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The structure of definable sets and maps in dense elementary pairs of o-minimal expansions of ordered abelian groups is described. It turns out that a certain notion of “small definable set” plays a special role in this description. Introduction. In a classical paper [8] A. Robinson proved the completeness of the theory of real closed fields with a… (More)

- Lou van den Dries
- Ann. Pure Appl. Logic
- 1989

- Lou van den Dries
- J. Symb. Log.
- 1997

- Philip Scowcroft, Lou van den Dries
- J. Symb. Log.
- 1988

We construct a model complete and o-minimal expansion of the field of real numbers in which each real function given on [0, 1] by a series ∑ cnxn with 0 ≤ αn → ∞ and ∑ |cn|rαn < ∞ for some r > 1 is definable. This expansion is polynomially bounded.

Theorems of Ax, Kochen and Ersov tell us that the elementary theory of a henselian valuation ring of equal characteristic 0 is completely determined by the elementary theories of its value group and residue eld, see 1], 3], 4], and the references therein. This elementary classiication goes through even when a predicate is added for a eld of representatives… (More)

- Lou van den Dries
- J. Symb. Log.
- 1984

- Lou van den Dries, Adam H. Lewenberg
- J. Symb. Log.
- 1995

- Lou van den Dries, Yiannis N. Moschovakis
- Bulletin of Symbolic Logic
- 2004

Much more is known about cε(a, b), but this simple-to-prove upper bound suggests the proper formulation of the Euclidean’s (worst case) optimality among its peers—algorithms from rem: Conjecture. If an algorithm α computes gcd(x, y) from rem with time complexity cα(x, y), then there is a rational number r > 0 such that for infinitely many pairs a > b > 1,… (More)

- Lou van den Dries, Angus Macintyre, David Marker
- Ann. Pure Appl. Logic
- 2001