We present a study of the Hausdorff Core problem on simple polygons. A polygon Q is a k-bounded Hausdorff Core of a polygon P if P contains Q, Q is convex, and the Hausdorff distance between P and Q is at most k. A Hausdorff Core of P is a k-bounded Hausdorff Core of P with the minimum possible value of k, which we denote kmin. Given any k and any ε > 0, we describe an algorithm for computing a...

We study the following problem: given a real number k and integer d, what is the smallest ε such that any fractional (k+ε)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k + ε)coloring of the whole graph? The exact values of ε were known for k ∈ {2} ∪ [3,∞) and any d. We determine the exact values of ε for k ∈ (2, 3)...

A (k, ε)-non-malleable extractor is a function nmExt : {0, 1}×{0, 1} → {0, 1} that takes two inputs, a weak source X ∼ {0, 1} of min-entropy k and an independent uniform seed s ∈ {0, 1}, and outputs a bit nmExt(X, s) that is ε-close to uniform, even given the seed s and the value nmExt(X, s′) for an adversarially chosen seed s′ 6= s. Dodis and Wichs (STOC 2009) showed the existence of (k, ε)-no...

We consider the dynamics of N boson systems interacting through a pair potential N−1Va(xi− xj) where Va(x) = a V (x/a). We denote the solution to the N -particle Schrödinger equation by ψN,t. Recall that the Gross-Pitaevskii (GP) equation is a nonlinear Schrödinger equation and the GP hierarchy is an infinite BBGKY hierarchy of equations so that if ut solves the GP equation, then the family of ...

We study the following problem: given a real number k and an integer d, what is the smallest ε such that any fractional (k+ε)-precoloring of vertices at pairwise distances at least d of a fractionally k-colorable graph can be extended to a fractional (k+ ε)coloring of the whole graph? The exact values of ε were known for k ∈ {2} ∪ [3,∞) and any d. We determine the exact values of ε for k ∈ (2, ...