Mathematics seminars 2005
Below is a list of seminars presented during 2005 in the Seminar Program of the then Department of Mathematical and Statistical Sciences at La Trobe University's campus in Bendigo, Victoria.
Inequalities Day – Three seminars on inequalities were presented on Monday 19 December 2005, as follows.
Bill Helton
Inequalities on polynomials with matrix variables, by Bill Helton (University of California, San Diego)
Scheduled: 11.00 am, Monday 19 December 2005, in Room B1.29
Abstract: Many problems in control and systems engineering, and in an expanding number of areas, directly convert to collections of matrix inequalities. If these are collections of convex or (better yet) collections of linear matrix inequalities, then they are amenable to reliable numerical solution. To get matrix inequalities into such a form requires trickery and often much cleverness. The talk will describe progress on theory underlying how one might make this mystical process systematic.
There is a classical theory of positive polynomials on Rn going back to Hilbert's 17-th problem which concerns the expression of positive polynomials as a sum of squares; it initiated the subject called real algebraic geometry. What is needed for matrix inequalities is a corresponding theory for polynomials in n-noncommuting variables whose properties we test by plugging in n-tupples of matrices; such values of polynomials are matrices. A "positive" noncommuting polynomial is one which always takes positive semidefinite matrix values. Which ones are sums of squares? An analogous notion of convexity for a noncommutative polynomial is common (and important) in systems engineeering. Can we classify them? The talk will describe the beginnings of a noncommutative real algebraic geometry which settles the problems described above.
Alex Rubinov
Abstract convexity with application to global optimization and inequalities, by Alex Rubinov (University of Ballarat)
Scheduled: 12 noon, Monday 19 December 2005, in Room B1.29
Abstract: Development of a theory of global optimization is one of the most challenging problems of modern optimization theory. Currently there are tractable conditions for a global optimum only for very narrow classes of problems, including convex problems. Calculus is not suitable for investigation of global optima, so some new techniques should be invented. One of them is abstract convexity. Recall that a lsc function is convex if and only if this function can be represented as the upper envelope of a set of affine functions. Replacing affine functions with another set of sufficiently simple functions we get abstract convex functions with respect to this set. In this talk I will discuss some main ideas of abstract convexity and application of abstract convexity to examination of global optimality conditions and some inequalities.
Sever Dragomir
Reverse inequalities for vectors and linear operators in Hilbert spaces, by Sever S. Dragomir (Victoria University, Melbourne)
Scheduled: 2.00 pm, Monday 19 December 2005, in Room B1.29
Abstract: The purpose of this talk is to survey recent results due to the author concerning reverses of the Schwarz inequality in real or complex inner product spaces that generalise the classical inequalities due to Polya-Szego (1925), Cassels (1955), Greub-Reinboldt (1959), Shisha-Mond (1967) and Klamkin-McLenaghan (1977). These results are then employed to provide various reverses for the numerical radius of bounded linear operators in complex Hilbert spaces under appropriate norm restrictions.
Zdenek Ryjacek
Closure concepts and stability of graph properties and graph classes, by Zdenek Ryjacek (University of West Bohemia, Pilsen, Czech Republic)
Scheduled: 12 noon, Monday 12 December 2005, in Room B2.05
Abstract: Closure techniques are a powerful tool in studying hamiltonian properties of graphs. In this talk, two closure concepts will be observed, namely, the Bondy-Chvátal closure based on degree sum of pairs of independent vertices, and the closure concept in claw-free graphs based on local completions of neighborhoods of locally connected vertices, introduced by the speaker. Behavior of several hamiltonian-type graph properties with respect to these closure concepts will be studied. Some open questions will be also mentioned.
Mark Mackay
Compartmental Modelling and Health: Learnings from Research on the Application of Compartmental Models for Modelling Bed Occupancy, by Mark Mackay (Department of Health, South Australia)
Scheduled: 12 noon, Friday 14 October 2005, in Room B2.30
Abstract: The use of compartmental models to describe bed occupancy was initiated by Millard and Harrison using geriatric data from the United Kingdom. The potential to improve strategic decision-making in relation to acute care provision in Australia was identified as a result of a bid for substantial additional resources by a hospital on the basis of needing to fund extra beds without any mechanism to support the bid. This presentation will detail the reasons for considering the use of the compartmental model, the validation of its use in acute care setting using Australian and New Zealand data, and some suggested modification of the model. Discussion about impediments for adoption of alternative models will be provided as well as opportunities that should encourage the use of better decision-making in the near future.
