Mathematics seminars 2008
Below is a list of seminars presented during 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.