Uniqueness in Harper’s vertex-isoperimetric theorem

An isoperimetric comparison theorem

The Vertex Isoperimetric Problem on Kneser Graphs

For a simple graph G = (V,E), the vertex boundary of a subset A ⊆ V consists of all vertices not in A that are adjacent to some vertex in A. The goal of the vertex isoperimetric problem is to determine the minimum boundary size of all vertex subsets of a given size. In particular, define μG(r) as the minimum boundary size of all vertex subsets of G of size r. Meanwhile, the vertex set of the Kn...

Vertex isoperimetric parameter of a Computation Graph

Let G = (V, E) be a computation graph, which is a directed graph representing a straight line computation and S ⊂ V . We say a vertex v is an input vertex for S if there is an edge (v, u) such that v ∕∈ S and u ∈ S. We say a vertex u is an output vertex for S if there is an edge (u, v) such that u ∈ S and v ∕∈ S. A vertex is called a boundary vertex for a set S if it is either an input vertex o...

An Isoperimetric Theorem in Plane Geometry

An isoperimetric theorem in plane geometry Alan Siegel1 COURANT INSTITUTE OF MATHEMATICAL SCIENCES NEW YORK UNIVERSITY New York Abstract Let be a simple polygon. Let the vertices of be mapped, according to a counterclockwise traversal of the boundary, into a strictly increasing sequence of real numbers in . Let a ray be drawn from each vertex so that the angle formed by the ray and a horizontal...

Kamke's Uniqueness Theorem

A generalization of Kamke's uniqueness theorem in ordinary differential equations is obtained for the limit Cauchy problem, viz x'{t) = f(t, x(t)), x{t) -> x0 as 1J10, where / and x take values in an arbitrary normed linear space X and the initial point {t0, x0) is permitted to be on the boundary of the domain of/. Kamke's hypothesis that \\f(t,x)-f{t,y)\\ < <(>(\t-to\, ||x-,y||) is replaced by...