Multiplicity of analytic hypersurface singularities under bi-Lipschitz homeomorphisms

We give partial answers to a metric version of Zariski's multiplicity conjecture. In particular, we prove the multiplicity of complex analytic surface (not necessarily isolated) singularities in $\mathbb{C}^3$ is a bi-Lipschitz invariant.Comment: Accepted for publication by the Journal of Topolog

Bell inequality violations under reasonable and under weak hypotheses

Given a sequence of pairs of spin-one half particles in the singlet state, assume that Alice measures the normalized projections along some vector of the spins of one vector per pair along that vector while Bob measures the normalized projections along some vector of the spins of the other member of each pair. Then Quantum Mechanics, or QM, lets one evaluate the correlation of the projections along these two vectors as minus the cosinus of the angle between said vectors; we assume that all vectors are chosen in a fixed plane. Assuming Classical Microscopic Realism, or CMR, there exist also normalized projection pairs of the spins of the pairs of particles along some other pair of vectors. Assuming QM and MR, we also have that the correlations of the projections along the other vectors as minus the cosinus of the angle between the extra vectors. Assuming Locality,i.e., the impossibility of any effect of an event on another event when said events are spatially separated, beside QM and MR, the theory of Bell lets one deduce various violations of some inequalities at some choices of quadruplets of the vectors that have been chosen. Our main result is the existence of quadruplets where at least one of the said inequalities is violated if one only assumes QM, MR and some very mild further hypotheses. These weak hypotheses only concern the behavior of correlations that we use near special quadruplets. We thus get versions of Bell's theorem that are strictly stronger than the original one and in particular do not assume Locality.Comment: 4 pages, 1 figur