Type: Seminar

Distribution: World

Expiry: 14 Sep 2010

Auth: wm@p628.pc (assumed)

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 ***** 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. ***** 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. ***** 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. ***** 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. ***** 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.