Ricardo Bianconi

Nondefinability Results for Expansions of the Field of Real Numbers by the Exponential Function and by the Restricted Sine Function

We prove that no restriction of the sine function to any (open and nonempty) interval is definable in (R, +, ., <, exp, constants), and that no restriction of the exponential function to an (open andnonempty) intervalisdefinablein (R, <, <sin0, constants), wheresino(x) = sin(x) forx E [-7t, 7r], and sino (x) = O for all x f [ , 7T]. ?

Model completeness results for elliptic and abelian functions

Abstract 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.

Undefinability results in o-minimal expansions of the real numbers

We show that if β∈R is not in the field generated by α1,…,αn, then no restriction of the function xβ to an interval is definable in 〈R,+,−,⋅,0,1,<,xα1,…,xαn〉. We also prove that if the real and imaginary parts of a complex analytic function are definable in Rexp or in the expansion of R (definitions in the text) by functions xα, for irrational α, then they are already definable in R. We conclude with some conjectures and open questions.

Some remarks on Schanuel's conjecture

Abstract Schanuels Conjecture is the statement: if x 1 ,…,x n ∈ C are linearly independent over Q , then the transcendence degree of Q (x 1 ,…,x n , exp (x 1 ),…, exp (x n )) 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.

A Daniell integral approach to nonstandard Kurzweil-Henstock integral

A workable nonstandard definition of the Kurzweil-Henstock integral is given via a Daniell integral approach. This allows us to study the HL class of functions from [9]. The theory is recovered together with a few new results.

Definability in o-minimal expansions of the real numbers

Abstract We present some recent results and problems concerning definable sets and functions in o-minimal expansions of the field of real numbers by analytic or smooth functions. We deal with geometry (cell decomposition), Schanuels conjecture and complex analytic varieties.

Schottky type groups and Kleinian groups acting on S3 with the limit set a wild Cantor set

We construct geometrically finite free Kleinian groups acting onS3 whose limit sets are wild Cantor sets.