Existence, Continuity, and Computability of Unique Fixed Points
in Analog Network Models

			by Nick D. James


The thesis consists of three research projects concerning mathematical
models for analog computers, originally developed by John Tucker and
Jeff Zucker.  The models are capable of representing systems that
essentially "diverge," exhibiting no valid behaviour--much the way
that digital computers are capable of running programs that never
halt.  While there is no solution to the general Halting Problem,
there are certainly theorems that identify large collections of
instances that are guaranteed to halt.  For example, if we use a
simplified language featuring only assignment, branching, algebraic
operations, and loops whose bounds must be fixed in advance (i.e. at
"compile time"), we know that all instances expressible in this
language will halt.

In this spirit, one of the major objectives of all three thesis
projects is identify a large class of instances of analog computation
(analog computer + input) that are guaranteed to "converge."  In our
semantic models, this convergence is assured if a certain operator
(representing the computer and its input) has a unique fixed point.
The first project is based on an original fixed point construction,
while the second and third projects are based on Tucker and Zucker's
construction.  The second project narrows the scope of the model to a
special case in order to concretely identify a class of operators with
well-behaved fixed points, and considers some applications.  The third
project goes the opposite way: widening the scope of the model in
order to generalize it.