Euclidian group

The existence of at least one invariant mean characterizes a class of groups, the amenable groups, which include translation groups R", rotation groups O", and Euclidian groups E . Rem. the reason is R" abelian, G compact, and E semi-direct product of R and O") However, the Loren tz group is not amenable. 

Note that in most physical applications, si is taken as the algebra of quasi-local observables on S (see, e.g., the next section). In this case, the translation group (and hence the euclidian group) does act in an r -abelian manner on all on as a consequence of locality (i.e, the fact that two observables relative to disjoint regions of space commute). The question of whether the time-evolution acts in the same manner is much more delicate, and on the basis of explicit solutions of known models, we know that this property is by no means guaranteed in general,... 

The //-space Euclidian distances for every object relative to the group averages A and B are calculated and plotted. 

If, in the second step, the elemental contents of the unknown hair sample (Table 5.7) are inserted into the discriminant functions of Eqs. 5.119 and 5.120, then the following values are obtained for the scalar values = 4.416 and 2 = -7.15. If compared to the centroids of the classes (Figure 5.27), we obtain the values 15.66,4.93, and 10.52 for the Euclidian distances of the classes A, B, and C, respectively. The shortest distance is found between the data of the unknown hair and class B, that is, the hair has to be assigned to the subject group B. 

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