DEFINABLE SETS OF BERKOVICH CURVES

Definable Sets in Ordered Structures

This paper introduces and begins the study of a well-behaved class of linearly ordered structures, the ^-minimal structures. The definition of this class and the corresponding class of theories, the strongly ©-minimal theories, is made in analogy with the notions from stability theory of minimal structures and strongly minimal theories. Theorems 2.1 and 2.3, respectively, provide characterizati...

Definable Sets in Valued Fields

Ind- and pro- definable sets

We describe the indand procategories of the category of definable sets, in some first order theory, in terms of points in a sufficiently saturated model.

Vapnik-chervonenkis Classes of Definable Sets

We show that a class of subsets of a structure uniformly definable by a first-order formula is a Vapnik-Chervonenkis class if and only if the formula does not have the independence property. Via this connection we obtain several new examples of Vapnik-Chervonenkis classes, including sets of positivity of finitely subanalytic functions.

Approximating the Volume of Definable Sets

The rst part of this paper deals with nite-precision arithmetic. We give an upper bound on the precision that should be used in a Monte-Carlo integration method. Such bounds have been known only for convex sets; our bound applies to almost any \rea-sonable" set. In the second part of the paper, we show how to construct in polynomial time rst-order formulas that approximately deene the volume of...