Bruce Berndt
Ramanujan's Life and Notebooks, by Professor Bruce Berndt (University of Illinois at Urbana-Champaign) – Professor Berndt is the 2005 Mahler Lecturer of The Australian Mathematical Society
Scheduled: 12 noon, Friday 7 October 2005, in Room B2.30
Abstract: Ramanujan was born in southern India in 1887 and died there in 1920 at the age of 32. He had only one year of college, but his mathematical discoveries, made mostly in isolation, have made him one of this century's most influential mathematicians. An account of Ramanujan's life will be presented. Most of Ramanujan's mathematical discoveries were recorded without proofs in notebooks, and a description and history of these notebooks will be provided. The lecture will be accompanied by overhead transparencies depicting Ramanujan, his home, his school, his notebooks, and those influential in his life, including his mother and wife.
An article about the seminar, titled "Studying a maths genious (sic) at Bendigo", appeared in the Bendigo Weekly newspaper of Friday September 30, 2005.
Jason Giri
Finding the Low Points: A Journey Through Constrained Nonconvex Minimisation, by Jason Giri (University of Ballarat)
Scheduled: 12 noon, Friday 16 September 2005, in Room B2.30
Abstract: Constrained nonconvex optimisation is a field of mathematics which hopes to provide general techniques for solving a large class of practical optimisation problems. Unfortunately, most of these problems are very difficult to solve because they exhibit qualities such as nonsmoothness or multiple constraints along with their nonconvexity. In this seminar I will discuss some of the current minimisation techniques which I have investigated in my PhD thesis and give an overview of their strengths and weaknesses. The seminar will culminate with the presentation of a solution for the well-known Rosen-Suzuki function which improves on that previously published in the literature.
Joe Ryan
The train marshalling problem, by Dr Joe Ryan (University of Ballarat)
Scheduled: 12 noon, Friday 26 August 2005, in Room B2.30
Abstract: The problem considered in this paper arose in connection with the rearrangement of railroad cars in China. In terms of sequences, the problem reads as follows.
Train marshalling problem: Given a partition S of {1,
2, …, n} into disjoint sets S1, S2,
…, St, find the smallest number
The sequence of numbers 1,2,…,n, 1,2,…,n,
… , 1,2,…,n, where the interval 1,2,…,n
is repeated k times, contains all the elements of
Gary Bloom
Crossing numbers, EODERMDROMEs, and the authenticity of Shakespeare's work, by Professor Gary Bloom (City College and the Graduate Center of the City University of New York)
Scheduled: 12 noon, Friday 19 August 2005, in Room B2.30
Abstract: A major technique in linguistics for attributing authorship of Shakespeare's work and other early writings (before copywriting) has been counting word frequencies in the work in question. This seems overly crude to us. We also questioned whether there was an "easy" way to distinguish between languages, other authors, styles of writing, etc.
We chose a type of graph, spelling nets, that are easily generated by text and then chose a graph parameter, crossing number, that we hoped would capture the essential differences in comparitive texts. We discuss why, after writing our third published paper in this area, we realized that we failed.
The silver lining that remained is an intriguing word game of the easy-to-state-but-moderately-hard-to-solve variety. We show a number of solutions in English and various other languages, and we show how even this game gives rise to a number of unanswered quesitons.
Finally, we return to the original questions and consider some more recent work that shows promise for answering our original questions.
Terry Mills
Lost in Space, by Professor Terry Mills (La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 29 July 2005, in Room B2.30
Abstract: If you were to step out into space, and wander around randomly, what are the chances that you would return to your starting point?
The answer depends on the space!
In this lecture, I will discuss a classical result of Georg Pólya (1921) in the theory of random walks.
John Crossley
What is mathematical logic?, by Professor John Crossley (Monash University)
Scheduled: 12 noon, Friday 3 June 2005, in Room B2.30 [This seminar is presented jointly with the Department of Computer Science and Computer Engineering.]
Abstract: Mathematical logic is the application of mathematical techniques to logic.
What is logic? I believe I am following the ancient Greek philosopher Aristotle when I say that logic is the (correct) rearranging of facts to find the information that we want.
