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Tuesday, February 28, 2006

**Abstract:** We are interested in expanding the real field by oscillatory functions, in particular, those that arise in solving certain kinds of ODEs. Certainly, in some cases (e.g., sin x), we wind up defining Z (the set of all integers), hence any hope of obtaining useful information via model-theoretic methods is lost. On the other hand, there are some oscillatory functions that do generate well-behaved (in a sense that can be made precise) structures; e.g., sin(log x). I will give a necessary condition on functions f such that (R, ,x,f) is o-minimal and (R, ,x,sin f) does not define Z.