In a recent paper Samek-Lodovici and Prince (1999) show: (i) that all the potential winners (harmonically unbounded candidates) can be determined in a ranking independent way, and (ii) that this set of potential winners is finite in number. However, they did not consider the influence of sympathetic constraints (McCarthy, 1999, 2003) on their results. These constraints can promote perpetual los...

An ideal K of R is a subset that is both a left ideal and a right ideal of R. For emphasis, we sometimes call it a two-sided ideal but the reader should understand that unless qualified, the word ideal will always refer to a two-sided ideal. The zero ideal (0) and the whole ring R are examples of two-sided ideals in any ring R. A (left)(right) ideal I such that I 6= R is called a proper (left)(...

1.1. Strictly Cyclic Modules and Modular Right Ideals. For a ring A with identity, cyclic modules are precisely those of the form a\A where a is a right ideal. What might be a useful analogous statement for a ring without identity? This question motivates what follows in this subsection. A module M is strictly cyclic if there exists m in M such that mA = M (such an m is called a generator); a r...

Notation Definition t 0 [0, 1] Time within the software life cycle [0,1] q Quality of the old perpetual software product ρ New perpetual software quality improvement ratio over the old version θ The SaaS initial quality improvement ratio over the old perpetual software, 1 < θ < ρ α Rate of software quality improvement for the SaaS product pu One-time upgrade price for existing users to upgrade ...