welcomes enquiries from potential postgraduate students.

Recently-completed Research Projects

An Application of Queueing Theory to Aged Care – Prof Terry Mills

A few years ago, the Australian government introduced a "transition care" program for hospitals. Some elderly patients may have completed their treatment in a hospital, but for some reason or other, they cannot return home just yet. Perhaps their home needs modification as a result of their illness, or perhaps they are waiting for a bed in an aged care hostel or nursing home. Transition care provides appropriate treatment for patients in this situation. Hospitals were invited to apply for funding for a number of beds under this new scheme. The question facing hospital managers is "How many beds should we request?". Terry Mills collaborated with colleagues from Bendigo Health (A. Crombie, J. Ham, K. Masman) to develop an approach to answering this question based on queueing theory.

Fibonacci and multiplication – Dr Christopher Lenard and Prof Terry Mills

Cathrine Yaneff (BEd(Hons) candidate), Diane Itter (School of Education, LTU), Christopher Lenard and Terry Mills completed a project that was centred on the 13th century work "Liber Abaci" by Fibonacci. The aim was to examine Fibonacci's approach to multiplication and assess its potential for adaptation in the classroom in the 21st century. A paper on this project was presented at the annual conference of the Mathematical Association of Victoria in December 2007 and appeared in the proceedings of that conference. The project was the basis of the thesis: Yaneff, C. (2007) Fibonacci and multiplication, BEd(Hons) thesis, La Trobe University.

Marketing Research for Bendigo Easter Festival 2004 – Prof Terry Mills and Ms Lex Milne

A team consisting of Wenbin Guo (Business, La Trobe, Bendigo), Terry Mills, Lex Milne and Rhett Walker (Business, La Trobe, Bendigo) conducted a marketing research project for the Bendigo Easter Festival 2004.

The Festival dates back to 1869 and is a major event in Bendigo's calendar. Further details can be found here.

The Bendigo Easter Festival Committee of Management asked the team to construct and administer a suitable quality assurance questionnaire. The aims of the project included:

• Determine the level of satisfaction of attendees at the Festival.
• Find out how people react to the major events associated with the Festival.
• Develop a broad demographic profile of people who attend the Festival.
• Examine the effectiveness of the various methods used to promote the Festival.
• Solicit suggestions from attendees for improving the Festival.

The project involved interviewing over 260 attendees of the Festival and was faciliated by a team of students who collected the data. A final report was submitted to the Committee of Management of the Festival in June 2004.

Predicting Emergency Department Demand – Dr Robert Champion and Prof Terry Mills

A team of researchers demonstrated how statistical forecasting methods can be used to make short-term predictions for monthly attendances at the emergency department of one hospital in regional Victoria using data from 2000 to 2005. The ability to predict the demand for attendance at an emergency department of a hospital is valuable – at micro level for such things as planning rosters for staff, and at a macro level for financial and strategic planning for the hospital. Improved resource allocation and strategic planning are potential outcomes of the study. The research team consisted of Robert Champion, Leigh Kinsman, Geraldine Lee, Kevin Masman, Elizabeth May, Terry Mills, Michael Taylor, Paulett Thomas and Ruth Williams.

Further information about the project is given in this media release from the Australian Healthcare Association.

Random Walks – Prof Terry Mills and Assoc Prof Simon Smith

Len Champion, Terry Mills and Simon Smith developed an elementary approach to a classical result of Georg Pólya in the theory of random walks.

Socio-economic Evaluation of Community Banking – Dr Graeme Byrne

Graeme was part of a research team (with Mr Ian Pinge and Dr Jill Francis from LTU, Bendigo, and Mr Jason McGovern from Bendigo Bank) that was awarded a La Trobe University Industry Collaborative Grant for the project "A socio-economic evaluation of community banking". The grant was matched dollar for dollar by an industry partner (in this case, Bendigo Bank).

The purpose of the project was to conduct a socio-economic evaluation of the effects of the initial phase of the development of Community Banking on those communities that had established their own Community Bank branch of Bendigo Bank. The project was significant because, while there is no shortage of communities wishing to establish their own Community Bank branches, there had been no research undertaken into the longer-term impact of such action on the socio-economic foundation of these communities. This research provided useful information for policy makers and stakeholders.

For information on the outcomes of the project, please contact Dr Graeme Byrne or La Trobe University's Centre for Sustainable Regional Communities.

Variational Splines – Dr Robert Champion, Dr Christopher Lenard and Prof Terry Mills

Spline functions were introduced to the mathematical literature in the 1940s by I. J. Schoenberg (1903-1990).

The basic idea of a spline function is to use simple functions to construct a smooth piecewise function. For example, cubic splines are smooth, piecewise cubic polynomial functions.

Splines have proved to be very useful tools in applied mathematics, as well as having deep theoretical interest. The study of splines is now a rich mixture of mathematical problems with applications in engineering, the physical sciences, and social sciences.

Our research has been focussed on variational splines. Although splines were introduced as piecewise functions, they often exhibit certain optimal properties. In such cases, they are solutions of variational (or optimisation) problems. Roughly speaking, these splines minimize some sort of "energy" functional. We call such splines "variational splines".

This variational point of view leads to a totally different approach to splines – and a corresponding beautiful theory. Our two survey papers below give an introduction to variational splines.

Champion, R., Lenard, C. T. and Mills, T. M., A variational approach to splines, The ANZIAM Journal, 42(1) (2000), 119-135

Champion, R., Lenard, C. T. and Mills, T. M., An introduction to abstract splines, The Mathematical Scientist, 21(1) (1996), 8-26

VCE Mathematics: Teachers' Responses to Changes in School-based Assessment – Ms Lex Milne

Lex was part of a research team (with Steve Tobias and Dr Chris Brew, both from the Institute for Education at La Trobe University) that was awarded a La Trobe University Intercampus Research Incentive Scheme (IRIS) Grant for 2000.

The team investigated the views of secondary school mathematics teachers on the implementation of the recent assessment changes to the Victorian Certificate of Education (VCE). Over 300 VCE mathematics teachers from across the school sectors (State, Catholic, Independent and TAFE) were surveyed during July–September 2000. The focus of the project was on:

• student and teacher workloads,
• issues of authentication,
• quality of learning outcomes,
• choice of assessment tasks and procedures, and
• the impact on teaching and assessment in Years 7 to 11.

Results from the project appeared in three papers in 2001 – for references, see the list of publications from the Department of Mathematics and Statistics at Bendigo. A complete version of the paper From CATs to coursework: Teacher feedback on the VCE Mathematics 2000 is available here.