# DE Seminars: Spring 2012

#### Monday February 6

Title |
Periodic Solutions of Isotone Hybrid Systems |

Speaker |
Thomas Seidman Department of Mathematics and Statistics UMBC |

**Abstract:**

Inspired by conversations in 1991 with Mark Krasnoselski and Alexei Pokrovski, this paper generalizes an earlier paper of theirs by defining a setting of hybrid systems with isotone switching rules (feedback) for a partially ordered set of modes and then obtaining a periodicity result in that context. An application is given to a partial differential equation modeling calcium release and diffusion in cardiac cells.

#### Monday February 13

Title |
Dynamics of pulsed optical fiber lasers that operate using nonlinear polarization rotation |

Speaker |
Brian Marks UMBC |

**Abstract:**

Fiber lasers have been considered for decades as a means of producing very stable short pulses, and they are making a resurgence recently in high-energy physics and medicine as well as time and frequency measurement and control. I will discuss the physics and dynamics of nonlinear polarization rotation in optical fiber. Based on this phenomenon, l will show how optical fiber lasers may be constructed. I will discuss the underlying dynamical problems related to the components within the laser and how they contribute to pulse formation and stability.

#### Monday February 27

Title |
Comments Concerning Models of Myelinated Fibers |

Speaker |
Jonathan Bell Department of Mathematics and Statistics UMBC |

**Abstract:**

I will introduce three relatively simple models for myelinated neural fibers, and discuss what has, and has not, been done on developing and analyzing traveling wave solutions to such problems. Such solutions must satisfy nonlinear functional differential equations with both forward and backward delays that must be determined along with the wave solution.

#### Monday March 5

Title |
Mathematical Models of Atherosclerosis |

Speaker |
Pak-Wing Fok University of Delaware |

**Abstract:**

Atherosclerotic plaques are fatty deposits that grow mainly in arteries and develop as a result of a chronic inflammatory response. Although plaques are frequently thought of as fatty deposits with little or no internal organization, they are actually commonly characterized according to their composition and morphology. In this talk, I will present models for two types of plaque: thin-cap fibroatheromas (TCFAs) and pathological intimal thickenings (PITs).

TCFAs are characterized by inflammation and the presence of necrotic cores. By solving coupled reaction-diffusion equations for macrophages and dead cells, we explore the joint effects of hypoxic cell death and chemoattraction to oxidized low-density lipoprotein (Ox-LDL), a molecule that is strongly linked to atherosclerosis. The model predicts cores that have approximately the right size and shape when compared to ultrasound images. Normal mode analysis and calculation of the smallest eigenvalue enable us to compute the formation times of the core. An asymptotic analysis reveals that the distribution of Ox-LDL within the plaquedetermines not only the placement and size of cores, but their time of formation.

PITs are characterized by the absence of endothelial cells and negative remodeling whereby the vessel lumen decreases in size. I will present some work in progress on PIT, described as a free-boundary problem. The model couples the diffusion of Platelet-Derived Growth Factor, governed by a Helmholtz equation, to cell migration and proliferation. The predictions are compared with data from animal studies.

#### Monday March 26

Title |
Limiting behavior of a model of an epidemic in a network |

Speaker |
Stephen Thompson, Department of Mathematics and Statistics UMBC |

**Abstract:**

In this talk, I will introduce a model of an epidemic in a network, and explain how the limiting behavior of the model can be described by an evolution equation in L^2. The idea is in some ways similar to the derivation of the heat equation as a the limiting model of transport on a lattice. However, there are important differences, because my model involves nonlocal interactions. I will describe some challenges caused by this, and will also explain some of the results that I have been able to prove.

#### Monday April 9

Title |
Optimal Control of a Free Boundary Problem |

Speaker |
Patrick Sodre UMCP |

**Abstract:**

We consider a PDE constrained optimization problem governed by a free boundary problem. The state system is based on coupling the Laplace equation in the bulk with a Young- Laplace equation on the free boundary to account for surface tension. Our analysis provides a convex constraint on the control such that the state constraints are always satisfied. Using only first order regularity we show that the control to state operator is twice Frechet differentiable. We demonstrate how to slightly improve the regularity of the state variables and under this regime show existence of a control together with second order sufficient optimality conditions.

#### Monday April 16

Title |
Parametric sensitivity estimation of stochastic chemical systems via the Girsanov transformation approach |

Speaker |
Ting Wang Department of Mathematics and Statistics UMBC |

**Abstract:**

Parametric sensitivity analysis is a powerful tool in the modeling and analysis of biochemical network models. There are several different methods to compute the sensitivity in current use: finite difference, path wise derivative, and the Girsanov transformation. All of these involve Monte Carlo simulation and the efficiency of a method depends on the variance of the estimator. In this talk, I will describe the Girsanov change of measure for counting processes and derive the variance of the Girsanov estimator for some simple reaction systems which have linear propensities. To do this, I will apply the Ito formula for counting processes to derive ODE systems describing the time evolution of variance.

#### Monday April 23

Title |
Differentially flat systems |

Speaker |
Muruhan Rathinam Department of Mathematics and Statistics UMBC |

**Abstract:**

Some underdetermined systems of ODEs possess an intriguing property that allows the set of all solutions to be uniquely characterized by a collection of $k$ free functions where $k$ is the number by which the system is underdetermined. For each choice of these $k$ functions a solution to the system is obtained by differentiation alone and all solutions can be obtained in this way. This property was noticed and investigated by David Hilbert and Elie Cartan. Elie Cartan who developed a geometric approach to studying ODE systems via differential forms was able to completely characterized such systems for $k=1$.

Much later, in the 90’s, control theorists recognized that the above property allowed very effective ways for motion planning. They coined the term {\em differentially flat} systems and embarked on a vigorous study. Both differential algebraic as well as differential geometric approaches were used to formulate and study these systems.

In this talk we shall provide a brief overview of differential flatness providing examples, some basics on the theory as well as some results.