# Math Colloquia: Spring 2007

#### Wednesday February 07

*2:30–3:30*

Title |
Blowup rates for single component local control of coupled systems |

Speaker |
Thomas I. Seidman |

Department of Mathematics and Statistics | |

UMBC | |

http://www.math.umbc.edu/~seidman/ |

**Abstract:**

I just got a nice result this weekend—a simpler proof replacing a suboptimal estimate by a couple of other people (with a bit of gossip about the earlier results). This is about the asymptotic blowup rate in coupled systems (as the thermoelastic system with hinged boundary conditions) using local control in a single component.

#### Friday February 09

No talk today

#### Friday February 16

*2:30–3:30*

Title |
Accurate State Estimation and Tracking of a Non-Cooperative Target Vehicle |

Speaker |
Julie Thienel |

Aerospace Engineering Department | |

U.S. Naval Academy | |

http://www.usna.edu/AcDean/facultyinfo/faculty&staff.html#aero |

**Abstract:**

Autonomous space rendezvous scenarios require knowledge of the target vehicle state in order to safely dock with the chaser vehicle. Ideally, the target vehicle state information is derived from telemetered data, or with the use of known tracking points on the target vehicle. However, if the target vehicle is non-cooperative and does not have the ability to maintain attitude control, or transmit attitude knowledge, the docking becomes more challenging. This work presents a nonlinear approach for estimating the body rates of a non-cooperative target vehicle, and coupling this estimation to a tracking control scheme. The approach is tested with the robotic servicing mission concept for the Hubble Space Telescope (HST). Such a mission would not only require estimates of the HST attitude and rates, but also precision control to achieve the desired rate and maintain the orientation to successfully dock with HST.

#### Friday February 23

*2:30–3:30*

Title |
Voltage-Controlled Oscillations in Neurons |

Speaker |
Frank Hoppensteadt |

New York University | |

http://www.nyu.edu/public.affairs/leadership/hoppensteadt.html |

**Abstract:**

Computer studies of the van der Pol model of neural activity revealed a rich structure of phase-locking behavior (Flaherty and Hoppensteadt 1978). This motivated experiments with forced rhythms in squid axons that revealed remarkably similar phase-locking behavior (Guttman et al. 1980). VCON emerged from this (serial) collaboration in 1980; it was derived using phase-locked loop methodologies (Horowitz and Hill 1989), which are important in electronics. VCONs share various behaviors with neurons, but they are more amenable to mathematical analysis than other models theretofore (Hoppensteadt 1997). While a VCON is consistent with numerous observations in neuroscience, it is also constructible as an electronic circuit on scales ranging from single electron transistors to phase-locked loop integrated circuits. This talk will concentrate on the mathematical dynamics of a VCON rather than its uses in neuroscience or engineering.

#### Friday March 02

*2:30–3:30*

Title |
Arbitrage and Geometry |

Speaker |
Daniel Q. Naiman |

Department of Applied Mathematics and Statistics | |

Johns Hopkins University | |

http://www.mts.jhu.edu/~dan/naiman.html |

**Abstract:**

Arbitrage is a fundamental notion in mathematical finance, and making the “no free lunch” assumption that arbitrage opportunities in the marketplace are unavailable has played a fundamental role in financial economics. The Arbitrage Theorem, an important example of a *theorem of the alternative*, will be explained for the case of an *m × n* payoff matrix corresponding to *m* scenarios and *n* investments. For an appropriate definition of a *random* payoff matrix, we use geometric reasoning to show that the probability of an arbitrage opportunity is * Σ _{i=0}^{n-1} binom{m-1}{i}/2^{m-1}. *. As a corollary, we conclude that if the number of scenarios

*m*is even, and the number of available investments

*m/2*then the probability of an arbitrage opportunity is

*1/2*.

(This is joint work with Edward R. Scheinerman)

#### Friday March 09

*2:30–3:30*

Title |
Frictional Contact Models with Local Compliance: Semismooth Formulation |

Speaker |
Jong-Shi Pang |

Dept. of Mathematical Sciences | |

Rensselaer Polytechnic Institute | |

http://www.rpi.edu/~pangj |

**Abstract:**

A 3-dimensional frictional contact model with local compliance and damping was introduced in the recent Ph.D. thesis of Peng Song and the paper by Song and Kumar and was subsequently studied extensively in two papers by Song, Kumar, and the speaker. This is the first of a two-part paper in which we examine a variant of this model where there is no damping in the normal contact forces but there is coupled stiffness between the normal and tangential forces via body deformations. We show that this frictional contact model admits a formulation as an ordinary differential equation with a boundedly Lipschitz continuous, albeit implicitly defined, semismooth right-hand side with global linear growth. Several major consequences follow from such a formulation: (a) existence and uniqueness of a continuously differentiable solution trajectory originated from an arbitrary initial state, (b) finite contact forces that are semismooth functions of the system state, (c) semismooth dependence of the trajectory on the initial state, and (d) convergence of a shooting method for solving two-point boundary problems. The derived results are valid for both a dynamic model and a quasistatic model, the latter being one in which inertia effects are ignored, and for a broad class of friction cones that include the well-known quadratic Coulomb cone and its polygonal approximations. An ongoing work aims at establishing the absence of Zeno states in such a frictional contact model.

#### Thursday March 15

*2:00–3:00pm, SOND 414*

Title |
Preferential Attachment Random Graphs with General Weight Function |

Speaker |
Krishna Athreya |

Department of Mathematics | |

Iowa State University | |

http://orion.math.iastate.edu/kba/ |

**Abstract:**

#### Friday March 16

*2:30–3:30*

Title |
Contingent Entropies in Elasticity and Electromagnetism |

Speaker |
Constantine M. Dafermos |

Division of Applied Mathematics | |

Brown University | |

http://www.dam.brown.edu/people/facultypage.dafermos.html |

**Abstract:**

For a class of hyperbolic systems of conservation laws in which the traditional entropy fails to be convex, the lecture will introduce an extension of the notion of entropy and will discuss its implications on existence and stability of solutions.

