Consider a smooth connected solvable group G over a field k. If k is algebraically closed then G = T nRu(G) for any maximal torus T of G. Over more general k, an analogous such semi-direct product structure can fail to exist. For example, consider an imperfect field k of characteristic p > 0 and a ∈ k−kp, so k′ := k(a1/p) is a degree-p purely inseparable extension of k. Note that k′ s := k ′ ⊗k...

A (d, k) set is a subset of R containing a translate of every k-dimensional disc of diameter 1. We show that if (1 + √ 2)k−1 + k > d and k ≥ 2, then every (d, k) set has positive Lebesgue measure. This improves a result of Bourgain, who showed that the analogous statement holds when 2k−1 +k ≥ d and k ≥ 2. We obtain this improvement in two parts. First, we replace Bourgain’s main estimate with a...

Let k be a positive integer, a subset Q of the set of vertices of a graph G is k-dependent in G if each vertex of Q has no more than k neighbours in Q. We present a parallel algorithm which computes a maximal k-dependent set in a graph on n nodes in time O(log 4 n) on an EREW PRAM with O(n 2) processors. In this way, we establish the membership of the problem of constructing a maximal k-depende...

We show a Θ(n log n) bound on the maximal number of occurrences of primitively-rooted k-th powers occurring in a string of length n for any integer k, k ≥ 2. We also show a Θ(n) bound on the maximal number of primitively-rooted powers with fractional exponent e, 1 < e < 2, occurring in a string of length n. This result holds obviously for their maximal number of occurrences. The first result co...

Sub-dominant theory provides efficient tools for clustering. However it classically works only for ultrametrics and ad hoc extensions like Jardine and Sibson’s 2ultrametrics. In this paper we study the extension of the notion of sub-dominant to other distance models in classification accounting for overlapping clusters. We prove that a given dissimilarity admits one and only one lower-maximal q...

We study the spectral radius of graphs with n vertices and k cut edges. In this paper, we show that of all the connected graphs with n vertices and k cut edges, the maximal spectral radius is obtained uniquely at Kk n , where K k n is a graph obtained by joining k independent vertices to one vertex of Kn−k . We also discuss the limit point of the maximal spectral radius. © 2004 Elsevier Inc. Al...