# Applied/ACMS/absF13

## Contents

# ACMS Abstracts: Fall 2013

### Rob Sturman (Leeds)

*Lecture 1: The ergodic hierarchy & uniform hyperbolicity*

Ergodic theory provides a hierarchy of behaviours of increasing complexity, essentially covering dynamics from indecomposable, through mixing, to apparently random. Demonstrating that a (real) system possesses any of these properties is typically difficult, but one important class of system - uniformly hyperbolic systems - make the ergodic hierarchy immediately accessible. Uniform hyperbolicity is a strong condition, and so we also describe its weaker counterpart, non-uniform hyperbolicity. Our chief example in this lecture is the Arnold Cat Map, an example of a hyperbolic toral automorphism.

*Lecture 2: Non-uniform hyperbolicity & Pesin theory*

The connection between non-uniform hyperbolicity and the ergodic hierarchy is more difficult, but is made possible by the work of Yakov Pesin in 1977, and extensions due to Katok & Strelcyn. Here we illustrate the theory with a linked twist map, a paradigmatic example of non-uniformly hyperbolic system.

*Seminar: Rates of mixing in models of fluid flow*

The exponential complexity of chaotic advection might be reasonably assumed to produce exponential rates of mixing. However, experiments suggest that in practice, boundaries slow mixing rates down. We give rigorous results from smooth ergodic theory which establishes polynomial mixing rates for linked twist maps, a class of simple models of bounded flows.

### Reed Ogrosky (UW)

*Long-wave modeling of a viscous liquid film inside a vertical tube*

Viscous film flows that coat the interior of a cylinder arise naturally in industrial and biological settings and have a free surface that is unstable to long-wave disturbances. I will discuss a series of recently derived long-wave asymptotic models of these flows, and an accompanying set of experiments, in the (i) absence and (ii) presence of a steady, upwards pressure-driven airflow. In the absence of airflow, the long-wave models show excellent agreement with experiments. In the presence of airflow, the agreement is primarily qualitative in nature, though modifications to the modeling of the free surface stress improve the agreement. In both cases, the long-wave models capture qualitative features of the flow seen in the experiments that are not captured by their popular thin-film model counterparts.

### Shamgar Gurevich (UW)

*The incidence and cross methods for efficient radar detection*

I will explain a model of radar detection and its digital form. The latter enables us to introduce techniques from Applied Algebra (construction of specific vectors using commutative groups of operators, and generalizations of Fast Fourier Transform techniques) to suggest new efficient algorithms for radar detection. I will explain these methods, and I will demonstrate an application to the Inhomogeneous Radar Scene Problem, formulated in our interaction with engineers from General Motors (GM), who want to develop sensitive radar devices for cars.

This is a joint work with Alexander Fish (Mathematics, Sydney) and is part from a joint project with Igal Bilik (GM), Akbar Sayeed (ECE, Madison), Oded Schwartz (EECS, Berkeley), and Kobi Scheim (GM).

TBA

### Shilpa Khatri (UNC)

TBA