Let f(x) be a differentiable function on the real line R, and let P be a point not on the graph of f(x). Define the illumination index of P to be the number of distinct tangents to the graph of f which pass through P . We prove that if f ′′ is continuous and nonnegative on R, f ′′ ≥ m > 0 outside a closed interval of R, and f ′′ has finitely many zeros onR, then any point P below the graph of f...

Any vertex labeling f : V → {0, 1} of the graph G = (V, E) induces a partial edge labeling f∗ : E → {0, 1} defined by f∗(uv) = f(u) if and only if f(u) = f(v). The balance index set of G is defined as {|f∗−1(0) − f∗−1(1)| : |f−1(0) − f−1(1)| ≤ 1}. In this paper, we first determine the balance index sets of rooted trees of height not exceeding two, thereby completely settling the problem for tre...

0. Let G be a reductive group over Z. For any field F we can consider the group Gp of F-points on G. At first glance, the groups Gp for different fields F appear to have little in common with each other. I. Gelfand has conjectured that (1) The structure of the representations of GF has fundamental features which do not depend on a choice of F. (2) Moreover, it is possible to define representati...

With the proliferation of XML data and applications on the Internet, efficient XML query processing techniques are in great demand. Answering queries using XML indexes is a natural approach. A number of XML indexes have been proposed in the literature; among them, F&B Index is one powerful index as it is the smallest index that answers all twig queries. However, an F&B Index suffers from the fo...