Rapid diversification and secondary sympatry in Australo-Pacific kingfishers (Aves: Alcedinidae: Todiramphus)

Todiramphus chloris is the most widely distributed of the Pacific's ‘great speciators’. Its 50 subspecies constitute a species complex that is distributed over 16 000 km from the Red Sea to Polynesia. We present, to our knowledge, the first comprehensive molecular phylogeny of this enigmatic radiation of kingfishers. Ten Pacific Todiramphus species are embedded within the T. chloris complex, rendering it paraphyletic. Among these is a radiation of five species from the remote islands of Eastern Polynesian, as well as the widespread migratory taxon, Todiramphus sanctus. Our results offer strong support that Pacific Todiramphus, including T. chloris, underwent an extensive range expansion and diversification less than 1 Ma. Multiple instances of secondary sympatry have accumulated in this group, despite its recent origin, including on Australia and oceanic islands in Palau, Vanuatu and the Solomon Islands. Significant ecomorphological and behavioural differences exist between secondarily sympatric lineages, which suggest that pre-mating isolating mechanisms were achieved rapidly during diversification. We found evidence for complex biogeographic patterns, including a novel phylogeographic break in the eastern Solomon Islands that separates a Northern Melanesian clade from Polynesian taxa. In light of our results, we discuss systematic relationships of Todiramphus and propose an updated taxonomy. This paper contributes to our understanding of avian diversification and assembly on islands, and to the systematics of a classically polytypic species complex.This project was funded in part by an American Museum of Natural History Chapman Fellowship (M.J.A.), an American Ornithologists' Union Research Award (M.J.A.), a University of Kansas Doctoral Student Research Fund (M.J.A.) and NSF DEB-1241181 and DEB-0743491 (R.G.M.

On the order of a non-abelian representation group of a slim dense near hexagon

We show that, if the representation group $R$ of a slim dense near hexagon $S$ is non-abelian, then $R$ is of exponent 4 and $|R|=2^{\beta}$, $1+NPdim(S)\leq \beta\leq 1+dimV(S)$, where $NPdim(S)$ is the near polygon embedding dimension of $S$ and $dimV(S)$ is the dimension of the universal representation module $V(S)$ of $S$. Further, if $\beta =1+NPdim(S)$, then $R$ is an extraspecial 2-group (Theorem 1.6)

Near polygons and Fischer spaces

In this paper we exploit the relations between near polygons with lines of size 3 and Fischer spaces to classify near hexagons with quads and with lines of size three. We also construct some infinite families of near polygons

The Veldkamp space of multiple qubits

We introduce a point-line incidence geometry in which the commutation relations of the real Pauli group of multiple qubits are fully encoded. Its points are pairs of Pauli operators differing in sign and each line contains three pairwise commuting operators any of which is the product of the other two (up to sign). We study the properties of its Veldkamp space enabling us to identify subsets of operators which are distinguished from the geometric point of view. These are geometric hyperplanes and pairwise intersections thereof. Among the geometric hyperplanes one can find the set of self-dual operators with respect to the Wootters spin-flip operation well-known from studies concerning multiqubit entanglement measures. In the two- and three-qubit cases a class of hyperplanes gives rise to Mermin squares and other generalized quadrangles. In the three-qubit case the hyperplane with points corresponding to the 27 Wootters self-dual operators is just the underlying geometry of the E6(6) symmetric entropy formula describing black holes and strings in five dimensions.Comment: 15 pages, 1 figure; added references, corrected typos; minor change

Properties of field functionals and characterization of local functionals

Functionals (i.e. functions of functions) are widely used in quantum field theory and solid-state physics. In this paper, functionals are given a rigorous mathematical framework and their main properties are described. The choice of the proper space of test functions (smooth functions) and of the relevant concept of differential (Bastiani differential) are discussed. The relation between the multiple derivatives of a functional and the corresponding distributions is described in detail. It is proved that, in a neighborhood of every test function, the support of a smooth functional is uniformly compactly supported and the order of the corresponding distribution is uniformly bounded. Relying on a recent work by Yoann Dabrowski, several spaces of functionals are furnished with a complete and nuclear topology. In view of physical applications, it is shown that most formal manipulations can be given a rigorous meaning. A new concept of local functionals is proposed and two characterizations of them are given: the first one uses the additivity (or Hammerstein) property, the second one is a variant of Peetre's theorem. Finally, the first step of a cohomological approach to quantum field theory is carried out by proving a global Poincar\'e lemma and defining multi-vector fields and graded functionals within our framework.Comment: 32 pages, no figur