We prove that no restriction of the sine function to any (open and nonempty) interval is definable in $\langle\mathbf{R}, +, \cdot, , and that no restriction of the exponential function to an (open and nonempty) interval is definable in $\langle \mathbf{R}, +, \cdot, , where $\sin_0(x) = \sin(x)$ for x ∈ [ -π,π], and $\sin_0(x) = 0$ for all $x \not\in\lbrack -\pi,\pi\rbrack$.

We show that if is not in the field generated by α1,…,αn, then no restriction of the function xβ to an interval is definable in . We also prove that if the real and imaginary parts of a complex analytic function are definable in Rexp or in the expansion of by functions xα, for irrational α, then they are already definable in . We conclude with some conjectures and open questions.

We prove the model completeness of expansions of the reals by restricted elliptic and abelian functions. We make use of an auxiliary structure admitting quantifier elimination, where the basic relations are strongly definable in the original structure.

We prove the existence of a bound to the number of components of an ∀-definable set in the reals, using Pfaffian functions, and give some applications.

We prove the existence of a bound to the number of components of an $\forall$-definable set in the reals, using Pfaffian functions, and give some applications.

Schanuel's Conjecture is the statement: if x 1 ,…,x n ∈ C are linearly independent over Q , then the transcendence degree of Q ,…, exp ) over Q is at least n . Here we prove that this is true if instead we take infinitesimal elements from any ultrapower of C , and in fact from any nonarchimedean model of the theory of the expansion of the field of real numbers by restricted analytic functions.