Abstract
Genetic algorithms and other evolutionary algorithms have been successfully applied to solve constrained minimum spanning tree problems in a variety of communication network design problems. In this paper, we enlarge the application of these types of algorithms by presenting a multi-population hybrid genetic algorithm to another communication design problem. This new problem is modeled through a hop-constrained minimum spanning tree also exhibiting the characteristic of flows. All nodes, except for the root node, have a nonnegative flow requirement. In addition to the fixed charge costs, nonlinear flow dependent costs are also considered. This problem is an extension of the well know NP-hard hop-constrained Minimum Spanning Tree problem and we have termed it hop-constrained minimum cost flow spanning tree problem. The efficiency and effectiveness of the proposed method can be seen from the computational results reported.

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We use Adler, Tresser and Worfolk decomposition of Anosov automorphisms to give an explicit construction of the stable and unstable C1+ self-renormalizable sequences.

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We study a Cournot duopoly model using Ferreira-Oliveira-Pinto's R&D investment function. We find the multiple perfect Nash equilibria and we analyse the economical relevant quantities like output levels, prices, consumer surplus, profits and welfare.