Mathematics seminars 2006-2008
Below is a list of seminars presented during the years 2006-2008 in the Seminar Program of the Department of Mathematics and Statistics at La Trobe University's campus in Bendigo, Victoria.
Simon Smith
Staying within the bounds: Curved majorants and interpolation, by Dr Simon Smith (La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 24 October 2008, in Room B2.15
Abstract: In 1889 a classic result of A. A. Markov answered the following question: 'Given a polynomial p of degree n whose graph on [-1,1] lies in the unit square, how large can its derivative p' be on the same interval?'. Much later this question was modified by replacing 'unit square' with 'unit circle', and, more generally, by asking about the size of p' if |p(x)| is bounded by c(x). Here c is a given function, called a curved majorant.
This seminar will begin with an overview of results for polynomials with curved majorants, highlighting the role played by the Chebyshev polynomials in the solutions to many of the problems. I will then discuss Lagrange and weighted polynomial interpolation methods, particularly when the interpolation is based on the zeros of Chebyshev polynomials. Finally, there will be a brief look at some recent work on the interpolation of functions with curved majorants.
Lun Zhang
Cross Disciplinary Thoughts: Graph Theory, Statistics and Engineering, by Dr Lun Zhang (Tongji University, Shanghai, China)
Scheduled: 11.00 am, Friday 22 August 2008, in Room B2.15
Abstract: A snapshot of my current work in related areas will be given. 1. Within Networking, Topological Control is much broader than Graph Theory – rather, it is a cross-disciplinary field involving Intelligent Computation, Cognitive Communication techniques, and Game Theory. 2. Meanwhile, since paradoxes might not be clearly explained in purely mathematically ways, could computers and other experimental methods help us to solve the problems? 3. Also, in the statistical area, more challenges turn out due to the fact that the Spatial and Nonparametric Statistics are penetrating engineering and sociological areas both in theory and methodology.
Eder Kikianty
Hermite-Hadamard's inequality and the p-HH-norm on the Cartesian product of two copies of a normed space, by Eder Kikianty (Victoria University, Melbourne)
Scheduled: 12 noon, Friday 2 May 2008, in Room B2.15
Abstract: The Cartesian product of two copies of a normed space is naturally equipped with the well-known p-norm. Another notion of norm is introduced, and will be called the p-HH-norm. This norm is an extension of the generalised logarithmic mean and is connected to the p-norm by the Hermite-Hadamard's inequality. The Cartesian product space (with respect to both norms) is complete, when the (original) normed space is. A proof for the completeness of the p-HH-norm via Ostrowski's inequality is provided. This space is embedded as a subspace of the well-known Lebesgue-Bochner function space (as a closed subspace, when the norm is a Banach norm). Consequently, its geometrical properties are inherited from those of Lebesgue-Bochner space. An explicit expression of the superior (inferior) semi-inner product associated to both norms is considered. Several norm inequalities of Ostrowski type, which involve the p-HH-norm, are also derived using the convexity and the absolute continuity of the norm. Some of these inequalities are proven to be sharp.
Christopher Lenard
Paths in Graphs: the long and the short, by Dr Christopher Lenard (Department of Mathematics and Statistics, La Trobe University)
Scheduled: 12 noon, Friday 11 April 2008, in Room B2.15
Abstract: Finding short paths in graphs is relatively easy, while finding longest paths is considerably more time consuming. Little is known about intersections of longest paths. It is easy to show that each pair of longest paths in a graph must share a common vertex, but is this also true of triplets of longest paths? We still don't have a complete answer, even after four decades.
David Yost
Decomposable Polyhedra, by Dr David Yost (School of Information Technology and Mathematical Sciences, University of Ballarat)
Scheduled: 12 noon, Friday 29 February 2008, in Room B2.15
Abstract: We use graph theoretic methods to solve a problem in combinatorial geometry, namely we complete the classification, in terms of Minkowski decomposability, of the 260 types of polyhedra with 15 or fewer edges. That is, for each such polyhedron P, we can say whether or not it can be expressed as a sum of two polyhedra which are not similar to P. The novelty of our approach is the use of 4-cycles which are not faces, in the graph of the polyhedron.
Diane Itter (left) and Cathrine Yaneff
Fibonacci and multiplication, by Diane Itter and Cathrine Yaneff (Faculty of Education, La Trobe University, Bendigo)
Scheduled: 12 noon, Monday 3 December 2007, in Room B2.09
Abstract: Fibonacci's seminal work "Liber Abaci", first published in 1202, made an enormous contribution to the introduction and dissemination of the Hindu number system and arithmetic throughout the western world. In this paper, we will explore Fibonacci's treatment of multiplication and examine Fibonacci's explanation of the method of checking by casting out nines. Furthermore, we will consider implications of Fibonacci's ideas for teaching multiplication in schools today. This is joint work with Christopher Lenard and Terry Mills.
