Distance functions with dense singular sets

Singular sets of Sobolev functions

Level Sets and Distance Functions

This paper is concerned with the simulation of the Partial Diierential Equation (PDE) driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian propose to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately t...

Dense sets of integers with prescribed representation functions

Let A be a set of integers and let h ≥ 2. For every integer n, let rA,h(n) denote the number of representations of n in the form n = a1 + · · ·+ah, where ai ∈ A for 1 ≤ i ≤ h, and a1 ≤ · · · ≤ ah. The function rA,h : Z→ N, where N = N∪{0,∞}, is the representation function of order h for A. We prove that every function f : Z → N satisfying lim inf|n|→∞ f(n) ≥ g is the representation function of ...

Reconciling Distance Functions and Level Sets

This paper is concerned with the simulation of the Partial Diierential Equation (PDE) driven evolution of a closed surface by means of an implicit representation. In most applications, the natural choice for the implicit representation is the signed distance function to the closed surface. Osher and Sethian propose to evolve the distance function with a Hamilton-Jacobi equation. Unfortunately t...

Different-Distance Sets in a Graph

A set of vertices $S$ in a connected graph $G$ is a different-distance set if, for any vertex $w$ outside $S$, no two vertices in $S$ have the same distance to $w$.The lower and upper different-distance number of a graph are the order of a smallest, respectively largest, maximal different-distance set.We prove that a different-distance set induces either a special type of path or an independent...