Arithmetic of the Length Function

On the other hand, we know from Corollary 10.1.4 that s G is equiva- 

Since p, q, r, s S -i(L), we obtain from Lemma 10.3.3 that 

We are assuming that r G Si q). Thus, there exists an element u in rq such 

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Fig. 2. Relationship between the bond lengths of neighboring bonds in sulfur rings indicating strong bond-bond interaction. The distance, d, of a bond between two-coordinated atoms is. a function of the arithmetic mean of the lengths of the two neighboring bonds, V2(di dj). The values were taken from the compounds indicated in the Figure. The curve ends on the right side at d2 = 189 pm, the bond length of the diatomic molecule Sj...

In graphic presentation, Dp,q is the arithmetic mean when plotting x as a function of x " after extraction of a proper root. For a particle size distribution, when the exponential term of d" is equal to 1, 2, or 3, the term d is length, surface, or volume weighted, respectively. The same is true for d . Figure 1.15 shows a particle yze distribution plotted in two forms and their corresponding arithmetic mean Dp,q. 

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