Robert Hunting
Project One: Uncovering the Seeds of Mathematics in Children Aged 12-24 Months, by Dr Robert Hunting (School of Education, La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 5 October 2007, in Room B2.15
Abstract: Mathematics is a human activity whose origins in the learner have both genetic and social roots. This presentation will discuss selected segments of video data of a toddler's spontaneous play showing evidence of pre-mathematical actions and activity. Such data, including commentary provided by the Investigator, is to be submitted for further analyses by an expert panel of mathematicians, if funded.
Chris Cope
Improving teaching and learning about confidence intervals, by Dr Chris Cope (Department of Computer Science and Computer Engineering, La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 24 August 2007, in Room B2.15
Abstract: Threshold concepts are the keys to meaningful learning progression in a discipline but are particularly difficult to teach and learn. The seminar will report on a study which improved understanding of teaching and learning about confidence intervals (CIs), a threshold concept in the statistics discipline. The responses to an exam question of 100 first year undergraduate students were analysed qualitatively using a phenomenographic technique. The outcome space consisted of 7 distinctly different, but hierarchically related, ways of experiencing CIs. A logical analysis of the outcome space identified three educationally critical aspects of learning about CIs at the introductory level: CIs are based on repeated sampling, CIs relate to the success of the method, and CIs are an estimation tool for improving decision making in the real world. The seminar will include a discussion on the types of learning activities most likely to help students come to terms with these educationally critical aspects.
Aleesha Keogh and Adam Rosenow
Government expenditure on the Pharmaceutical Benefits Scheme, by Aleesha Keogh and Adam Rosenow (La Trobe University, Bendigo)
Scheduled: 12 noon, Monday 4 June 2007, in Room B2.32
Abstract: We will discuss recent trends in the cost of the Pharmaceutical Benefits Scheme (PBS) and the application of time series analysis for modelling and forecasting monthly PBS cost data. This work was carried out for the third-year undergraduate unit Topics in Statistics.
Robert Champion
Nonlinear springs – where Hooke got it wrong, by Dr Robert Champion (La Trobe University, Bendigo) [Co-author: Len Champion]
Scheduled: 12 noon, Friday 23 March 2007, in Room B2.15
Abstract: Models derived on the assumption that springs exhibit linear elastic behaviour are found to be inadequate for precisely fitting measurements of the static extension and periods of oscillation of a vertically suspended, loaded, helical spring. A nonlinear spring model, derived from the elasticity equations for a helical wire, is found to overcome the deficiencies in the linear models.
Terry Mills
Bernstein's inequality in probability, by Professor Terry Mills (La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 23 February 2007, in Room B2.05
Abstract: In 1924, S. N. Bernstein (1880-1968) sought to improve an inequality in the theory of probability established by P. L. Chebyshev (1821-1894). Although Bernstein's inequality is a decided improvement of the result of Chebyshev, and was published 82 years ago, it is not as well known as the result of Chebyshev. Even books on probability do not give it much prominence. In this seminar I will outline Bernstein's result and its proof, and describe various improvements and extensions of this inequality. The story surrounding this inequality will illustrate how the theory of probability in general has developed over the last 80 years.
Joachim Gwinner
Variational Inequalities – a way to treat inequality-constrained problems in operations research and mechanics, by Professor Joachim Gwinner (Universität der Bundeswehr München)
Scheduled: 12 noon, Tuesday 13 February 2007, in Room B1.29
Abstract: Variational Inequalities (VI) are now established as a versatile mathematical model to treat inequality-constrained problems in various fields like operations research and mechanics. Here I discuss (dis)equilibria of distributed markets, traffic equilibria in networks, and elliptic boundary value problems with Signorini boundary conditions that arise in unilateral contact mechanics and in fluid flow in porous media.
By the VI approach not only can existence and stability results be obtained, but also efficient numerical methods can be developed in combination with variational discretisation methods for partial differential equations such as the Finite Element Method or the Boundary Element Method.
In this seminar, I will present a recent extension of the VI methodology to inequality-constrained problems with random data.
Deane Arganbright
Innovative Techniques in Mathematical Visualization with Microsoft Excel, by Dr Deane Arganbright
Scheduled: 12 noon, Thursday 18 January 2007, in Room B1.29 [This seminar is also part of the Seminar Program of the Department of Computer Science & Computer Engineering.]
Abstract: Excel is a natural tool for learning and implementing a wide range of mathematical concepts in a way that closely parallels the way one naturally approaches mathematics. It provides an easy-to-use graphic platform for interactive ways to visualize mathematical concepts and algorithms. Furthermore, since Excel is the principal mathematical tool of the workplace, one does not always need to use specialized software to do significant mathematics. Finally, this approach allows a number of topics ordinarily considered quite advanced to be understood by a very wide audience of people. Thus, Excel assists in making mathematical ideas more accessible.
