The real rank two locus of an algebraic variety is the closure of the union of all secant lines spanned by real points. We seek a semi-algebraic description of this set. Its algebraic boundary consists of the tangential variety and the edge variety. Our study of Segre and Veronese varieties yields a characterization of tensors of real rank two.

An isometric path is merely any shortest path between two vertices. If the vertices of the hypercube Qn are represented by the set of 0–1 vectors of length n, an isometric path is obtained by changing the coordinates of a vector one at a time, never changing the same coordinate more than once. The minimum number of isometric paths required to cover the vertices of Qn is at least 2=(n+1). We sho...

We show (under mild topological assumptions) that small oscillation of the unit normal vector implies Reifenberg flatness. We then apply this observation to the study of chord-arc domains and to a quantitative version of a two-phase free boundary problem for harmonic measure previously studied by Kenig-Toro [KT06].

In a previous paper an incomplete investigation into regular two-graphs on 36 vertices established the existence of at least 227. Using a more efficient algorithm, the two authors have independently verified that in fact these 227 comprise the complete set. An immediate consequence of this is that all strongly regular graphs with parameters (35,16,6,8), (36,14,4,6), (36,20,10,12) and their comp...

A k-fault-tolerant gossip graph is a (multiple) graph whose edges are linearly ordered such that for any ordered pair of vertices u and v, there are k + 1 edge-disjoint ascending paths from u to v. Let τ(n, k) denote the minimum number of edges in a k-fault-tolerant gossip graph with n vertices. In this paper, we present upper and lower bounds on τ(n, k) which improve the previously known bound...