We recall several results of zero divisor graphs of commutative rings. We then examine the preservation of diameter and girth of the zero divisor graph under extension to polynomial and power series rings.

The objective of this paper is to present a mathematically grounded description of the two topological spaces for the design problem and the design solution. These spaces are derived in a generalized form such that they can be applied by researchers studying engineering design and developing new methods or engineers seeking to understand the influence that changes in the problem space have on the solution space. In addition to the formal definitions of the spaces, including assumptions and limitations, three types of supported reasoning are presented to demonstrate the potential uses. These include similarity analysis to compare spaces, an approach to sensitivity analysis of the solution space to changes in the problem space, and finally a distance measure to determine how far a current proposal is to the feasible solution space. This paper is presented to establish a common vocabulary for researchers when discussing, studying, and supporting the dyadic nature of engineering design (problem-solution co-evolution).