Sortable Elements for Quivers with Cycles

Each Coxeter element c of a Coxeter group W defines a subset of W called the c-sortable elements. The choice of a Coxeter element of W is equivalent to the choice of an acyclic orientation of the Coxeter diagram of W. In this paper, we define a more general notion of Omega-sortable elements, where Omega is an arbitrary orientation of the diagram, and show that the key properties of c-sortable elements carry over to the Omega-sortable elements. The proofs of these properties rely on reduction to the acyclic case, but the reductions are nontrivial; in particular, the proofs rely on a subtle combinatorial property of the weak order, as it relates to orientations of the Coxeter diagram. The c-sortable elements are closely tied to the combinatorics of cluster algebras with an acyclic seed; the ultimate motivation behind this paper is to extend this connection beyond the acyclic case.Comment: Final version as published. An error corrected in the previous counterexample, other minor improvement

Sortable elements in infinite Coxeter groups

In a series of previous papers, we studied sortable elements in finite Coxeter groups, and the related Cambrian fans. We applied sortable elements and Cambrian fans to the study of cluster algebras of finite type and the noncrossing partitions associated to Artin groups of finite type. In this paper, as the first step towards expanding these applications beyond finite type, we study sortable elements in a general Coxeter group W. We supply uniform arguments which transform all previous finite-type proofs into uniform proofs (rather than type by type proofs), generalize many of the finite-type results and prove new and more refined results. The key tools in our proofs include a skew-symmetric form related to (a generalization of) the Euler form of quiver theory and the projection \pidown^c mapping each element of W to the unique maximal c-sortable element below it in the weak order. The fibers of \pidown^c essentially define the c-Cambrian fan. The most fundamental results are, first, a precise statement of how sortable elements transform under (BGP) reflection functors and second, a precise description of the fibers of \pidown^c. These fundamental results and others lead to further results on the lattice theory and geometry of Cambrian (semi)lattices and Cambrian fans.Comment: This is essentially the final version, which will appear in Transactions of the AMS. Minor changes have been made in response to comments by referee

Comparative Morphology of Two Sympatric Species of Hedgehog in Ikh Nart Nature Reserve, Mongolia

Daurian hedgehog (Mesechinus dauuricus) and long-eared hedgehog (Hemiechinus auritus) inhabit Ikh Nart Nature Reserve, Mongolia and both species appear to occupy similar niches. We gathered morphological measurements to test for differences between species and sexes. We hoped to gain insight into the mechanisms that allow the species to coexist. We collected morphometric data from 10 long-eared hedgehogs (6 male, 4 female) and 18 Daurian hedgehogs (10 male, 7 female, and 1 unknown sex). Only total body length in Daurian hedgehogs differed significantly between males and females. Otherwise, we found no significant differences between morphometric measures of male and female hedgehogs of either species. Daurian hedgehogs were larger than long-eared hedgehogs for every measurement except for ear length; however, only girth, weight, and length of ear differed significantly. A general linear model found that animal girth best differentiated the two species. Coexistence of species that occupy similar niches generally occurs through the differential use of resources and may be inferred from morphological differentiation. We plan continued research to better explore resource partitioning between the two species

Comparative Morphology of Two Sympatric Species of Hedgehog in Ikh Nart Nature Reserve, Mongolia

Daurian hedgehog (Mesechinus dauuricus) and long-eared hedgehog (Hemiechinus auritus) inhabit Ikh Nart Nature Reserve, Mongolia and both species appear to occupy similar niches. We gathered morphological measurements to test for differences between species and sexes. We hoped to gain insight into the mechanisms that allow the species to coexist. We collected morphometric data from 10 long-eared hedgehogs (6 male, 4 female) and 18 Daurian hedgehogs (10 male, 7 female, and 1 unknown sex). Only total body length in Daurian hedgehogs differed significantly between males and females. Otherwise, we found no significant differences between morphometric measures of male and female hedgehogs of either species. Daurian hedgehogs were larger than long-eared hedgehogs for every measurement except for ear length; however, only girth, weight, and length of ear differed significantly. A general linear model found that animal girth best differentiated the two species. Coexistence of species that occupy similar niches generally occurs through the differential use of resources and may be inferred from morphological differentiation. We plan continued research to better explore resource partitioning between the two species

Comparing different types of patagial tags for use on vultures

Raptor research often requires identifying individuals. Researchers place patagial tags on raptors to facilitate such identification. Researchers in southern African use two main types of patagial tags: hard plastic ear tags originally designed for cattle and soft vinyl tags. We deployed both types of tags on vultures in Botswana.  Based on our observations, we recommend using soft vinyl tags as they appear to be more aerodynamic and  can be read from below when a raptor is soaring, as well as when the bird is perched. Cattle ear tags  sometimes flutter when raptors fly and can only be read when the dorsal surface of the wing is visible

A multiphase seismic investigation of the shallow subduction zone, southern North Island, New Zealand

The shallow structure of the Hikurangi margin, in particular the interface between the Australian Plate and the subducting Pacific Plate, is investigated using the traveltimes of direct and converted seismic phases from local earthquakes. Mode conversions take place as upgoing energy from earthquakes in the subducted slab crosses the plate interface. These PS and SP converted arrivals are observed as intermediate phases between the direct P and S waves. They place an additional constraint on the depth of the interface and enable the topography of the subducted plate to be mapped across the region. 301 suitable earthquakes were recorded by the Leeds (Tararua) broad-band seismic array, a temporary line of three-component short-period stations, and the permanent stations of the New Zealand national network. This provided coverage across the land area of southern North Island, New Zealand, at a total of 17 stations. Rays are traced through a structure parametrized using layered B-splines and the traveltime residuals inverted, simultaneously, for hypocentre relocation, interface depth and seismic velocity. The results are consistent with sediment in the northeast of the study region and gentle topography on the subducting plate. This study and recent tectonic reconstructions of the southwest Pacific suggest that the subducting plate consists of captured, oceanic crust. The anomalous nature of this crust partly accounts for the unusual features of the Hikurangi margin, e.g. the shallow trench, in comparison with the subducting margin further north

A Correction for Overestimation Bias in Estimates of Black-Tailed Prairie Dog Abundance Based on Aerial Surveys of Colony Sites in Colorado and Montana

Estimates of abundance of black-tailed prairie dogs (Cynomys ludovicianus) are obtained by estimating the area occupied by colonies. An approach for estimating this area used in Colorado and Montana was based on aerial survey transects that recorded the end points where transects intercepted and exited colony sites. Line intercept mathematical techniques were applied to these intercept data to obtain estimates of occupied area. We define a “colony site” as an aggregation of prairie dog burrows while a prairie dog “colony” is defined as the portion of a colony site that is occupied by living prairie dogs. Because of poisoning, plague and other factors, colony sites are commonly not completely occupied by colonies. In both Colorado and Montana, however, estimates obtained were estimates of the area occupied by colony sites that had some undetermined level of occupancy by colonies. We show for Colorado that the difference between estimates of area occupied by colonies was much less than the area occupied by colony sites. We provide an approach to correct estimates based on the extent of colony sites. This approach requires ground surveys of a sample of aerial intercepts of colony sites to document the proportion that is actually occupied by colonies of living black-tailed prairie dogs. Black-tailed prairie dogs were found as not-warranted for listing in 2004 in part because of inflated estimates of abundance obtained in Colorado that incorrectly equated the extent of colony-sites as equivalent to the extent of colonies in that state