# Past Seminars

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Cycles and patterns in permutations, lattice paths, and exclusion processes

*Speaker:* Robert Parviainen

*Institution:* The University of Melbourne

*Date:* Fri 9 Jun 2006

*Time:* 3:15 pm

*Location:* Theatre 1, Ground Floor, 111 Barry Street, Carlton*Abstract:* Recently a lattice path representation of the stationary distributions for some exclusion processes was derived. This enables us to write the generating function for the stationary distributions as a continued fraction.
These results in turn may be used to show deep connections between these exclusion processes and permutation statistics. More specifically, parameters in the stationary distribution generating function correspond to, for example, a) the number of cycles, and b) the number of occurrences of a certain pattern in permutations.
This bi-statistic of cycles and patterns in permutations appears, despite high interest in patterns in permutations, to be previously unstudied.

Weights on Walls & Combinatorial Calculations with Orthogonal Polynomials

*Speaker:* Judy-anne Osborn

*Institution:* The University of Melbourne

*Date:* Fri 26 May 2006

*Time:* 3:15 pm

*Location:* Theatre 1, Ground Floor, 111 Barry Street, Carlton*Abstract:* Recently an open problem from the 1970's, that of enumerating directed lattice paths in a slit, each wall of which has an independent weight associated with it, was solved using the Constant Term Method. In this
talk I will describe that solution, as well as presenting its generalization to the case where each wall has any finite number of independent weights layered against it.
The method utilizes rational functions which come from orthogonal polynomials subject to a geometrically apt variable change. I will also present a combinatorial method to directly calculate these transformed orthogonal polynomials.

PDE Approach to valuation and hedging of credit derivatives

*Speaker:* Marek Rutkowski

*Institution:* University of New South Wales,

*Date:* Fri 12 May 2006

*Time:* 3:15 pm

*Location:* Theatre 2, Ground Floor, 111 Barry Street, Carlton.*Abstract:* Our aim is to examine the PDE approach to the valuation and hedging
of a defaultable claim in various settings; this allows us to emphasize the importance of the choice of the traded assets. We start with a general model for the dynamics of the traded primary assets. Subsequently, we further specify our model and we deal with particular defaultable claims such as, for instance, survival claims. For the sake of notational simplicity, we deal throughout with three primary traded assets only. A generalization to the case of any number of primary assets is straightforward, however.
First, we examine the no-arbitrage property of a model in terms of a
martingale measure. Next, we study the PDE approach to valuation of
defaultable claims and we derive the formulae for hedging strategies of a defaultable claim under the assumption that prices of all primary assets are strictly positive. Finally, we show how to adapt the valuation PDEs if some primary assets are defaultable securities with zero recovery, so that their prices vanish after default.

Long cycles in random graphs

*Speaker:* Nick Wormald

*Institution:* University of Waterloo, Canada

*Date:* Fri 5 May 2006

*Time:* 3:15 pm

*Location:* Theatre 2, Ground Floor, 111 Barry Street, Carlton.*Abstract:* When a random graph on n vertices has enough edges (lines joining the vertices), i.e. about (1/2) n log n edges, it almost surely has a cycle containing all of its vertices, and the circumference (length of longest cycle) is n. What happens in the sparser case? When the graph has much less than n/2 edges, so the average degree of vertices (number of incident edges) is less than 1, there are few cycles and the answer is well understood. I will discuss old results on circumference as well as a new lower bound applying to the intermediate range, where the average degree of a vertex is at least 1 but can be regarded as bounded. (Joint work with Jeong Han Kim of Microsoft Research.)

Complexity of a System as a Key to Its Optimisation

*Speaker:* Galina Korotkikh and Victor Korotkikh

*Institution:* Central Queensland University

*Date:* Fri 28 Apr 2006

*Time:* 3:15 pm

*Location:* Theatre 2, Ground Floor, 111 Barry Street, Carlton.*Abstract:* In the talk we discuss results of computational experiments offering the possibility of a general optimality condition of complex systems.
“A complex system demonstrates the optimal performance for a problem, when the structural complexity of the system is in a certain relationship with the structural complexity of the problem.”
The optimality condition presents the structural complexity of a system [2] as a key to its optimisation. Indeed, according to the optimality condition the optimal result can be obtained as long as the structural complexity of the system is properly related with the structural complexity of the problem.
The computational results also indicate that the performance of a complex system may behave as a concave function of the structural complexity and thus reduce the optimisation of a complex system to a one-dimensional concave optimisation. This would become possible irrespective of the number of variables involved in the description of a complex system, because the structural complexity of the system, considered as one variable, could control the performance in such a remarkable way.

