An Approach to Hopf Algebras via Frobenius Coordinates I

In Section 1 we introduce Frobenius coordinates in the general setting that includes Hopf subalgebras. In Sections 2 and 3 we review briefly the theories of Frobenius algebras and augmented Frobenius algebras with some new material in Section 3. In Section 4 we study the Frobenius structure of an FH-algebra H \cite{Par72} and extend two recent theorems in \cite{EG}. We obtain two Radford formulas for the antipode in H and generalize in Section 7 the results on its order in \cite{FMS}. We study the Frobenius structure on an FH-subalgebra pair in Sections 5 and 6. In Section 8 we show that the quantum double of H is symmetric and unimodular.Comment: 24 pages. To appear: Beitrage Alg. Geo

Population need and geographical access to general practitioners in rural New Zealand

To use a geographical information system (GIS) approach to demonstrate the extent to which different areas in New Zealand vary in their geographical access to GPs, and to analyse the extent to which spatial access varies in relation to different population groups. Methods Three methods; population/GP ratios, least cost path analysis (LCPA), and an allocation method (which considered the capacity constraint of GPs) were used to demonstrate differences in geographic accessibility to GPs. Travel time, and distance to the closest GP, was calculated for every census enumeration district in New Zealand (n=38336)—thus enabling population-based accessibility statistics to be calculated and aggregated to the territorial local authority level. These calculations include the average travel time if everybody visited a GP once and the population more than 30 minutes from a GP. The composition of this population is analysed according to three criteria of need: the level of deprivation (NZDep2001), ethnicity (%Maori), and age (% <5 years, and %65 years and over). Results There are significant regional variations in geographical accessibility in New Zealand, and these differences are dependent upon the method to calculate accessibility. Ratio measures give a different picture of GP access than the other two indicators, reflecting the fact that TAs with similar ratios often have wide variations in travel times as well as the size and proportion of the population living more than 30 minutes from the closest GP. TAs with larger numbers and a higher proportion of their populations living in such areas tend to be more deprived and have a higher proportion of Maori, especially in the North Island. There appears to be no significant trend by age. Conclusion Given the health and service consequences of poor access, the results suggest that more attention needs to be paid to extending the spatial information base in primary care, in order to achieve more effective planning of services for disadvantaged populations

A transmission problem for quasi-linear wave equations

We prove the local existence and uniqueness of solutions to a system of quasi-linear wave equations involving a jump discontinuity in the lower order terms. A continuation principle is also established.Comment: The results of this article are essential to a local existence proof of solutions to the Einstein field equations coupled to elastic matter that represent gravitating compact bodies. The full existence proof will be presented in a separate articl

Threshold games and cooperation on multiplayer graphs

Objective: The study investigates the effect on cooperation in multiplayer games, when the population from which all individuals are drawn is structured - i.e. when a given individual is only competing with a small subset of the entire population. Method: To optimize the focus on multiplayer effects, a class of games were chosen for which the payoff depends nonlinearly on the number of cooperators - this ensures that the game cannot be represented as a sum of pair-wise interactions, and increases the likelihood of observing behaviour different from that seen in two-player games. The chosen class of games are named "threshold games", and are defined by a threshold, $M > 0$, which describes the minimal number of cooperators in a given match required for all the participants to receive a benefit. The model was studied primarily through numerical simulations of large populations of individuals, each with interaction neighbourhoods described by various classes of networks. Results: When comparing the level of cooperation in a structured population to the mean-field model, we find that most types of structure lead to a decrease in cooperation. This is both interesting and novel, simply due to the generality and breadth of relevance of the model - it is likely that any model with similar payoff structure exhibits related behaviour. More importantly, we find that the details of the behaviour depends to a large extent on the size of the immediate neighbourhoods of the individuals, as dictated by the network structure. In effect, the players behave as if they are part of a much smaller, fully mixed, population, which we suggest an expression for.Comment: in PLOS ONE, 4th Feb 201

Tate (co)homology via pinched complexes

For complexes of modules we study two new constructions, which we call the pinched tensor product and the pinched Hom. They provide new methods for computing Tate homology and Tate cohomology, which lead to conceptual proofs of balancedness of Tate (co)homology for modules over associative rings. Another application we consider is in local algebra. Under conditions of vanishing of Tate (co)homology, the pinched tensor product of two minimal complete resolutions yields a minimal complete resolution.Comment: Final version; 23 pp. To appear in Trans. Amer. Math. So

The Metallicities of Stars With and Without Transiting Planets

Host star metallicities have been used to infer observational constraints on planet formation throughout the history of the exoplanet field. The giant planet metallicity correlation has now been widely accepted, but questions remain as to whether the metallicity correlation extends to the small terrestrial-sized planets. Here, we report metallicities for a sample of 518 stars in the Kepler field that have no detected transiting planets and compare their metallicity distribution to a sample of stars that hosts small planets (Rp < 1.7 R_Earth). Importantly, both samples have been analyzed in a homogeneous manner using the same set of tools (Stellar Parameters Classification tool; SPC). We find the average metallicity of the sample of stars without detected transiting planets to be [m/H]_SNTP,dwarf = -0.02 +- 0.02 dex and the sample of stars hosting small planets to be [m/H]_STP = -0.02 +- 0.02 dex. The average metallicities of the two samples are indistinguishable within the uncertainties, and the two-sample Kolmogorov-Smirnov test yields a p-value of 0.68 (0.41 sigma), indicating a failure to reject the null hypothesis that the two samples are drawn from the same parent population. We conclude that the homogeneous analysis of the data presented here support the hypothesis that stars hosting small planets have a metallicity similar to stars with no known transiting planets in the same area of the sky.Comment: Accepted for publication in Ap