NDSU Applied Mathematics Seminar - Spring 2010
Loyola University Chicago
Department of Mathematics and Statistics


Time: 4:00PM-4:50PM  
BVM Hall - Room 1102 (6349 N. Kenmore Avenue, Chicago, IL 60626) [map]

Organizers: Marian Bocea and Rafal Goebel

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Tuesday, August 25  -  ROBERT JENSEN (Loyola University Chicago) - The Vibrating Inverted Pendulum
Abstract: Click here.

Thursday, September 10 - DANIEL SPECTOR (National Chiao Tung University, Taiwan) -  A new embedding theorem in "Sobolev" spaces
Abstract: Specialists in PDEs and Harmonic Analysis oftentimes have different perspectives on the same question.  In this talk I will discuss some of these viewpoints in the context of embedding theorems. In particular, I will present a new result in this area which is surprising in several ways, raising fundamental questions regarding what we understand about integration.

Tuesday, November 10 - RAFAL GOEBEL (Loyola University Chicago) -  A (mildly biased) survey of mathematical models of hybrid systems
Abstract: Classical approach to dynamical systems categorizes each dynamical system as a continuous-time dynamical system or a discrete-time dynamical system. For example, systems of planets that evolve under Newton's laws or biological systems where populations of species grow or decay exponentially are continuous-time systems and have solutions parameterized by time. Your savings account balance that increases every month, recursively-defined sequences, binary values in an algorithm, etc. are examples of discrete-time systems and have solutions parameterized by natural numbers that count the number of transitions. Broadly understood, hybrid dynamical systems are dynamical systems that exhibit features characteristic of continuous-time dynamical systems and features characteristic of discrete-time systems. Modeling of cyber-physical systems that combine analog and digital components; of mechanical systems with impacts, where velocities may change instantaneously; of biological systems with impulses; etc. motivate the interest in hybrid systems in control theory and control engineering literature. There, one can find piecewise affine systems, complementarity systems, differential or hybrid automata, dynamical systems on time scales, hybrid inclusions, impulsive differential equations, measure-driven differential equations, switched systems, etc., all of which have hybrid features. The talk will present several of these modeling frameworks, discuss some concepts of solutions to the models, and relate selected frameworks to hybrid inclusions.

Tuesday, November 17 - CHARLES SMART (University of Chicago) - Rates of convergence for the random conductance model
I will survey some of the recent work on the homogenization of the random conductance model.  This will include some work of mine with Armstrong and some work of Armstrong-Kuusi-Mourrat and Gloria-Otto.  There are now two approaches to obtaining optimal rates in for random elliptic equations in divergence form.  The first obtains Green's function estimates by appealing to the Efron-Stein concentration inequality.  The second uses regularity theory to localize the dependence of the solution on the coefficients.  I will discuss both of these methods.