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Lídia Rejtő

Lídia Rejtő
I: Mathematical Institute of the Hungarian Academy of Sciences; II: University of Delaware Statistics Program I: H-1364 Budapest, P.O. Box 127 Hungary; II: 214 Townsend Hall, Newark, DE 19717-1303 USA

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Gábor Tusnády

Gábor Tusnády
Mathematical Institute of the Hungarian Academy of Sciences Reáltanoda u. 13-15. H-1364 Budapest, P.O. Box 127 Hungary 1053 Budapest

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*clustering tree with a given length* of a given finite set of points is the spanning tree of an appropriately chosen other set of points approximating the given set of points with minimal sum of square distances among all spanning trees with the given length. DCM A matrix of real numbers is said to be *column monotone orderable* if there exists an ordering of columns of the matrix such that all rows of the matrix become monotone after ordering. The {\emph{monotone sum of squares of a matrix}} is the minimum of sum of squares of differences of the elements of the matrix and a column monotone orderable matrix where the minimum is taken on the set of all column monotone orderable matrices. *Decomposition clusters of monotone orderings* of a matrix is a clustering ofthe rows of the matrix into given number of clusters such that thesum of monotone sum of squares of the matrices formed by the rowsof the same cluster is minimal.DCP A matrix of real numbers is said to be *column partitionable* if there exists a partition of the columns such that the elements belonging to the same subset of the partition are equal in each row. Given a partition of the columns of a matrix the *partition sum of squares of the matrix* is the minimum of the sum of square of differences of the elements of the matrix and a column partitionable matrix where the minimum is taken on the set of all column partitionable matrices. *Decomposition of the rows of a matrix into clusters of partitions* is the minimization of the corresponding partition sum of squares given the number of clusters and the sizes of the subsets of the partitions.