Let S be a subset of R with finite positive Lebesgue measure. The Beer index of convexity b(S) of S is the probability that two points of S chosen uniformly independently at random see each other in S. The convexity ratio c(S) of S is the Lebesgue measure of the largest convex subset of S divided by the Lebesgue measure of S. We investigate the relationship between these two natural measures of...

Lemma 1. f is submodular function. Proof. Let X, Y, Z ⊆ V , and u, v ∈ V be X = Z ∪ {u}, Y = Z ∪ {v}, v ∈ V , and e′(v) be the number of outgoing edges of u toward any vertices of Z. Let m be the number of connected components which are connected to u, but not connected to Z and b, and n be the number of connected components which are connected to v, but not connected to Z and u. 1. There are n...

Let r be the fundamental group of a hyperbolic surface of genus g; for I ::; p ::; q, PSU(p, q) is the group of isometries of a certain Hermitian symmetric space Dp , q of rank p. There exists a characteristic number c : Hom(r, PSU(p, q)) --+ lR, which is constant on each connected component and such that Ic(p)1 ::; 4pn(g I) for every representation p. We show that representations with maximal ...

We prove that given two metrics g+ and g− with curvature κ < −1 on a closed, oriented surface S of genus τ ≥ 2, there exists an AdS manifold N with smooth, space-like, strictly convex boundary such that the induced metrics on the two connected components of ∂N are equal to g+ and g−. Using the duality between convex space-like surfaces in AdS3, we obtain an equivalent result about the prescript...

Let p1, p2, p3 be three noncollinear points in the plane, and let P be a set of n other points in the plane. We show that the number of distinct distances between p1, p2, p3 and the points of P is Ω(n), improving the lower bound Ω(n) of Elekes and Szabó [4] (and considerably simplifying the analysis).

Consider an asymmetric wireless network represented by a digraph G = (V , E). A subset of vertices U is called a strongly connected dominating set (SCDS) if the subgraph induced by U is strongly connected and every vertex not in U has both an in-neighbor in U and an out-neighbor in U . SCDS plays an important role of the virtual backbone in asymmetric wireless networks. Motivated by the constru...

We determine the set of connected components of closed affine Deligne-Lusztig varieties for special maximal compact subgroups of split connected reductive groups. We show that there is a transitive group action on this set. Thus such an affine Deligne-Lusztig variety has isolated points if and only if its dimension is 0. We also obtain a description of the set of these varieties that are zero-d...

Let B be a building over a type set I . Suppose J ⊆ I meets every connected component of the diagram of B and let (P,L) be the point-line truncation of the J -Grassmann geometry of B. We present a theorem that allows one to identify shadows of apartments of residues of B in the point-collinearity graph of (P,L). Also, other results related to shadows of apartments of residues and shadows of res...

The problem of packing k vertex-disjoint copies of a graph H into another graph G is NP-complete if H has more than two vertices in some connected component. In the framework of parameterized complexity we analyze a particular family of instances of this problem, namely the packing of stars. We prove that packing k copies of H = K1,s is fixed-parameter tractable and give a quadratic kernel for ...