George Collins' discovery of Cylindrical Algebraic Decomposition (CAD) as a method for
Quantifier Elimination (QE) for the elementary theory of real closed fields brought a major …

Abstract The Vapnik–Chervonenkis dimension (VC-dim) characterizes the sample learning
complexity of a classification model and it is often used as an indicator for the generalization …

The arrangement of a finite collection of geometric objects is the decomposition of the space
into connected cells induced by them. We survey combinatorial and algorithmic properties of …

Unlike the classical Sturm theorems on the zeros of solutions of second-order ODEs, Sturm's
evolution zero set analysis for parabolic PDEs did not attract much attention in the 19th …

This paper is a survey of the upper bounds on the complexity of basic algebraic and
geometric operations with Pfaffian and Noetherian functions, and with sets definable by …

Abstract We show that the complement of a subanalytic set defined by real analytic functions
from any subalgebra closed under differentiation is a subanalytic set defined by the …

A bstract Inspired by the notion that physical systems can contain only a finite amount of
information or complexity, we introduce a framework that allows for quantifying the amount of …

" Based on a lecture series given by the authors at a satellite meeting of the 2006
International Congress of Mathematicians and on many articles written by them and their …

We consider the structure R^RE obtained from (R,\lt,+,⋅) by adjoining the restricted
exponential and sine functions. We prove Wilkie's conjecture for sets definable in this …

An effective estimate for the local multiplicity of a complete intersection of complex algebraic
and Pfaffian varieties is given, based on a local complex analog of the Rolle-Khovanskii …