Applied Maths Honours Seminar

Applied Maths Honours Seminar 2010 

Tuesday, September 14, AGR Carslaw 829: 

2:00 Tarek Elgindy 
2:30 Zach Berry-Porter 3:00 Stephanie Wang 

3:30 Break 

3:45 Lucy Cao 
4:15 Thuc Tran 

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Speaker 1: Tarek Elgindy 
Title: Transformations and discrete Painleve equations 

Abstract: The discrete versions of Painleve equations have been derived using methods
including singularity confinement which have allowed links to discrete integrability.
However it is not possible to explicitly write all solutions as special function
solutions to these equations.  By linearizing these discrete equations for special
cases, a large amount of information can be extracted about their behavior.  This also
allows the linearized equations to be expressed as cellular automata.  Furthermore the
linearized version of qP3 can be transformed into q-Bessel functions which provides a
special function relation for these linearized equations.  

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Speaker 2: Zach Berry-Porter 
Title: Laplace’s shallow water equations on a rotating sphere 

Abstract: The Laplace Shallow Water Equations (LSWE’s) govern the motion of a thin,
uniform layer of fluid on the surface of a rotating sphere.  This problem is of
fundamental importance in meteorology and geophysical fluid dynamics when considering
dynamic tides in the Earth’s ocean and atmosphere.  The focus here will be on the
oceanic case with an emphasis on free mode oscillations.  Solution is via an expansion
of the associated fields in terms of spherical harmonics.  This particular approach has
the advantage of reducing the LSWE’s to a linear algebraic eigenvalue problem with nice
symmetry properties.  

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Speaker 3: Stephanie Wang 
Title: Integrability of the discrete generalised sine-Gordon equation 

Abstract: A discrete system on a quad-graph is called integrable if it can be embedded
consistently into three or more dimension.  Scalar discrete integrable systems on
quad-graphs were classified completely under certain conditions by Adler et al (2003).
We considered a discrete system that does not fit into this classification.  The system
studied is the n-dimensional generalised sine-Gordon equation (gSGE), discovered by
differential geometers.  For arbitrary n, we showed the discrete gSGE evolves on a
quad-graph in R2 and is consistent in R3.  We also deduced a discrete linear problem for
this discrete system.  

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Speaker 4: Lucy Cao 
Title: A game theory analysis of Google Adwords auction 

Abstract: In Google’s Adwords program, advertisements are sold based on a cost-per-click
(CPC) model, where each ad “unit” sold is measured “per click”.  Pricing of each click
is determined by a General Second Price (GSP) auction mechanism.  The advertisers’
bidding behaviours in the GSP auction can be analysed as a game of incomplete
information.  Within the set of static game Nash equilibria, one maximises the
advertisers’ payoffs.  The extension of this analysis allows us to find the balanced
bidding (BB) strategy.  If every advertiser plays BB strategy in a repeated game, the
result converges to the static game Nash equilibrium.  

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Speaker 5: Thuc Tran 
Title: Various techniques on estimation of Greeks on
financial securities 

Abstract: The partial derivatives of an option price with respect to the parameters of
the underlying model are called the Greeks of the security.  When closed form solutions
for the prices of securities exist, then closed form solutions for Greeks are also
available.  However, there are many cases of prices of exotic securities that do not
have closed form solutions, so Greeks are then often calculated using Monte Carlo
simulations.  I will be implementing a variety of Monte Carlo simulations to estimate
Greeks across a range of different securities and comparing their advantages and
disadvantages.