### Ramification and units of the cubic number field Q(zeta) generated by a root of x3+abx+B=0

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Sufficient conditions for a map having nonwandering critical points to be Ω-stable are introduced. It is not known if these conditions are necessary, but they are easily verified for all known examples of Ω-stable maps. Their necessity is shown in dimension two. Examples are given of Axiom A maps that have no cycles but are not Ω-stable.

Let X and Y be Banach spaces. A subset M of (X,Y) (the vector space of all compact operators from X into Y endowed with the operator norm) is said to be equicompact if every bounded sequence (xₙ) in X has a subsequence $\left({x}_{k\left(n\right)}\right)\u2099$ such that $\left(T{x}_{k\left(n\right)}\right)\u2099$ is uniformly convergent for T ∈ M. We study the relationship between this concept and the notion of uniformly completely continuous set and give some applications. Among other results, we obtain a generalization of the classical Ascoli theorem and a compactness criterion...

For p ≥ 1, a set K in a Banach space X is said to be relatively p-compact if there exists a p-summable sequence (xₙ) in X with $K\subseteq \sum \u2099\alpha \u2099x\u2099:\left(\alpha \u2099\right)\in {B}_{{\ell}_{{p}^{\text{'}}}}$. We prove that an operator T: X → Y is p-compact (i.e., T maps bounded sets to relatively p-compact sets) iff T* is quasi p-nuclear. Further, we characterize p-summing operators as those operators whose adjoints map relatively compact sets to relatively p-compact sets.

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