We study the multiplicity mod 2 of real algebraic hypersurfaces. We prove that under some assumptions on the singularity it is preserved through a semi-algebraic bi-Lipschitz homeomorphism of R. In a first part we find a part of the tangent cone enclosing the multiplicity mod 2 and prove that it is an equivariant subset of S. Studying equivariant submanifolds of S we are able to conclude about ...

— We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a p-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our ...

— We study the geometry of germs of definable (semialgebraic or subanalytic) sets over a p-adic field from the metric, differential and measure geometric point of view. We prove that the local density of such sets at each of their points does exist. We then introduce the notion of distinguished tangent cone with respect to some open subgroup with finite index in the multiplicative group of our ...

We extend Mora’s tangent cone or the écart division algorithm to a homogenized ring of differential operators. This allows us to compute standard bases of modules over the ring of analytic differential operators with respect to sufficiently general orderings which are needed in the D-module theory.

We provide an example of two closed sets S1, S2 ⊂ IR such that S1∩S2 = {0}. Yet, at the origin, a Boltyanskii tangent cone C1 to S1 and the Clarke tangent cone C2 to S2 are strongly transversal. This settles a question originally proposed by H. Sussmann. 1 Introduction Let t 7→ x∗(t) and t 7→ u∗(t) be respectively a trajectory and an optimal control for the Mayer problem    minimize φ ( x(...

Given a flat, projective morphism Y → T from an equidimensional scheme to a nonsingular curve and a subscheme Z of Y , we give conditions under which specialization of the Segre class s(NZY ) of the normal cone of Z in Y implies flatness of the normal cone. We apply this result to study when the relative tangent star cone of a flat family is flat.

Error bounds for cone inclusion problems in Banach spaces are established under conditions weaker than Robinson’s constraint qualification. The results allow the cone to be more general than the origin and therefore also generalize a classical error bound result concerning equality-constrained sets in optimization. Applications in finite-dimensional differentiable inequalities and in tangent co...

1. The tangent cone. We can estimate the directional derivative and gradient of a smooth function quickly and easily using finite difference formulas. While rather inaccurate, such estimates have some appeal, needing neither calculus rules nor even a closed-form expression for the function. In the variational geometry of sets, the role of derivatives and gradients are played by the cones of “ta...