Statistical hyperbolicity in groups

In this paper, we introduce a geometric statistic called the "sprawl" of a group with respect to a generating set, based on the average distance in the word metric between pairs of words of equal length. The sprawl quantifies a certain obstruction to hyperbolicity. Group presentations with maximum sprawl (i.e., without this obstruction) are called statistically hyperbolic. We first relate sprawl to curvature and show that nonelementary hyperbolic groups are statistically hyperbolic, then give some results for products, for Diestel-Leader graphs and lamplighter groups. In free abelian groups, the word metrics asymptotically approach norms induced by convex polytopes, causing the study of sprawl to reduce to a problem in convex geometry. We present an algorithm that computes sprawl exactly for any generating set, thus quantifying the failure of various presentations of Z^d to be hyperbolic. This leads to a conjecture about the extreme values, with a connection to the classic Mahler conjecture.Comment: 14 pages, 5 figures. This is split off from the paper "The geometry of spheres in free abelian groups.

Regrowing the past to preserve the future: Cultural heritage preservation through the Pawnee Seed Preservation Project

Traditional foodways of indigenous peoples worldwide face the risk of loss in an increasingly interconnected and modernizing world. To prevent further loss of culture, reclaim food sovereignty, and improve community health, many indigenous groups are taking steps to protect these foodways. Currently, no academic literature focuses on the Pawnee Nation of Oklahoma's Pawnee Seed Preservation Project which has brought back 18 varieties of ancestral Pawnee corn since the project officially began in 1998 through collaborative efforts between both Pawnee in Oklahoma, and non-Pawnee growers in Nebraska. This paper examines the affect of the project on preserving Pawnee culture and traditions through semi-structured interviews conducted with five Pawnee Elders with ties to the project. The findings indicate a strong correlation between corn grown through the project continuing to help shape Pawnee identity, traditions, spirituality, ceremonial uses, and influence a younger generation to get involved in the growing process. Supplementary findings also show optimism towards health improvements related to the reintroduction of this corn into the diet of the tribe, possible business and scientific endeavors associated with growing the corn, as well as a strengthening of relationships with non-Pawnee growers who have been essential to the project's success

Arthroscopic Internal Fixation of Coracoid Fractures: Surgical Technique Guide

Fractures of the coracoid process are uncommon injuries and are usually the result of high-energy trauma or avulsion-type injuries. Typically coracoid fractures treated with nonoperative management have yielded good results. Operative treatment of coracoid fractures is reserved for a subset of clinical situations, including fracture nonunion. We detail our technique for arthroscopic debridement of a Type II coracoid fracture nonunion, as well as the use of arthroscopic-assisted percutaneous fixation for a Type II coracoid fracture

Why Pad\'e Approximants reduce the Renormalization-Scale dependence in QFT?

We prove that in the limit where the beta function is dominated by the 1-loop contribution (``large beta_0 limit'') diagonal Pad\'e Approximants (PA's) of perturbative series become exactly renormalization scale (RS) independent. This symmetry suggest that diagonal PA's are resumming correctly contributions from higher order diagrams that are responsible for the renormalization of the coupling-constant. Non-diagonal PA's are not exactly invariant, but generally reduce the RS dependence as compared to partial-sums. In physical cases, higher-order corrections in the beta function break the symmetry softly, introducing a small scale and scheme dependence. We also compare the Pad\'e resummation with the BLM method. We find that in the large-N_f limit using the BLM scale is identical to resumming the series by a $x[0/n]$ non-diagonal PA.Comment: 25 pages, LateX. Replaced so that the figures would fit into the page siz

Calcium Bistriflimide-Mediated Sulfur(VI)–Fluoride Exchange (SuFEx): Mechanistic Insights toward Instigating Catalysis

We report a mechanistic investigation of calcium bistriflimide-mediated sulfur(VI)–fluoride exchange (SuFEx) between sulfonyl fluorides and amines. We determine the likely pre-activation resting state─a calcium bistriflimide complex with ligated amines─thus allowing for corroborated calculation of the SuFEx activation barrier at ∼21 kcal/mol, compared to 21.5 ± 0.14 kcal/mol derived via kinetics experiments. Transition state analysis revealed: (1) a two-point calcium-substrate contact that activates the sulfur(VI) center and stabilizes the leaving fluoride and (2) a 1,4-diazabicyclo[2.2.2]octane additive that provides Brønsted-base activation of the nucleophilic amine. Stable Ca–F complexes upon sulfonamide formation are likely contributors to inhibited catalytic turnover, and a proof-of-principle redesign provided evidence that sulfonamide formation is feasible with 10 mol % calcium bistriflimide

Coral development: from classical embryology to molecular control

The phylum Cnidaria is the closest outgroup to the triploblastic metazoans and as such offers unique insights into evolutionary questions at several levels. In the post-genomic era, a knowledge of the gene complement of representative cnidarians will be important for understanding the relationship between the expansion of gene families and the evolution of morphological complexity among more highly evolved metazoans. Studies of cnidarian development and its molecular control will provide information about the origins of the major bilaterian body axes, the origin of the third tissue layer, the mesoderm, and the evolution of nervous system patterning. We are studying the cnidarian Acropora millepora, a reef building scleractinian coral, and a member of the basal cnidarian class, the Anthozoa. We review ourwork on descriptive embryology and studies of selected transcription factor gene families, where our knowledge from Acropora is particularly advanced relative to other cnidarians. We also describe a recent preliminary whole genome initiative, a coral EST database.Eldon E. Ball, David C. Hayward, John S. Reece-Hoyes, Nikki R. Hislop, Gabrielle Samuel, Robert Saint, Peter L. Harrison and David J. Mille