Distances from Bα functions to F(p,q,s) space

On images of continuous functions from a compact manifold to Euclidean space

Repeated distances in space

For i = 1, . . . . n let C(x;, r,) be a circle in the plane with centre .x i and radius r; . A repeated distance graph is a directed graph whose vertices are the centres and where (xi, x;) is a directed edge whenever x; lies on the circle with centre x, . Special cases are the nearest neighbour graph, when ri is the minimum distance between x, and any other centre, and the furthest neighbour gr...

Popular distances in 3-space

Let m(n) denote the smallest integer m with the property that any set of n points in Euclidean 3-space has an element such that at most m other elements are equidistant from it. We have that cn log log n6m(n)6n (n); where c¿0 is a constant and (n) is an extremely slowly growing function, related to the inverse of the Ackermann function. c © 1999 Elsevier Science B.V. All rights reserved

on images of continuous functions from a compact manifold to euclidean space

Computing Geodesic Distances in Tree Space

We present two algorithms for computing the geodesic distance between phylogenetic trees in tree space, as introduced by Billera, Holmes, and Vogtmann (2001). We show that the possible combinatorial types of shortest paths between two trees can be compactly represented by a partially ordered set. We calculate the shortest distance along each candidate path by converting the problem into one of ...