Counting zero kernel pairs over a finite field

Helmke et al. have recently given a formula for the number of reachable pairs of matrices over a finite field. We give a new and elementary proof of the same formula by solving the equivalent problem of determining the number of so called zero kernel pairs over a finite field. We show that the problem is, equivalent to certain other enumeration problems and outline a connection with some recent results of Guo and Yang on the natural density of rectangular unimodular matrices over F-qx]. We also propose a new conjecture on the density of unimodular matrix polynomials. (C) 2016 Elsevier Inc. All rights reserved.

Molecular phylogeny of Glyphochloa (Poaceae, Panicoideae), an endemic grass genus from the Western Ghats, India

The genus Glyphochloa (Poaceae: Panicoideae: Andropogoneae: Rottboellinae) is endemic to peninsular India and is distributed on lateritic plateaus of low and high altitude in and around Western Ghats and the Malabar Coast. The genus presumably originated and diversified in the Western Ghats. Species relationships in the genus Glyphochloa were deduced here based on molecular phylogenies inferred using nuclear ribosomal ITS sequences and plastid intergenic spacer regions (atpB-rbcL, trnT-trnL, trnL-trnF), and new observations were made of spikelet morphology, caryopsis morphology and meiotic chromosome counts. We observed two distinct clades of Glyphochloa s.l. One of these (group I') includes Ophiuros bombaiensis, and is characterized by a single-awned lower glume and a base chromosome number of 6; it grows in low elevation coastal areas. The other clade (group II') has a double-awned lower glume, a base chromosome number of 7, and is restricted to higher elevation lateritic plateaus; G. ratnagirica may belong to the group II clade, or may be a third distinct lineage in the genus. A sister-group relationship between group I and II taxa (with or without G. ratnagirica) is not well supported, although the genus is recovered as monophyletic in shortest trees inferred using ITS or concatenated plastid data. We present a key to species of Glyphochloa and make a new combination for O. bombaiensis.