This paper studies notions of scalar relations and crispness of relations.

Let X be a Banach space. Set, for x ∈ X and r ≥ 0 U(x, r) = {y ∈ X : ||x− y|| = r}. Given a nonempty, bounded set A ⊂ X, we set r(A, x) = sup{||x−a|| : a ∈ A} (x ∈ X) (radius of A with respect to x); r(A) = inf{r(A; x) : x ∈ X} (radius of A); δ(A) = sup{||a− b|| : a, b ∈ A} (diameter of A). Clearly, δ(A) ≤ 2r(A) always. Define, for A, the following properties: A is diametral: r(A, x) = δ(A) for...

Call a domain R an sQQR-domain if each simple overring of R, i.e., each ring of the form R[u] with u in the quotient field of R, is an intersection of localizations of R. We characterize Prüfer domains as integrally closed sQQR-domains. In the presence of certain finiteness conditions, we show that the sQQR-property is very strong; for instance, a Mori sQQR-domain must be a Dedekind domain. We ...

Now, the statement (i) follows from (S.1.3) by setting y = x. (ii) Since the vectors ci1 , . . . , ciK−1 are linearly independent in R , it follows that there exists a vector y such that det [ ci1 . . . ciK−1 y ] 6= 0. Hence, by (S.1.3), the (i1, . . . , iK−1)-th column of B(C) is nonzero. (iii) follows from (S.1.3) and the fact that det [ ci1 . . . ciK−1 y ] = 0 if and only if y ∈ span{ci1 , ....

A novel explanation of judgments by experts about the likely outcome of events is presented. It is proposed that these judgments are derived from a combination of the likelihood of every known risk factor that could affect the outcome of the uncertain event, each weighted by the chance recalling that risk. To test this proposal, an ACT-R model was developed in a study of Royal Navy officers. Ea...

Let f s : R R a be a real-valued scalar field, and let x x x s T s = ... ∈ R ( , , ) 1 be partitioned into t subsets of non-overlapping variables as x X X t T = ... ( , , ) 1 , with X i p i ∈ R , for i = 1, ...,

There is no simple formula for developing and implementing successful business strategies across national borders. A popular approach to this challenge is to “think global, act local.” Some argue, however, that maximum consistency across global markets is critical, citing examples such as CocaCola, whose emphasis on quality, brand recognition, and a small world theme has been successful in a nu...