Let T be the theory of divisible ordered Abelian groups in a language L
where T has quantifier elimination. Let f be a new unary function symbol. We
would like to consider the L(f)-theory T(a) expanding T together with axioms
for “f is an automorphism”. Unfortunately it is well known that T(a) does not
have a model companion and generally is not easy to analyze. Rather we look at
a weaker theory T(l) once again expanding T but with axioms for “l is a linear
bijection”. T(l) has a model companion and we provide a detailed analysis of
this theory.

Let $U$ be an open subset of $R^n$ and $f$, a function from $U$ to $R$, be $C^m$. We call the collection of $f$ and its derivatives, the jet of order $m$ of $f$. In 1934, H. Whitney asked how can we determine whether a collection of continuous functions on a closed subset of $R^n$ is a jet of order $m$ of a $C^m$-function and also gave a solution to this question which is known as Whitney’s Extension Theorem.
In this talk, let $R$ be an o-minimal expansion of the real field. We discuss whether a collection of continuous functions on a closed subset of $R^n$ is a jet of order $m$ of a $C^m$-function which is definable in $R$.