We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly… (More)

We consider some classical maps from the theory of abelian varieties and their moduli spaces, and prove their definability on restricted domains in the o-minimal structure Ran,exp. In particular, we… (More)

Let M = 〈M, +, <, 0, {λ}λ∈D〉 be an ordered vector space over an ordered division ring D, and G = 〈G,⊕, eG〉 an n-dimensional group definable in M. We show that if G is definably compact and definably… (More)

In the case of the field C, the algebraic structure is insufficient to determine the Euclidean topology; another topology, Zariski, is associated with the field but this will be too coarse to give a… (More)

For any o-minimal expansion e IR of the ordered additive group of real numbers, the expansion of e IR by the usual multiplication is o-minimal. Recent developments concerning o-minimal expansions of… (More)

Theorem 1. Let K = 〈K, +, ·, . . .〉 be an expansion of K all of whose atomic relations are definable in R. If K is a proper expansion of 〈K, +, ·〉 then the field R is definable in K. By “proper… (More)

We recast the problem of calculating Vapnik-Chervonenkis (VC) density into one of counting types, and thereby calculate bounds (often optimal) on the VC density for some weakly o-minimal, weakly… (More)

The proof of the above theorem, derived from ideas of A. Dolich, is more direct then the original proof of Theorem 0.1. Unfortunately, Theorem 0.2 fails in the case when φ(C, ā) is not closed and… (More)