Computational Methods for Analyzing Phylogenetic Trees

*Speaker:* Katherine St. John

*Institution:* City University of New York

*Date:* Fri 21 Apr 2006

*Time:* 3:15 pm

*Location:* Theatre 3, 1st Floor, 111 Barry Street, Carlton.*Abstract:* Evolutionary histories, or phylogenies, form an integral part of much work in biology. In addition to the intrinsic interest in the interrelationships between species, phylogenies are used for drug design, multiple sequence alignment, and even as evidence in a recent criminal trial. Much work has been done on designing algorithms that build phylogenetic trees given representative sequences of their DNA. The optimization criteria preferred by biologists for building trees is NP-hard. So, heuristics are often used that return many possible trees, instead of single optimal tree. This talk concentrates on the heuristics used for tree reconstruction, as well as how to summarize, analyze, and visualize these sets of trees. In particular, we will focus on fast reconstruction methods with provably nice properties, calculating biologically meaningful distances between trees quickly, and visualizing large sets of trees using the treecomp package designed by our group, as a module for the Mesquite system (developed by Wayne and David Maddison).
(This work is joint with Nina Amenta, David Hillis, Tamara Munzner, and Tandy Warnow and is supported by grants from the National Science Foundation.)

The Volume Weighted Average Price Option

*Speaker:* Antony Stace

*Institution:* University of Queensland

*Date:* Thu 16 Mar 2006

*Time:* 11:00 am

*Location:* Room 216, Prentice Building (42), University of Queensland*Abstract:* In this talk I introduce the volume weighted average price (VWAP) option. VWAP options have a payoff that is dependent on both the stock price and volume of stock traded over the lifetime of the option. I give some real work examples of both the VWAP price and VWAP options. Following this, I describe several different methods that I have developed to price VWAP options. I concentrate on the valuation of European style, fixed and floating strike call options. Little additional effort is required to adapt the results of this work to value puts.

Bounds for the decay parameter of a general birth-death process

*Speaker:* David Sirl

*Institution:* University of Queensland

*Date:* Thu 9 Mar 2006

*Time:* 11:00 am

*Location:* Room 216, Prentice Building (42), University of Queensland*Abstract:* In the study of continuous-time Markov chains with state spaces consisting of an absorbing state, which is accessible from an irreducible transient class, the so-called decay parameter is of fundamental importance. Aside from being of interest in its own right, the decay parameter plays a central role in the theory of the quasi-stationary distributions of such processes. Despite its importance, it is notoriously difficult to evaluate or even approximate. We outline its significance in the aforementioned contexts, then introduce an alternative characterisation and indicate how this leads to explicit bounds for the value of the decay parameter of a general birth-death process in terms of the birth and death rates only. An immediate corollary is a necessary and sufficient condition for positivity of the decay parameter. We illustrate the power of these results with several examples.
This is joint work with Hanjun Zhang and Phil Pollett

An exactly solvable model of recombination, mutation and selection

*Speaker:* Professor Michael Baake

*Institution:* University of Bielefeld, Germany

*Date:* Wed 1 Mar 2006

*Time:* 11:00 am

*Location:* Richard Berry Building, The University of Melbourne*Abstract:* Population genetics is concerned with the time evolution of a large ensemble of individuals, represented by their genetic sequences. The latter, in the course of reproduction, change under the influence of mutation, selection, recombination and various other factors. Though this is a problem of stochastic nature, important and relevant information can be extracted from its deterministic infinite population limit, which leads to a system of nonlinear differential equations.
In this talk, starting from the case of pure recombination, a relevant continuous time case is discussed that can be solved exactly. In a second step, this exact solution is extended to include arbitary mutation and additive selection. This could serve as the starting point for further extensions of perturbative type and should be more realistic than the models of neutral evolution used so far.

Maritime Path Planning in Minefield Threat Environments

*Speaker:* Ranga Muhandiramge

*Institution:* University of Western Australia

*Date:* Thu 2 Feb 2006

*Time:* 2:15 pm

*Location:* Theatre 2, Ground Floor, 111 Barry Street, Carlton.*Abstract:* The Defense Science and Technology Organization (DSTO) have funded a project to investigate the ways in which mathematics, and in particular operations research, may be helpful in the problem of minefield path
transit at sea. These include finding minimum risk paths and detection/clearance of mines.
Many different models of the problem will be formulated and solved using different network algorithms. Of particular interested is network weight constrained shortest path problem (WCSPP) formulation and its relation to the continuous version of the problem. Improvements to the WCSPP algorithm will also be presented.

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