We consider two very different types of mixing in this thesis. In the first chapter, we discuss the mixing of powders in a slowly rotating drum. In the second, we talk of a type of topological mixing, known as scrambling, as it relates to 'chaotic' functions. Below are brief descriptions of each problem.

While the mixing of granular solids, or powders, is of great interest to the pharmaceutical industry and has been studied quite extensively, very little is known about the basic principles of powder mixing. One common industrial mixing process consists of placing the granular material in a very slowly rotating drum. As the drum turns, successive avalanches occur in the material which generally yield a uniform mixture for collections of particles which are homogeneous with respect to size, shape, and density. We develop a mathematical model of the mixing in this situation. We then generalize this model to describe the mixing that occurs when the drum is rotated quickly enough to induce a continuous avalanche in the powder but slowly enough to avoid significant inertial effects.

The second type of mixing that we consider, unrelated to the first, regards the nature of chaos in discrete dynamical systems. Although there are many commonly used definitions of chaos which utilize such concepts as topological entropy and the Lyapunov exponent, no standard definition of 'chaos' has been universally accepted. In their paper, Period Three Implies Chaos (1975), Li and Yorke introduce the notion of a scrambled pair. For a function f on a metric space (X, d), two points p and q form a scrambled pair iff $/limsup/limits/sb[n/to/infty]/ d (f/sp[n](p), f/sp[n](q))>0,$ and $/liminf/limits/sb[n/to/infty]/ d (f/sp[n] (p),f/sp[n](q))=0.$ We show that, with certain modifications, this concept can be a good indicator of chaos. In particular, our definition of chaos implies positive topological entropy and sensitivity to initial conditions in the sense of Devaney.