General Relativity, the Cosmological Constant and Modular Forms

Strong field (exact) solutions of the gravitational field equations of General Relativity in the presence of a Cosmological Constant are investigated. In particular, a full exact solution is derived within the inhomogeneous Szekeres-Szafron family of space-time line element with a nonzero Cosmological Constant. The resulting solution connects, in an intrinsic way, General Relativity with the theory of modular forms and elliptic curves. The homogeneous FLRW limit of the above space-time elements is recovered and we solve exactly the resulting Friedmann Robertson field equation with the appropriate matter density for generic values of the Cosmological Constant %Lambda and curvature constant K. A formal expression for the Hubble constant is derived. The cosmological implications of the resulting non-linear solutions are systematically investigated. Two particularly interesting solutions i) the case of a flat universe K=0, Lambda not= 0 and ii) a case with all three cosmological parameters non-zero, are described by elliptic curves with the property of complex multiplication and absolute modular invariant j=0 and 1728, respectively. The possibility that all non-linear solutions of General Relativity are expressed in terms of theta functions associated with Riemann-surfaces is discussed.Comment: LaTeX file, 34 pages plus 9 EPS figures, Accepted for Publication in Classical and Quantum Gravit

The thermodynamics of collapsing molecular cloud cores using smoothed particle hydrodynamics with radiative transfer

We present the results of a series of calculations studying the collapse of molecular cloud cores performed using a three-dimensional smoothed particle hydr odynamics code with radiative transfer in the flux-limited diffusion approximation. The opacities and specific heat capacities are identical for each calculation. However, we find that the temperature evolution during the simulations varies significantly when starting from different initial conditions. Even spherically-symmetric clouds with different initial densities show markedly different development. We conclude that simple barotropic equations of state like those used in some previous calculations provide at best a crude approximation to the thermal behaviour of the gas. Radiative transfer is necessary to obtain accurate temperatures.Comment: 8 pages, 9 figures, accepted for publication in MNRA

Reduction in livestock losses following placement of livestock guarding dogs and the impact of herd species and dog sex

Livestock guarding dogs have been placed on South African farms by the not-for-profit organisation, Cheetah Outreach Trust, since 2005, and have been proven to be an efficient form of non-lethal predator control against jackal, caracal, leopards, cheetahs and other predators found in South Africa. However, the impact that herd species (sheep, goat, cattle or mixed) or the sex of the dog may have on the observed reduction in livestock losses following placement of a livestock guarding dogs has not been investigated. To address this, the reduction in livestock losses following placement of an Anatolian livestock guarding dogs was measured in two South African provinces over a nine year period and data simultaneously collected on herd type and dog sex. Dogs comprised of 78 males and 49 females. Farms consisted of 68 sheep, 37 goats, 23 cattle, and two exotic game farms. Effectiveness was measured as the difference between farmer-reported livestock losses before and after the placement of a dog and was calculated as percentage change in stock loss after introduction of a livestock guarding dog according herd species and dog sex. This study determined the impact of herd type or dog sex on the difference between livestock loss before versus after livestock guarding dog s placement. This study indicates that the use of this breed of livestock guarding dog is an effective means of reducing perceived livestock losses due to predation, regardless of dog sex, and may be used with equal effectiveness with a range of herd species

Constraints on the thermal and tectonic evolution of Greymouth coalfield

The southern end of the Paparoa Range in Westland, South Island, New Zealand, comprises an asymmetrical, southward plunging, faulted (Brunner-Mt Davy) anticline, the eastern limb of which is common with the western limb of an asymmetrical (Grey Valley) syncline forming a Neogene foreland basin (Grey Valley Trough). The faulted anticline is a classic inversion structure: compression during the Neogene, associated with the development of the modern Australia-Pacific plate boundary, caused a pre-existing normal fault zone, about which a late Cretaceous-Oligocene extensional half graben had formed (Paparoa Trough), to change its sense of displacement. The resulting basement loading formed the foreland basin, containing up to 3 km of mainly marine sedimentary section. Fission track results for apatite concentrates from 41 shallow drillhole and outcrop samples from the Greymouth Coalfield part of the Brunner-Mt Davy Anticline are reported and interpreted, to better establish the timing and amount of inversion, and hence the mechanism of inversion. The fission track results integrated with modelling of vitrinite reflectance data, show that the maximum paleotemperatures experienced during burial of the Late Cretaceous and mid-Eocene coal-bearing succession everywhere exceeded 85deg.C, and reached a peak of 180deg.C along the axis of the former basin. Cooling from maximum temperatures occurred during three discrete phases: 20-15 Ma, 12-7 Ma, and c. 2 Ma to the present. The amount of denudation has been variable across the inverted basin, decreasing westward from a maximum of c. 2.5 km during the first deformation phase, c. 1.2 km during the second phase, and 1.4 km during the third phase. It appears that exhumation over the coalfield continued for about 2 m.y. beyond the biostratigraphically determined time ranges of each of two synorogenic unconformities along the western limb of the Grey Valley Syncline. Stick-slip behaviour on the range front fault that localised the inversion is inferred. The tectonic evolution of the anticline-syncline pair at the southern end of the Paparoa Range, is therefore identical in style, and similar in timing, to the development of the Papahaua Range-Westport Trough across the Kongahu Fault Zone, in the vicinity of Buller Coalfield

A simplified model of the Martian atmosphere - Part 1: a diagnostic analysis

In this paper we derive a reduced-order approximation to the vertical and horizontal structure of a simplified model of the baroclinically unstable Martian atmosphere. The original model uses the full hydrostatic primitive equations on a sphere, but has only highly simplified schemes to represent the detailed physics of the Martian atmosphere, e.g. forcing towards a plausible zonal mean temperature state using Newtonian cooling. Three different norms are used to monitor energy conversion processes in the model and are then compared. When four vertical modes (the barotropic and first three baroclinic modes) are retained in the reduced-order approximation, the correlation norm captures approximately 90% of the variance, while the kinetic energy and total energy norms capture approximately 83% and 78% of the kinetic and total energy respectively. We show that the leading order Proper Orthogonal Decomposition (POD) modes represent the dominant travelling waves in the baroclinically-unstable, winter hemisphere. In part 2 of our study we will develop a hierarchy of truncated POD-Galerkin expansions of the model equations using up to four vertical modes

A simplified model of the Martian atmosphere - Part 2: a POD-Galerkin analysis

In Part I of this study Whitehouse et al. (2005) performed a diagnostic analysis of a simplied model of the Martian atmosphere, in which topography was absent and in which heating was modelled as Newtonian relaxation towards a zonally symmetric equilibrium temperature field. There we derived a reduced-order approximation to the vertical and the horizonal structure of the baroclinically unstable Martian atmosphere, retaining only the barotropic mode and the leading order baroclinic modes. Our objectives in Part II of the study are to incorporate these approximations into a Proper Orthogonal Decomposition-Galerkin expansion of the spherical quasi-geostrophic model in order to derive hierarchies of nonlinear ordinary differential equations for the time-varying coefficients of the spatial structures. Two different vertical truncations are considered, as well as three different norms and 3 different Galerkin truncations. We investigate each in turn, using tools from bifurcation theory, to determine which of the systems most closely resembles the data for which the original diagnostics were performed