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By A.K. Sharma

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We begin with the structures of multidimensional scaling and continue with the structures of classification. Most of these structures correspond to cones of V and most are included in the cone Voo of semi-distances. 1. 1. A dissimilarity d on J is said to be Euclidean iff there exists some finite-dimensional Euclidean space E and an embedding {Xi, i E I} of J in E such that: \;;I (i,j) E J2, IIxi - xjll = d (i,j). 7. Then we can give the following obvious remarks: - a Euclidean dissimilarity is necessarily a semi-distance (we will speak about Euclidean semi-distances).

E. the cone of all ultrametrics obtained from a given ultrametric by a positive nondecreasing transformation. More recently, Aschbacher, Baldi, Baum and Wilson (1987) seem to have rediscovered the embedding. Their proof is original and uses determinants. Let us observe that the embedding is an immediate consequence of inclusions established above. Indeed, if d is ultrametric so is d2 • Then d2 is in V t S;; ... S;; V qh . 9: dE Ve· Again the property for the dimensionality derives from one of two results mentioned above, but not proved.

K, d( i, e) = d(j, e) implies: Vi ::; i, d(e, k) = d(e, j). The dissimilarity and the order are strongly-compatible iff they are right and left strongly-compatible. 0 Clearly an order is right (resp. left) strongly-compatible iff its converse is left (resp. right: strongly-compatible. Therefore, when an order is strongly-compatible (with d), its converse is too. Durand (1989) has shown that when d admits a strongly-compatible order, then every compatible order is strongly-compatible. 13. A dissimilarity d on I is said to be strongly-Robinsonian iff it admits a strongly-compatible order.

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