The minimal fractions of the popular vote that could have elected a US President in the Electoral College in two-party elections in 1948–2004 are calculated by solving auxiliary knapsack problems. It is shown that under the rules of US presidential elections determined by Article 2 of the US Constitution, the values of these minimal fractions were within the range 16.072%–22.103%.
A US Federal election in which candidates from two major political parties compete for the votes of those undecided voters in a state who usually do not vote in US elections is considered. A mathematical model for evaluating the expectation of the margin of votes to be received from such voters by either candidate as a result of the election campaigns of all the competing candidates is proposed. On the basis of this model, finding the estimation under consideration is reducible to finding the minimum of the maximin function of the difference of two bilinear functions with one and the same first vector argument whose second vector arguments belong to a polyhedron of connected variables (strategies of the candidates), and this minimum is sought on another polyhedron.
Formulae for calculating the minimal number of homogeneous objects to constitute a plurality of objects within a finite heterogeneous system are presented. Such a system is searched for among all systems that can be formed by groups of available types of homogeneous objects and consist of one and the same total number of objects grouped into one and the same number of groups of homogeneous objects.