Most democratic countries use election methods to transform election results
into whole numbers which usually give the number of seats in a legislative body
the parties obtained. Which election method does this best can be specified by
measuring the error between the allocated result and the ideal proportion. We
show how to find an election method which is best suited to a given error
function. We also discuss several properties of election methods that have been
labelled paradoxa. In particular we explain the highly publicised ``Alabama''
Paradox for the Hare/Hamilton method and show that other popular election
methods come with their very own paradoxa
We are concerned with the Dirichlet eigenvalue problem Δu + λu = 0 G, u = 0 on Γ, where G is a bounded, two dimensional domain with sufficiently smooth boundary Γ. We deal with the “Ansatz” \documentclass{article}\pagestyle{empty}\begin{document}$ u\left({x,\lambda } \right) = \sum\limits_{m = 1}^M {a_m Y_0 } \left({\sqrt \lambda |x - y_m |} \right) $\end{document} to compute approximate eigenpairs (u*, λ*) by the collocation method. Lower and upper eigenvalue bounds are estimated by an inclusion theorem due to Kuttler‐Sigillito. In contrast to usual choices of trial functions, it is possible to control the numerical stability by placing the source points in dependence of M. The effect arises from the logarithmic singularity for x = ym and allows us to sharpen the eigenvalue bounds by increasing M. We give a strategy for placing the sources, present various application and give a comparison of results which indicates a high efficiency of the method.
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