This involves looking at syntax and semantics. Syntax is how we say things; semantics is what we mean. It should be quite surprising that a few basic rules suffice to yield all those syntactic expression that are always true – and no others.
Next we consider the variety of logics there now are.
We then turn to other parts of mathematical logic: computable or recursive functions, set theory and proof theory.
All of these developments from virtually nothing have taken place over little more than a century.
Christopher Lenard
Tutte's Flow Conjectures: Beyond the Four Colour Theorem, by Dr Christopher Lenard (La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 27 May 2005, in Room B2.30
Abstract: About 30 years ago the Four Colour Theorem (for planar graphs) was proved by a computational examination of thousands of special cases: today we seem to be no nearer a "short" proof.
Tutte's Flow Conjectures are set in a framework which exploits the duality between colouring faces of a map and colouring edges in a graph, and so these conjectures are a natural extension of ideas surrounding the Four Colour Theorem.
A flow on a graph is an integer weighting of the (directed) edges in such a way that Kirkhoff's laws are obeyed at each vertex, while a k-flow is a flow in which the weights are chosen from
Four Colour Theorem: Every planar graph is face-4-colourable, and equivalently, every bridgeless planar graph admits a nowhere-zero 4-flow.
Eliminating the constraint of planarity leads to the first of Tutte's conjectures:
5-flow Conjecture: Every bridgeless graph admits a nowhere-zero 5-flow.
In this talk we will survey the ideas surrounding this and various other of Tutte's conjectures.
Grant Cairns
Classification of a special family of graded Lie algebras, by Associate Professor Grant Cairns (La Trobe University, Bundoora)
Scheduled: 12 noon, Friday 20 May 2005, in Room B2.30
Abstract: We classify those finite dimensional Lie algebras which have a basis x1,...,xn with the following properties:
- [xi, xj] = ci, j xi+j for some constants ci, j ,
- c1, j is not zero, for all 1<j<n.
It turns out that there are only 6 such algebras of dimension n<7. In each of the dimensions 7,8,9,10,11, there are infinitely many algebras, while in dimension n>11, there are precisely 4 or 5 algebras, according to whether n is even or odd respectively. This is joint work with Barry Jessup (University of Ottawa).
George Grätzer
Representing finite distributive lattices as congruence lattices of lattices, by Professor George Grätzer FRSC (University of Manitoba, Canada)
Scheduled: 2.30 pm, Friday 13 May 2005, in Room B2.27
Abstract: In 1942, R. Dilworth proved the following theorem: Every finite distributive lattice D can be represented as the congruence lattice of a finite lattice L.
We want: Every finite distributive lattice D can be represented as the congruence lattice of a "nice" finite lattice L.
In 1960, G. Grätzer and E. T. Schmidt proved this with nice = sectionally complemented.
We shall review this field discussing, in addition, nice = semimodular, "small", uniform, isoform, and so on.
Simon Smith
Lebesgue Constants – from Bendigo to Budapest, by Associate Professor Simon Smith (La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 15 April 2005, in Room B2.30
Abstract: Lagrange interpolation is a well-known method for approximating a continuous function by a polynomial that agrees with the function at a number of chosen points (the "nodes"). However, the accuracy of the approximation can be greatly influenced by the location of these nodes.
In this talk I will explain why the Lebesgue constant is a useful way to measure a given set of nodes to determine whether its Lagrange polynomials are likely to provide good approximations. There will then be discussion of methods and results for the calculation of Lebesgue constants for some particular node systems and for a weighted interpolation method. Some of these methods and results are classical, while others follow from research carried out recently in Bendigo and while I was on OSP leave in Budapest.
John Schutz
A System of Axioms for Hyperbolic Geometry, by Associate Professor John Schutz (La Trobe University, Bendigo)
Scheduled: 12 noon, Friday 18 February 2005, in Room B2.32
Abstract: Three-dimensional hyperbolic geometry is characterized using axioms of order, incidence, dimension, continuity and, instead of an axiom of parallels, there is an axiom of "rigidity" and, rather than several axioms of congruence, there is one axiom of symmetry. It is claimed that this system of axioms is simpler than previous systems of axioms. If people are interested and if time permits, I will also discuss a characterisation of ellipsoids in projective geometry by a property of isotropy about a single point.