Inverse problem of a buried metallic object

[[abstract]]In this paper we address an inverse scattering problem whose aim is to discuss the CPU time for recovering a perfectly conducting cylindrical object buried in a half-space. First, we use Fourier-series or cubic-spline methods to describe the shape and reformulate the inverse problem into an optimization one. Then we solved it by the improved steady state genetic algorithm (SSGA) and simple genetic algorithm (SGA) respectively and compare the cost time in finding out the global extreme solution of the objective function. It is found the searching ability of SSGA is much powerful than that of the SGA. Even when the initial guess is far away from the exact one, the cost time for converging to a global extreme solution using by SSGA is much less than that by SGA. Numerical results are given to show that the inverse problem by using SSGA is much better than SGA in time costing.[[notice]]需補地點及國別[[conferencetype]]國際[[conferencedate]]20051004~2005100

Inverse problem and Bertrand's theorem

The Bertrand's theorem can be formulated as the solution of an inverse problem for a classical unidimensional motion. We show that the solutions of these problems, if restricted to a given class, can be obtained by solving a numerical equation. This permit a particulary compact and elegant proof of Bertrand's theorem.Comment: 11 pages, 3 figure

The Inverse Shapley Value Problem

For $f$ a weighted voting scheme used by $n$ voters to choose between two candidates, the $n$ \emph{Shapley-Shubik Indices} (or {\em Shapley values}) of $f$ provide a measure of how much control each voter can exert over the overall outcome of the vote. Shapley-Shubik indices were introduced by Lloyd Shapley and Martin Shubik in 1954 \cite{SS54} and are widely studied in social choice theory as a measure of the "influence" of voters. The \emph{Inverse Shapley Value Problem} is the problem of designing a weighted voting scheme which (approximately) achieves a desired input vector of values for the Shapley-Shubik indices. Despite much interest in this problem no provably correct and efficient algorithm was known prior to our work. We give the first efficient algorithm with provable performance guarantees for the Inverse Shapley Value Problem. For any constant \eps > 0 our algorithm runs in fixed poly$(n)$ time (the degree of the polynomial is independent of \eps) and has the following performance guarantee: given as input a vector of desired Shapley values, if any "reasonable" weighted voting scheme (roughly, one in which the threshold is not too skewed) approximately matches the desired vector of values to within some small error, then our algorithm explicitly outputs a weighted voting scheme that achieves this vector of Shapley values to within error \eps. If there is a "reasonable" voting scheme in which all voting weights are integers at most \poly(n) that approximately achieves the desired Shapley values, then our algorithm runs in time \poly(n) and outputs a weighted voting scheme that achieves the target vector of Shapley values to within error $\eps=n^{-1/8}.