# DE Seminars: Fall 2010

#### Monday September 20

Title |
Neuronal Cable Theory on Graphs: A Synopsis of Some Forward and Inverse Problems |

Speaker |
Jonathan Bell Department of Mathematics & Statistics UMBC |

**Abstract:**

Motivated by conduction of signals along dendritic trees of a neuron, I will introduce various problems associated with diffusion equations on metric graphs. I will mention some threshold results and bounds on propagation speed, and conduction block, as ‘forward’ problems, and recovery of parameters and morphology of the tree as inverse problems.

#### Monday October 4

Title |
Nonlinear Pulse Dynamics in a Fiber Laser |

Speaker |
John Zweck Department of Mathematics & Statistics UMBC |

**Abstract:**

We study the nonlinear dynamics of the high power femto-second pulses generated in fiber lasers that incorporate a normal dispersion fiber amplifier and anomalous dispersion compensation. Stable operation is permitted provided the total dispersion is normal and does not fall below a power-dependent threshold. Whereas the chirp of a stable pulse is always positive, unstable operation occurs when the chirp undergoes large swings about zero. We discuss implications for minimization of timing jitter and applications to accurate measurement of time and frequency.

#### Monday October 11

Title |
Convergence of tau leaping schemes for Markov processes on an integer lattice |

Speaker |
Muruhan Rathinam Department of Mathematics & Statistics UMBC |

**Abstract:**

Markov process models in continuous time with a multidimensional integer state space naturally arise in several applications. Examples include intracellular chemical kinetics, epidemiology, population dynamics and queuing theory. While exact Montecarlo simulation of sample paths of such systems is possible, for many situations in practice this may be computationally prohibitive. Tau leap methods were devised to accelerate such simulations and are analogous to time stepping methods such as Euler or Runge-Kutta for deterministic or stochastic ordinary differential equations.

Convergence results available for tau leaping methods so far only consider particular tau leap methods under the assumption that the system remains in a bounded domain with probability one. We present new convergence results which apply to fairly general class of tau leap methods as well as to unbounded systems that satisfy certain moment growth bounds.

#### Monday October 18

Title |
Chaos in a Predator-Prey Model with a Scavenger |

Speaker |
Kathleen Hoffman Department of Mathematics & Statistics UMBC |

**Abstract:**

The dynamics of the planar two-species Lotka-Volterra predator-prey model are well understood. In this talk, I will develop the dynamics of the planar system and onto such a predator-prey model, we introduce a third species, a scavenger of the prey. Our model allows for two scenarios, one where the scavenger is also a predator of the original prey, and one where the presence of the scavenger simply inhibits the prey. We show all trajectories are bounded in forward time, and numerically demonstrate persistent cascades of period-doubling orbits over a wide range of parameter values.

#### Monday November 1

Title |
Integral tau methods for stiff stochastic chemical systems |

Speaker |
Yushu Yang Department of Mathematics and Statistics UMBC |

**Abstract:**

Tau leaping methods can efficiently simulate models of stochastic chemical systems. Stiff stochastic systems are particularly difficult since implicit methods which are good for stiffness result in non-integer states. The occurrence of negative states is also a common problem in tau leaping. In this paper, we introduce the Implicit Minkowski-Weyl tau (IMW-tau) methods. Two updating schemes of the IMW-tau methods are presented: IMW-S (Implicit Minkowski-Wyel Sequential) and IMW-P (Implicit Minkowski-Wyel Parallel). The main desirable feature of these methods is that they are designed for stiff stochastic systems with molecular copy numbers ranging from small to large and that they produce integer states without rounding. We illustrate the IMW-S and IMW-P methods by some numerical examples, and compare them with the trapezoidal tau and REMM tau methods. For most cases, the IMW-S and IMW-P methods perform favorably.

#### Monday November 15

Title |
ADI as Preconditioner for Krylov Subspace Methods |

Speaker |
Kyle Stern Department of Mathematics & Statistics UMBC |

**Abstract:**

The alternating directions implicit (ADI) method is a classical iterative method for numerically solving linear systems arising from discretizations of partial differential equations. We propose to use ADI as a preconditioner for Krylov subspace methods for linear systems arising from the discretization of partial differential equations. This talk uses a finite difference approximation of an elliptic test problem as test problem for the computational experiments. The method is attractive because it allows highly efficient matrix-free implementations both in serial MATLAB and parallel C with MPI.

#### Monday November 29

Title |
#1: Modeling Sensory Input to the Lamprey Spinal Cord#2: A Mathematical Model of Vulnerable Plaque Growth and Rupture |

Speaker |
#1: Geoffrey Clapp #2: Alexandria Volkening Department of Mathematics & Statistics, UMBC |

**Abstract:**

(Geoffrey Clapp): We develop and evaluate a model of the lamprey.s central pattern generator (CPG) of locomotion, which will be used to study the relationship between sensory input from edge cells and vertebrate locomotion. A mathematical model has become essential to advancing our knowledge on this topic because of a lack of dependable methods for obtaining data from biological experiments. Previous biological experiments have demonstrated that edge cell input plays a significant role in modulating the lamprey swimming behavior. Beyond that, there are significant gaps in our knowledge about the connections between edge cells and the spinal cord, and about the intra- and intersegmental connections between neurons. Our specific model is an example of a neural model, which provides a more detailed representation of classes of neurons, in contrast to a phase model that uses only one variable per spinal cord segment. So far, we have successfully reproduced numerical results from both a biological experiment and from simulations using a phase model. Our study has revealed that the phenomenon of larger entrainment ranges produced when forcing at the middle is not a generic property of coupled oscillators; rather, it suggests that the CPG has some special underlying properties in its connections between individual neurons.

(Alexandria Volkening): Atherosclerosis, the hardening and narrowing of one’s arteries over time as a result of the accumulation of plaques within the blood vessels, is one of the leading causes of death in industrialized countries. Although many arterial plaques are completely harmless, a subset of lesions, referred to as vulnerable plaques or thin cap fibroatheromata (TCFAs), may grow to the point of rupture, resulting in thrombosis and myocardial infarction. Rupture is caused by many factors, but the thickness of the cap separating the plaque contents from the bloodstream plays a chief role. Simplifying the problem to a two-dimensional longitudinal slice of the artery, we developed a mathematical model of lesion growth to investigate the mechanical and biological processes leading up to plaque rupture. The lesion cap was modeled by a one-dimensional Euler-Bernoulli beam and differential equations were developed to track changes in loading, biological factors, cap rigidity, and cap thickness. Past studies focus mainly on the effects of cap thinning, but our results suggest that cap rigidity and the biological processes that weaken the cap play major roles in determining whether or not rupture will occur.

#### Monday December 13

Title |
The Geometry of the Quaternions and its Applications to Rigid Body Dynamics |

Speaker |
Rouben Rostamian Department of Mathematics & Statistics UMBC |

**Abstract:**

Vector algebra and vector analysis are such essential tools in mathematics that it is difficult to imagine where we will be without these. The standard undergraduate mathematics curriculum puts a great deal of emphasis on vectors (and rightly so). In contrast, quaternions, which are closely related to vectors, receive hardly any attention at all.

It is my objective in this seminar to: (i) illustrate the beauty of the quaternions by providing a review of the quaternionic algebra and analysis; (ii) relate these to the geometry of rotations and reflections in the three-dimensional space; and (iii) show some nontrivial applications of quaternions to the analysis of motion of rigid bodies in space.