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Proper gromov transforms of metrics are metrics

โœ Scribed by A. Dress

Publisher
Elsevier Science
Year
2002
Tongue
English
Weight
312 KB
Volume
15
Category
Article
ISSN
0893-9659

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โœฆ Synopsis

In phylogenetic analysis, a standard problem is to approximate a given metric by an additive metric. Here it is shown that, given a metric D defined on some finite set X and a nonexpansive map f : X + R, the one-parameter family of the Gromov transforms DApf of D relative to f and A that starts with D for large values of A and ends with an additive metric for A = 0 consists exclusively of metrics. It is expected that this result will help to better understand some standard tree reconstruction procedures considered in phylogenetic analysis.

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A metric transformation between two metric spaces M 1 and M 2 is defined to be a function f such that for some function p:~+ ~ ~+, called the scale function associated with f, p(dl(x, y)) = dz(f(x), f(y)), for all x, y~M 1 . Here E+ = {t~lt >~ 0} and neither of the functions f or p is assumed to be