In this paper we show how the newly introduced concept of superstable systems can be used to efficiently solve some traditionally hard design problems in control theory, such as stabilization with fixed–order controllers, rejection of bounded disturbances and robust controller design. Moreover, this approach also allows for formulating and treating some new problems such as the “linear-linear r...

In this paper we prove Vaught's conjecture for superstable theories in which each complete type has nite U? rank. The general idea is to associate with the theory an V ? deenable group G (called the structure group) which controls the isomorphism types of the models. We use module theory to show that when the theory has few countable models and M is a countable model there is a nice decompositi...

We propose a generalization of the graphical chip-firing model allowing for the redistribution dynamics to be governed by any invertible integer matrix while maintaining the long term critical, superstable, and energy minimizing behavior of the classical model.

I wrote this document while preparing for my qualifying exam in model theory. The goal was to clarify for myself some of the many equivalent definitions that appear in stability theory, including where countable language assumptions are necessary, and to collect the arguments that go into Shelah’s stability spectrum for countable theories in a compact, readable, and hopefully memorable form. In...

It is proved that all groups of finite U -rank that have the descending chain condition on definable subgroups are totally transcendental. A corollary is that any stable group that is definable in an o-minimal structure is totally transcendental of finite Morley rank. Motivation for this paper is a problem concerning stable groups definable in ominimal structures. One theorem is that any defina...