Parabolic Subsets

Throughout this section, the letter K stands for a subset of L. 

We are assuming that q G p K). Thus, there exists an element s in (K) such that q G ps. 

The following lemma generalizes Lemma 3.4.8 for Coxeter sets. 

On the other hand, we obtain from Lemma 3.4.8 that H)nSi(HnK) HnK) = (H), 

In this section, we shall see that (L is the direct product of simple closed subsets each of which is generated by the elements of L which it contains. As a consequence, we shall also look at the case where (L is simple. 

Prom Lemma 3.4.8 we know that = (L). Thus, fis / e L, there 

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The eleventh chapter is an outline of a general theory of Coxeter schemes. It turns out that most of the basic observations on Coxeter schemes are natural generalizations of the corresponding ones in the theory of Coxeter groups. In fact, the treatment of so-called parabolic subgroups can be taken over almost word-by-word to parabolic subsets. This close relationship between Coxeter schemes and Coxeter groups has a firm mathematical basis, and it leads us to a general question in scheme theory which we had postponed earlier and would like to briefly address here. 

In the first section, we compile a few general facts about closed subsets of (L) generated by subsets of L. Such closed subsets are called parabolic. The section can be viewed as a continuation of Section 3.6. 

Therefore, a subset of 24 compounds (A18, A32 - A34, A36, ASS -A63, A68 - A7S, A78 -A79) containing only a fluorine at position 6 and ethyl at position 1 was selected in order to better understand just what descriptors were important for activity at position 7. In contrast with the entire data set of 41 - 43 compounds, LFER models for all three bacterial test systems were derived from the subset as shown in Tables XIX, XX and XXI. Equation 9 (Table XIX) for S. aureus indicates that only the presence of an amide nitrogen, INCO(7), and a-p electronic interactions, API, are important determinants. An amide nitrogen at position 7 reduces activity, and there is a parabolic relationship of a-p electronic interactions between positions 6 and 7. The nonlinear result for the latter is probably due to the... 

The same parabolic relationship for a-p electronic interactions is seen with E. coli, but it may be biased for the reasons already discussed. In contrast, the LFER model for Ps. aeruginosa (Table XXII, Eq. 4) indicates that only lipophilicity and molar refractivity of the substituents at position 7 are important determinants of activity. In the initial analysis for this subset using the Ps. aeruginosa test system, A34 (norfloxacin) was an outlier. Because it is the only compound showing such high activity, it was deleted. The latter s calculated activity was 1.96 versus the observed 1.64. 

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