Solving Non-Uniqueness in Agglomerative Hierarchical Clustering Using Multidendrograms

In agglomerative hierarchical clustering, pair-group methods suffer from a problem of non-uniqueness when two or more distances between different clusters coincide during the amalgamation process. The traditional approach for solving this drawback has been to take any arbitrary criterion in order to break ties between distances, which results in different hierarchical classifications depending on the criterion followed. In this article we propose a variable-group algorithm that consists in grouping more than two clusters at the same time when ties occur. We give a tree representation for the results of the algorithm, which we call a multidendrogram, as well as a generalization of the Lance andWilliams’ formula which enables the implementation of the algorithm in a recursive way. The authors thank A. Arenas for discussion and helpful comments. This work was partially supported by DGES of the Spanish Government Project No. FIS2006–13321–C02–02 and by a grant of Universitat Rovira i Virgili.     

New Membranes Based on Polybenzimidazole for Polymer Fuel Cells

1 Results Acid-doped polybenzimidazoles[1] are particularly appealing because of high proton conductivity with no or low humidification and promising fuel cells performances. PBI, in fact, contains basic functional groups which can easily interact with strong oxo-acids, such as H3PO4 and H2SO4. The acid partially protonates the polymer and partially is freely dispersed in the polymer backbone, so allowing proton migration via Grotthuss mechanism along the anionic chains[2]. Anyway, a technological limit...