Eleven spherically symmetric constant density solutions with cosmological constant

Einstein's field equations with cosmological constant are analysed for a static, spherically symmetric perfect fluid having constant density. Five new global solutions are described. One of these solutions has the Nariai solution joined on as an exterior field. Another solution describes a decreasing pressure model with exterior Schwarzschild-de Sitter spacetime having decreasing group orbits at the boundary. Two further types generalise the Einstein static universe. The other new solution is unphysical, it is an increasing pressure model with a geometric singularity.Comment: 19 pages, 5 figures, 1 table, revised bibliography, corrected eqn. (3.11), typos corrected, two new reference

A Business Entity By Any Other Name: Corporation, Community and Kinship

Forty-five years ago, the Alaska Native Claims Settlement Act resolved outstanding land claims between the federal and state government and Alaska Natives. The fund created by the settlement was used as seed money to establish the Alaska Native Corporations. The Native corporations have particular features which make them distinct from other business entities, these differences have been lauded by some shareholders but have simultaneously drawn ire from others. In 2015 the Alaska legislature introduced H.B. 49, a benefit corporation bill that would allow entrepreneurs to pursue both profits and social ends. This note traces the rise of the modern Alaska Native Corporation. It then weighs the merits of each business entity and assesses which is best aligned to improve the lives of Alaska Natives

Bounds on M/R for static objects with a positive cosmological constant

We consider spherically symmetric static solutions of the Einstein equations with a positive cosmological constant $\Lambda,$ which are regular at the centre, and we investigate the influence of $\Lambda$ on the bound of M/R, where M is the ADM mass and R is the area radius of the boundary of the static object. We find that for any solution which satisfies the energy condition $p+2p_{\perp}\leq\rho,$ where $p\geq 0$ and $p_{\perp}$ are the radial and tangential pressures respectively, and $\rho\geq 0$ is the energy density, and for which $0\leq \Lambda R^2\leq 1,$ the inequality \frac{M}{R}\leq\frac29-\frac{\Lambda R^2}{3}+\frac29 \sqrt{1+3\Lambda R^2}, holds. If $\Lambda=0$ it is known that infinitely thin shell solutions uniquely saturate the inequality, i.e. the inequality is sharp in that case. The situation is quite different if $\Lambda>0.$ Indeed, we show that infinitely thin shell solutions do not generally saturate the inequality except in the two degenerate situations $\Lambda R^2=0$ and $\Lambda R^2=1$. In the latter situation there is also a constant density solution, where the exterior spacetime is the Nariai solution, which saturates the inequality, hence, the saturating solution is non-unique. In this case the cosmological horizon and the black hole horizon coincide. This is analogous to the charged situation where there is numerical evidence that uniqueness of the saturating solution is lost when the inner and outer horizons of the Reissner-Nordstr\"{o}m solution coincide.Comment: 14 pages; Improvements and corrections, published versio

Environmental Superstatistics

A thermodynamic device placed outdoors, or a local ecosystem, is subject to a variety of different temperatures given by short-tem (daily) and long-term (seasonal) variations. In the long term a superstatistical description makes sense, with a suitable distribution function f(beta) of inverse temperature beta over which ordinary statistical mechanics is averaged. We show that f(beta) is very different at different geographic locations, and typically exhibits a double-peak structure for long-term data. For some of our data sets we also find a systematic drift due to global warming. For a simple superstatistical model system we show that the response to global warming is stronger if temperature fluctuations are taken into account.Comment: 37 figures. Significantly extended version, to appear in Physica A. Added new material in section 6 quantifying the stronger response to global warming if temperature fluctuations are taken into account. Concluding section 7 and several new references adde

Large Deviations for Nonlocal Stochastic Neural Fields

We study the effect of additive noise on integro-differential neural field equations. In particular, we analyze an Amari-type model driven by a $Q$-Wiener process and focus on noise-induced transitions and escape. We argue that proving a sharp Kramers' law for neural fields poses substanial difficulties but that one may transfer techniques from stochastic partial differential equations to establish a large deviation principle (LDP). Then we demonstrate that an efficient finite-dimensional approximation of the stochastic neural field equation can be achieved using a Galerkin method and that the resulting finite-dimensional rate function for the LDP can have a multi-scale structure in certain cases. These results form the starting point for an efficient practical computation of the LDP. Our approach also provides the technical basis for further rigorous study of noise-induced transitions in neural fields based on Galerkin approximations.Comment: 29 page