Anyone with an interest in mathematics, mathematical modelling, or creative uses of spreadsheets in teaching might enjoy this seminar.
Deane Arganbright has just completed an appointment as visiting professor of mathematics at Korea Advanced Institute of Science and Technology, a leading technological research university. In the coming spring he will be a guest professor at the University of Vienna. He is a pioneer and an internationally recognized expert in the creative use of spreadsheets for mathematics and mathematics teaching. He has published three books and numerous articles on the subject and given many invited international presentations. He has taught at universities in the US, Austria, Australia, PNG and Korea.
An article about the seminar, titled "Learn to Excel at maths", appeared in the Bendigo Advertiser newspaper of Thursday January 18, 2007.
Barry Jessup
Does symmetry punch holes in space?, by Professor Barry Jessup (University of Ottawa)
Scheduled: 12 noon, Friday 4 August 2006, in Room B2.15
Abstract: Symmetry in a problem effectively reduces its dimension, making it easier to solve. An important question is then: "How much symmetry can the manifold M of configurations of a system support?"
The toral rank of M is the largest integer r such that a compact abelian symmetry group of dimension r acts freely on M, and is difficult to determine in general. Estimates are therefore useful, and in 1968 Steve Halperin conjectured (based on very little evidence, as he likes to admit) that the toral rank of a closed manifold is at most log (base 2) of the total number of "holes" in M.
We will survey this stubborn conjecture, which remains open in general. After being a bit more precise about how we shall count "holes", if we have time, we will focus on the case when M "comes from" a nilpotent Lie algebra L (e.g. the Heisenberg lie algebra), where the toral rank of M is known to be the dimension of the centre of L.
The principal tools we will use are those of rational homotopy, whose success essentially relies on systematically ignoring small groups, an approach the speaker does not support in every situation.
Terry Mills
Emergency Maths, by Professor Terry Mills (La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 7 July 2006, in Room B2.15
Abstract: Most of us have had to visit the emergency department of a hospital either as a patient, or accompanying a patient. The emergency departments are well known for long waits and short fuses while the staff try valiantly to meet the patients' needs.
In this seminar, I will describe ways in which mathematics can be used to describe various aspects of the emergency department of a hospital.
James McEwan
Egyptian Mathematics, by James McEwan (Bendigo Senior Secondary College)
Scheduled: 11.30 am, Friday 23 June 2006, in Room B1.29
Abstract: As part of the CSIRO Student Research Scheme, James McEwan has been working in the Department of Mathematics and Statistics. In this presentation, James will discuss his work on the history of Egyptian mathematics.
Robert Gray
Extracting discrete information from a continuous world: Quantization, Compression, and Classification, by Professor Robert M. Gray (Stanford University) – Professor Gray is a 2006 Distinguished Lecturer of the IEEE Signal Processing Society
Scheduled: 12 noon, Wednesday 22 March 2006, in Room B1.29
Abstract: Scientists and engineers often seek to measure, communicate, store, process, reproduce, or analyze signals encountered in the real world. Most such signals are inherently continuous or analog in nature, yet increasingly the means for communicating, storing, and manipulating such information are discrete or digital. Generally something is lost when continuous information is converted into discrete approximations, so a natural goal is to preserve as much of the original information as possible. This is the general problem of quantization, a technique that historically has cropped up in a variety of branches of signal processing, taxonomy, physics, mathematics, and statistics as well as playing a key role as the interface between a continuous world and digital processing. Quantization traditionally has been used to model analog to digital conversion, Shannon source coding, and data compression. Viewed generally, quantization also models the extraction of information from signals, including statistical classification, clustering methods, and machine learning. This talk will describe the fundamentals of quantization along with examples and recent research topics in theory and application.
Rachael Hamilton-Keene
Modelling Patient Satisfaction, by Rachael Hamilton-Keene (ICE-EM Vacation Scholar 2006)
Scheduled: 11.00 am, Wednesday 15 February 2006, in Room B2.15
Abstract: This presentation is the result of a 6-week research project carried out as an ICE-EM summer scholar. The project involved modelling hospital inpatient satisfaction at Bendigo Health. Patient satisfaction is an important measure of health care provider performance because preliminary research has linked it to better health outcomes, and because patients are paying customers. Patient satisfaction at Bendigo Health has been assessed over several years with patient satisfaction surveys. Binary logistic regression analysis was used to determine the key drivers of patient satisfaction. Discussed will be the process of data clean-up, binary logistic regression itself, and the results of the analysis.