By L. Beranek

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**Extra info for Acoustical Measurements (rev.)**

**Sample text**

The domain is the positive real numbers with their usual ordering and Θ defined by x®y = x+y + 2c(xy) 1/2 where c is a constant between —1 and 1. For each nonnegative value of c, the structure satisfies the axioms of Definition 1: weak ordering is immediate, monotonicity follows from the fact that the first partial derivatives of x Θ y are positive (since x and y are positive and c is nonnegative), and local definability is not an issue because the operation in fact is closed. For negative values of c, however, monotonicity fails.

If this can be done at all, it can be done in many ways: any strictly increasing function defined on the set of numerical labels gives a different set of numerical labels that is just as good, and any two such sets of labels must be related by a strictly increasing function since both preserve order. Second, the numerical operation is just the empirical operation O transferred to work on the labels. Since the original structure is a PCS, the resulting numerical structure must also be a PCS. The structures are exactly alike except that the empirical structure converts equivalence into equality of the representing labels.

In the 48 19. NONADDITIVE REPRESENTATIONS present instance, for right-concatenation, the domain D consists of all a for which a O b is defined. Thus, asserting that right-concatenation by b is upper semicontinuous means that for every c in A there is an open subset G(c) of A such that if a O b is defined, then c > a O b if and only if a is in G(c). Part of the proof of Theorem 6 shows that this is so if and only if the defining condition of Definition 5 is satisfied. Lower semicontinuity is handled similarly but more simply since we do not need to worry about the possible lower boundedness of the domain over which a O b is defined.

### Acoustical Measurements (rev.) by L. Beranek

by James

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