Real algebraic surfaces with isolated singularities

Given a real algebraic surface S in RP3, we propose a constructive procedure to determine the topology of S and to compute non-trivial topological invariants for the pair (RP3, S) under the hypothesis that the real singularities of S are isolated. In particular, starting from an implicit equation of the surface, we compute the number of connected components of S, their Euler characteristics and the weighted 2-adjacency graph of the surface

Geometry of the faithfulness assumption in causal inference

Many algorithms for inferring causality rely heavily on the faithfulness assumption. The main justification for imposing this assumption is that the set of unfaithful distributions has Lebesgue measure zero, since it can be seen as a collection of hypersurfaces in a hypercube. However, due to sampling error the faithfulness condition alone is not sufficient for statistical estimation, and strong-faithfulness has been proposed and assumed to achieve uniform or high-dimensional consistency. In contrast to the plain faithfulness assumption, the set of distributions that is not strong-faithful has nonzero Lebesgue measure and in fact, can be surprisingly large as we show in this paper. We study the strong-faithfulness condition from a geometric and combinatorial point of view and give upper and lower bounds on the Lebesgue measure of strong-faithful distributions for various classes of directed acyclic graphs. Our results imply fundamental limitations for the PC-algorithm and potentially also for other algorithms based on partial correlation testing in the Gaussian case.Comment: Published in at http://dx.doi.org/10.1214/12-AOS1080 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

A graph-based mathematical morphology reader

This survey paper aims at providing a "literary" anthology of mathematical morphology on graphs. It describes in the English language many ideas stemming from a large number of different papers, hence providing a unified view of an active and diverse field of research

Ambient Isotopic Meshing of Implicit Algebraic Surface with Singularities

A complete method is proposed to compute a certified, or ambient isotopic, meshing for an implicit algebraic surface with singularities. By certified, we mean a meshing with correct topology and any given geometric precision. We propose a symbolic-numeric method to compute a certified meshing for the surface inside a box containing singularities and use a modified Plantinga-Vegter marching cube method to compute a certified meshing for the surface inside a box without singularities. Nontrivial examples are given to show the effectiveness of the algorithm. To our knowledge, this is the first method to compute a certified meshing for surfaces with singularities.Comment: 34 pages, 17 Postscript figure