Evaluation of thresholds for power mean-based and other divisor methods of apportionment

For divisor methods of apportionment with concave up or concave down rounding functions, we prove explicit formulas for the threshold values—the lower and upper bounds for the percentage of population that are necessary and sufficient for a state to receive a particular number of seats. Among the rounding functions with fixed concavity are those based on power means, which include the methods of Adams, Dean, Hill–Huntington, Webster, and Jefferson. The thresholds for Dean’s and Hill–Huntington’s methods had not been evaluated previously. We use the formulas to analyze the behavior of the thresholds for divisor methods with fixed concavity, and compute and compare threshold values for Hill–Huntington’s method (used to apportion the US House of Representatives).     

Bitopologies on Totally Ordered Sets

We study the category of ray bispaces, that is, the category whose objects are totally ordered sets with two topologies, each having a subbase of rays and so that the resulting bitopological space is pairwise weakly symmetric, and whose morphisms are the pairwise continuous functions. In contrast with the purely topological results of [5], we show that, (1) such spaces are utterly normal and hence monotonically normal (in the sense of [6]), and (2) (Intermediate Value Theorem) the pairwise continuous image of a pairwise connected bitopological space in a selective ray bispace is an interval. We also obtain conditions for the equality of the de Groot dual (see [4]) and the ray dual (see [5]) of a ray topology and show that a selective ray topology is compact if and only if it is skew compact.