Friday, October 13, 2006
2:00pm – 3:00pm
Tam‡s Kalm‡r-Nagy,
Ph.D.
Department of
Aerospace Engineering
Texas A&M
University
ÒOf Random Time
Delays, Rabbits, and the Visible
Points of the
Integer LettuceÓ
Abstract:
The salient feature of today's ubiquitous embedded systems is the
coupling of dynamic components via the underlying communication network.
Embedded systems are common in cars, aircrafts, chemical/nuclear plants and
thus questions regarding stability of such systems should be rigorously
studied. Developing a theoretical and computational framework for stability
analysis for systems with random time delays (due to the communication) is
therefore of great theoretical and practical importance. Motivated by this
need, the author started to investigate digital control systems with random
time delays. This problem gives rise to discrete-time jump linear systems,
where the transition jumps are modeled with an underlying finite-state Markov
chain. One of the simplest, non-trivial such model is the so-called random
Fibonacci sequence. This is a second-order difference equation where the
coefficient matrix is randomly chosen (from a set of two integer valued
matrices) at every time step. In this presentation we show some beautiful and
far-reaching connections between the random Fibonacci recurrence, the visible
points of the plane, a random walk on the induced self-similar graph and
generalized Catalan numbers. In particular, we show that by suitably modifying
the rules of the random Fibonacci map, there is a unique correspondence between
all visible points and the nodes of a self-similar graph (what we call the
Fibonacci graph). The study of the random walk on this structure reveals a nice
interpretation of the generalized Catalan numbers.
Bio:
Tam‡s Kalm‡r-Nagy received his M.S. degree in Engineering Mathematics from the Technical
University of Budapest and his Ph.D. degree in Theoretical and Applied Mechanics from Cornell University in 1995 and 2002, respectively. During 2001-2005 he was a Postdoctoral
Associate and Lecturer at Cornell University, a Research Engineer at the United Technologies
Research Center and he is now an Assistant Professor in the Department of Aerospace Engineering at Texas A&M University. His research interests are in dynamics and control of autonomous vehicles, uncertain and stochastic systems, delay-differential equations, and nonlinear vibrations.