# Twenty-Fourth Annual University of Alabama System Applied Mathematics Meeting

## Saturday, November 5, 2011

### University of Alabama at Birmingham

#### Program

All sessions will be held on the fourth floor of Campbell Hall (CH) on the campus of the University of Alabama at Birmingham.

10:00 | Refreshments | CH 451 |

10:15 | Welcoming remarks Rudi Weikard (UAB) | CH 405 |

10:25 | Traveling Wave Solutions for a Class of Diffusive Predator-Prey Systems Wenzhang Huang (UAH) | CH 405 |

11:05 | Most Likely Path to The Shortfall Risk in Long-Term Hedging with Short-Term Futures Contracts Jing Chen (UA) | CH 405 |

11:25 | Distribution of spiral points on surfaces of revolution Eric Askelson (UAB) | CH 405 |

11:45 | Lunch | Area Restaurants |

1:00 | Random Walks at Random Times: A tool for constructing self-similar processes Paul Jung (UAB) | CH 405 |

1:40 | Global dynamics of four-stage-structured mosquito population models Junliang Lu (UAH) | CH 405 |

2:00 | Treecode-Accelerated Boundary Integral Poisson-Boltzmann Solver Weihua Geng (UA) | CH 405 |

2:40 | Refreshments | CH 451 |

3:00 | Faculty discussion | CH 405 |

3:15 | Student discussion | CH 445 |

#### Abstracts

*Random Walks at Random Times: A tool for constructing self-similar processes*, Paul Jung (UAB)

Random walks in random scenery (RWRS) were first introduced by Kesten and Spitzer (1979). Cohen and Samorodnitsky (2006) studied a certain renormalization of RWRS and proposed self-similar, symmetric alpha-stable processes, which generalize fractional Brownian motion, as their scaling limits. The limiting processes have self-similarity exponents H>1/α. We consider a modification in which a sign associated to the scenery alternates upon successive visits. The resulting process is what we call a random walk at random time. We will discuss their scaling limits, and show that the alternating scenery leads to processes which are stochastic integrals of indicator kernels. Our results complement the above results in that the processes have self-similarity exponents H<1/α.

*Distribution of spiral points on surfaces of revolution,* Eric Askelson (UAB)

Much research has been completed on the subject of evenly distributing points across the surface of a sphere, a problem which is both interesting from a purely mathematical perspective and important from a scientific one in the areas of chemistry and physics. One algorithm in particular has previously been shown to provide near-optimal results on the sphere with large N; this algorithm distributes points in a spiral across the surface of the sphere, forming a characteristic hexagonal lattice which maximizes the distance between neighboring points. In this work, the spiral points algorithm is extended to generalized surfaces of revolution through modification of the algorithm's iteration method; where the original algorithm iterated over z linearly, the modified algorithm iterates over arc length with a correction for uniformity. The spiral points algorithm includes a scaling factor, and this modification introduces another scaling factor. As such, numerical optimization over two parameters is required for the calculation of spiral point sets. This optimization is performed here on the sphere, oblate and prolate ellipsoids, and the torus, and the results presented.

*Traveling Wave Solutions for a Class of Diffusive Predator-Prey Systems,* Wenzhang Huang (UAH)

A shooting method, with the application of a Liapunov function, has been developed to show the existence of traveling wave fronts for a class of Lotka-Volterra diffusive predator-prey systems. In addition, the minimum wave speed has been identified.

*Global dynamics of four-stage-structured mosquito population models,* Junliang Lu (UAH)

Mosquitoes are the main vector for malaria. There are at least 350-500 million cases of malaria annually in the world, which results in about between 1.5 and 2.7 million deaths annually. An effective way to prevent these diseases is to control mosquitoes. In this paper, we construct and study continuous and discrete mosquito population models. In continuous model, we obtain the inherent net reproductive number r_{0}. If r_{0}<1, the continuous model has only one trivial equilibrium point, which is globally asymptotically stable. If r_{0}>1, besides a trivial equilibrium point, which is unstable, the continuous model has a unique positive equilibrium point, which is globally asymptotically stable. Similarly, for the discrete model, we obtain the inherent net reproductive number R_{0}: If R_{0}<1; the discrete model has a unique trivial fixed point, which is globally asymptotically stable. If R_{0}>1, the discrete model has a trivial fixed point, which is unstable, and a unique positive fixed point, which is locally asymptotically stable.

*Treecode-Accelerated Boundary Integral Poisson-Boltzmann Solver*, Weihua Geng (UA)

Implicit solvent models based on the Poisson-Boltzmann (PB) equation greatly reduce the cost of computing electrostatic potentials of solvated biomolecules, in comparison with explicit solvent models. Even so, PB solvers still encounter numerical difficulties stemming from the discontinuous dielectric constant across the molecular surface, boundary condition at spatial infinity, and charge singularities representing the biomolecule. To address these issues we present a linear PB solver employing a well-conditioned boundary integral formulation and GMRES iteration accelerated by a treecode algorithm. The accuracy and efficiency of the method are assessed for the Kirkwood sphere and a solvated protein. We obtain numerical results for both the Poisson-Boltzmann and Poisson equations. Results are compared with those obtained using the mesh-based APBS method. The present scheme offers the opportunity for relatively simple implementation, efficient memory usage, and straightforward parallelization.

*Most Likely Path to The Shortfall Risk in Long-Term Hedging with Short-Term Futures Contracts*, Jing Chen (UA)

In this talk, the most likely paths to the shortfall risk in long-term hedging with short-term futures contracts are discussed. Base on a simple model initially discussed in Culp and Miller, Mello and Parsons, Glasserman and a simple discussion about comparing risks of a cash shortfall and the most likely path to a shortfall by Glasserman, we calculate the most likely path for four basic cases: mean reverting or not, hedged or not. These "optimal" paths give information about how risky events occur and not just their probability of occurrence.

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