# DE Seminars: Spring 2008

#### Monday February 18

Title |
An optimization approach to coiling cables |

Speaker |
Tom Seidman Department of Mathematics and Statistics UMBC |

**Abstract:**

We consider a Cosserat rod model with a concern for the nonlocal selfinteraction prohibiting selfintersection or contact. We introduce an appropriate notion of `rod homotopy’ and then the principal result is the existence of a minimizer for the total potential energy within each nonempty homotopy class. [The intent is that the model will eventually be applicable to coiled DNA strands.]

This is joint work with K. Hoffman.

#### Monday February 25

Title |
G-convergence and homogenization of elliptic operators |

Speaker |
Alex Pankov Department of Mathematics Morgan State University |

**Abstract:**

This is a survey of the homogenization theory from the point of view of G-convergence. We discuss basic facts on G-convergence of linear elliptic operators and homogenization problems in periodic, almost periodic and random settings. Next, we stop by further results on nonlinear operators and some open problems.

#### Monday March 24

Title |
A Noncommutative Wiener’s Lemma and Applications to Gabor expansions |

Speaker |
Radu Balan Department of Mathematics and CSCAMM, University of Maryland, College Park |

**Abstract:**

In this talk I present results on various Banach *-algebras of time-frequency shift operators with absolutely summable coefficients, and two applications. The L^1 theory contains noncommutative versions of the Wiener lemma with various norm estimates. The L^2 theory is built around an explicit formula of the faithful trace in this algebra. The Zak transform turns out to be an effective computational tool in cases of practical interest.

One application is the Heil-Ramanathan-Topiwala conjecture that states that finitely many time-frequency shifts of one L^2 function are linearly independent. This turns to be equivalent to the absence of eigenspectrum for finite linear combinations of time-frequency shifts. I will prove a special case of this conjecture.

The second application is related to the channel equalization problem. I will show how to use the Wiener lemma and the Zak transform to effectively compute this inverse.

#### Monday April 7

Title |
Electrowetting on dielectric |

Speaker |
Andrea Bonito Department of Mathematics University of Maryland, College Park |

**Abstract:**

Electrowetting (EWOD) on dielectric refers to a parallel-plate micro-device that moves fluid droplets through electrically actuated surface tension effects. These devices have potential applications in biomedical `lab-on-a-chip’ devices (such as automated DNA testing and cell separation) and controlled micro-fluidic transport (e.g. mixing and concentration control).

We model the fluid dynamics using Hele-Shaw type equations (in 2-D) with a focus on including the relevant boundary phenomena. In one hand, the Laplace-Beltrami operator plays a crucial role in order to take into account surface tension effects. In the other hand, we model contact line pinning as a static (Coulombic) friction effect that effectively becomes a variational inequality for the motion of the liquid-gas interface.

We analyze this approach, present simulations and compare them to experimental videos of EWOD driven droplets.

This is joint work with R. Nochetto (University of Maryland) and S. Walker (Courant Institute).

#### Monday April 14

Title |
A Computational Model of Nutrient Transport and Acquisition by Diatom Chains in a Moving Fluid |

Speaker |
Magdalena Musielak Department of Mathematics George Washington University |

**Abstract:**

The role of fluid motion in the transport of solutes to and away from cells and aggregates is a fundamental question in biological and chemical oceanography. However, little is known about behavior of phytoplankton cells in well-defined flow fields. Experimental data to test the contribution of advection to nutrient acquisition by phytoplankton are scarce, mainly because of the inability to visualize, record and thus imitate fluid motions in the vicinities of cells in natural flows. Nutrient fluxes on the scale of interest are difficult to detect, and experimental errors in the measurements of enhancement of flux due to flow may often be comparable in magnitude to the predicted values, especially if the test organisms are very small. Thus, computational experiments are needed to analyze the contribution of advection to mass transfer and nutrient acquisition by phytoplankton. We present in this talk a mathematical model that describes the flexible diatom chain, the surrounding fluid, and the nutrient, by a coupled mechanical system. The chain is modeled as a collection of neutrally-buoyant cylinders connected by filaments. The motion of the fluid is governed by the incompressible Navier-Stokes equations. We use the immersed boundary method to couple the interaction of non-motile diatom chains with the viscous, incompressible, moving fluid, and with the nutrient that is advected by and diffusing in the fluid and also consumed by the cells. We apply our model to investigate the behavior of diatom chains in various flow regimes. We examine the impact of shape, length, and flexibility of chains on nutrient uptakes in a turbulent environment. Our numerical solutions for nutrient mass transfer to diatom cells fall within the bounds of the known analytic solutions for limiting cases. These results confirm intuitive predictions, and open the door to possible experimental work to measure the nutrient transport and acquisition for chains with different elasticities.

#### Monday April 21

Title |
Modeling Leakage of CO2 along a Fault for Risk Assessment |

Speaker |
Susan Minkoff Department of Mathematics and Statistics UMBC |

**Abstract:**

Geologic carbon sequestration is one promising tool in the arsenal needed to combat the ever increasing levels of greenhouse gases in the atmosphere. The process of storing carbon dioxide in aquifers and abandoned subsurface reservoirs over long time periods will only be successful if risk assessment indicates that some large proportion of these gases will remain contained underground over hundreds or even thousands of years after injection. Because CO2 is a less dense fluid than brine it will naturally tend to rise to the top of the storage aquifer. After this upward migration CO2 may move rapidly along the top surface of the compartment. Faults and fractures in the overlying stratum could provide a conduit for flow out of the storage container if encountered by the migrating buoyant fluid.

We will present a computational study of the situations in which CO2 would be likely to leak into assets (i.e., groundwater, other hydrocarbon reservoirs, or the earth’s surface) along a simple fault. Specifically we look at the extent of attenuation of the flux when the rising CO2 can enter permeable layers intersected by the fault.

#### Monday April 28

Title |
Steric hindrance effects in thin reaction zones |

Speaker |
David Edwards Department of Mathematical Sciences University of Delaware |

**Abstract:**

Many biological and industrial processes have reactions which occur in thin zones of densely packed receptors, and understanding the rate of such reactions is important. However, interpreting biosensor data correctly is difficult since large ligand molecules can block multiple receptor sites, thus skewing the kinetics. General mathematical principles are presented for handling this phenomenon, and a receptor layer model is presented explicitly. In the limit of small Damkohler number, the non-local nature of the system becomes evident in the association problem, while other experiments can be modeled using local techniques. Explicit and asymptotic solutions are constructed for large-molecule cases motivated by experimental design. The analysis provides insight into surface/volume reactions occurring in various contexts. In particular, this steric hindrance effect can often be quantified with a single dimensionless parameter.

#### Monday May 5

Title |
Period doubling cascades in high dimensional systems |

Speaker |
Evelyn Sander Department of Mathematical Sciences George Mason University |

**Abstract:**

Period doubling cascades were first discovered in 1962, and have since been a hallmark in the study of dynamical behavior. Feigenbaum’s famous results for rigorous demonstrations of cascades and the universality of their placement have been used for a large variety of one-dimensional maps which are similar to quadratic maps. Yorke and Alligood and Franks took a topological point of view for showing that a dynamical change in behavior of a map results as a parameter is varied results in period doubling cascades. We are able to extend these results and apply it to a much richer set of examples than previously done. In particular, we look at period doubling cascades for certain arbitrarily large coupled systems.