**Biosketch:** I received a Diploma in Civil Engineering from the National Technical University of Athens (1964) and a Ph.D. in Mechanics from the Johns Hopkins University (1967). I have served as Assistant Professor at Cornell University (1968–1971),and as Associate Professor (1971–1975) and Professor (1975–) in the Division of Applied Mathematics at Brown University. Since 1984, I have been the Alumni-Alumnae University Professor at Brown.

#### Friday March 23

Spring Break

#### Friday March 30

*2:30–3:30*

Title |
Non-oscillatory hierarchical reconstruction for DG, central DG and finite volume schemes |

Speaker |
Yingjie Liu |

School of Mathematics | |

Georgia Institute of Technology | |

http://www.math.gatech.edu/~yingjie |

**Abstract:**

Motivated by the moment limiter of Biswas, Devine and Flaherty [Appl. Numer. Math. 14 (1994)], we develop a general non-oscillatory hierarchical reconstruction (NOHR) procedure for removing the spurious oscillations in the high degree polynomial of a cell computed by the central DG scheme. This procedure is fully multidimensional and can be applied to any shapes of cells at least in theory. HR uses the most compact stencil and thus fitts very well with DG. Further more, it doesn’t need any charcteristic decomposition even for very high order, such as 5th order, even though there will be small overshoots/undershoots for very high order when there are interactions of discontinuities. HR can be applied to central and finite volume schemes as well resulting in a new finite volume approach. These finite volume schemes can be used for unstructured meshes with essentially no restriction on meshes, and can be done without charcteristic decomposition. We will demonstrate through numerical experiments the effectiveness of the HR. If there are more time, I will also discuss the recently developed back and forth error compensation and correction method (BFECC) with applications in level set interface computation and fluid simulations.

#### Friday April 06

*2:30–3:30*

Title |
Tau-leaping methods in stochastic chemical kinetics |

Speaker |
Muruhan Rathinam |

Department of Mathematics and Statistics | |

UMBC | |

http://www.math.umbc.edu/~muruhan |

**Abstract:**

Intracellular gene regulatory mechanisms involve small numbers of large molecules and are essentially discrete and stochastic in nature. An appropriate dynamic model involves a discrete state (lattice of non-negative integer vectors) and continuous time Markov process. Exact Montecarlo simulation of sample paths of such processes, though simple, is often prohibitively expensive for systems with several molecular species and several reaction channels.

In this talk we describe an ongoing project which aims to develop methods and rigorous mathematical theory for simulating these systems in an efficient way by “leaping over” several events at a time. We address certain important issues such as stiffness (i.e. the existence of vastly different time scales) as well as the preservation of the nonnegative integer values of the states.

More broadly these schemes would be applicable to other situations such as the dynamics of market microstructures and traffic models where the detailed models are discrete and stochastic.

#### Friday April 13

No talk today

#### Friday April 20

*2:30–3:30*

Title |
Resetting Behavior in a Model of Bursting in Secretory Pituitary Cells: Distinguishing Plateaus from Pseudo-plateaus |

Speaker |
Arthur Sherman |

National Institute of Health | |

http://mrb.niddk.nih.gov/sherman/ |

**Abstract:**

We study a recently discovered class of models for plateau bursting, inspired by models for endocrine pituitary cells. In contrast to classical models for fold-homoclinic (square-wave) bursting, the spikes of the active phase are not supported by limit cycles of the frozen fast subsystem, but are transient oscillations generated by unstable limit cycles emanating from a subcritical Hopf bifurcation around a stable steady state. Experimental timecourses are suggestive of such fold-subHopf models because the spikes tend to be small and variable in amplitude. We show here that distinct properties of the response to attempted resets from the silent phase to the active phase provide a clearer, qualitative criterion for choosing between the two classes of models. The fold-homoclinic class is characterized by induced active phases that increase towards the duration of the unperturbed active phase as resets are delivered later in the silent phase. For the fold-subHopf class, resetting is difficult and succeeds only in limited windows of the silent phase but, paradoxically, can dramatically exceed the native active phase duration.

#### Friday April 27

*2:30–3:30*

Title |
Numerical Analysis of Systems of Ordinary Stochastic Differential Equations |

Speaker |
Henri Schurz |

Department of Mathematics | |

Southern Illinois University | |

http://www.math.siu.edu/schurz/personal.html |

**Abstract:**

Systems of stochastic differential equations (SDEs) play an essential role in dynamic modeling in sciences, engineering, biology and economy due to Heisenberg’s uncertainty principle. Most of those equations cannot be solved analytically, and hence one has to resort to numerical methods. We start with the well-known Ito-formula and present stochastic Taylor expansions along functions of strong solutions of SDEs, known as Wagner-Platen expansions. This leads to the construction of an incredible rich pool of numerical methods for SDEs which we are going to indicate in a condensed form. Eventually, consistency, convergence, order bounds, contractivity, stability and nonnegativity are discussed as the key concepts of numerical approximations. If time permits, some applications to dynamic pricing theory in economy and random population ecology are presented.

The talk is focussing rather on key concepts and tools of stochastic numerics than all technical details. The audience is supposed to be familiar with some essential facts of ordinary SDEs and stochastic integration. However, it is also appropriate for those who want to grasp elements of modern stochastic-numerical analysis in a concise introduction.

#### Friday May 04

No talk today