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F. Rouessac, A. Rouessac's Chem Analysis - Modern Instrum. Methods and Techniques PDF

By F. Rouessac, A. Rouessac

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Proof. Necessity is obvious. Conversely, define Y as the measurable partition of X formed by all nonempty subsets of the type Y = niEICi where Ci is equal to Ai or Ai. For all Y E Y, let Wy = p,(y). For every i E I we define Bi to be {y E Yly ~ Ai}. Thus for every i and j in I, y E Bi~Bj iff y ~ Ai~Aj and so d( i, j) = p,(u{yly ~ Ai~Aj}) = :L{wyIY E Bi~Bj}. • Spaces of symmetric difference type correspond exactly to spaces of Ll-type, as mentioned by Assouad (1980,a) and Avis (1981) and proved in Avis (1977).

An exact visual display is therefore possible. The link between the two approaches is simple: it may be proved that the bijection of Fichet is the restriction to strongly-Robinsonian dissimilarities of the bijection of Diday. For this link and a unified presentation see Durand and Fichet (1988). 12. A dissimilarity d on I and order::; on I are said to be right strongly-compatible (resp. left strongly-compatible) iff: (i) d and ::; are right-compatible (resp. left-compatible). (ii) [i::; j ::; k, d( i, k) = d(j, k)] implies: (resp.

We have the obvious necessary and sufficient condition: (1, d) is embeddable in a weighted star-graph iff the quotient distance J is a star-distance. 0 F. CRlTCHLEY, 34 B. 13. The set Vs of star semi-distances is a closed convex polyhedral cone. For n > 2, its extreme rays are those spanned by the i-dichotomies d{i}' i E I and are linearly independent. Proof. 9: dE Vs Now let: do = ~aid{i} and a. 8. d = ~wid{i} . ~ai: i= i, Summing now over all i, gives: a. 6. Vs is empty. iff gives: (n - 2)ai + a.

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Chem Analysis - Modern Instrum. Methods and Techniques by F. Rouessac, A. Rouessac


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