Article history: Received 13 October 2007 Received in revised form 12 June 2008 Accepted 31 July 2008

In this paper, we derive some local a priori estimates for Ricci flow. This gives rise to some strong uniqueness theorems. As a corollary, let g(t) be a smooth complete solution to the Ricci flow on R, with the canonical Euclidean metric E as initial data, then g(t) is trivial, i.e. g(t) ≡ E.

In this paper we introduce metric-based means for the space of positive-definite matrices. The mean associated with the Euclidean metric of the ambient space is the usual arithmetic mean. The mean associated with the Riemannian metric corresponds to the geometric mean. We discuss some invariance properties of the Riemannian mean and we use differential geometric tools to give a characterization...

We consider the problem of traveling among random points in Euclidean space, when only a random fraction of the pairs are joined by traversable connections. In particular, we show a threshold for a pair of points to be connected by a geodesic of length arbitrarily close to their Euclidean distance, and analyze the minimum length Traveling Salesperson Tour, extending the BeardwoodHalton-Hammersl...

Short proofs are given to various characterizations of the (circum-)Euclidean squared distance matrices. Linear preserver problems related to these matrices are discussed.

Graphs are attached to the n-dimensional space Z 2r where Z2r is the ring with 2 r elements using an analogue of Euclidean distance. The graphs are shown to be non-Ramanujan for r=4. Comparisons are made with Euclidean graphs attached to Z pr for p an odd prime. The percentage of non-zero eigenvalues of the adjacency operator attached to these 1nite Euclidean graphs is shown to tend to zero as ...

Let f be a function that is analytic in the unit disc. We give new estimates, and new proofs of existing estimates, of the Euclidean length of the image under f of a radial segment in the unit disc. Our methods are based on the hyperbolic geometry of plane domains, and we address some new questions that follow naturally from this approach.