Ambient cosmology and spacetime singularities

We present a new approach to the issues of spacetime singularities and cosmic censorship in general relativity. This is based on the idea that standard 4-dimensional spacetime is the conformal infinity of an ambient metric for the 5-dimensional Einstein equations with fluid sources. We then find that the existence of spacetime singularities in four dimensions is constrained by asymptotic properties of the ambient 5-metric, while the non-degeneracy of the latter crucially depends on cosmic censorship holding on the boundary.

Erosion Damage Mapping: Assessing Current Soil Erosion Damage in Switzerland

Within the scope of a comprehensive assessment of the degree of soil erosion in Switzerland, common methods have been used in the past including test plot measurements, artificial rainfall simulation, and erosion modelling. In addition, mapping guidelines for all visible erosion features have been developed since the 1970s and are being successfully applied in many research and soil conservation projects. Erosion damage has been continuously mapped over a period of 9 years in a test region in the central Bernese plateau. In 2005, two additional study areas were added. The present paper assesses the data gathered and provides a comparison of the three study areas within a period of one year (from October 2005 to October 2006), focusing on the on-site impacts of soil erosion. During this period, about 11 erosive rainfall events occurred. Average soil loss rates mapped at each study site amounted to 0.7 t ha-1, 1.2 t ha-1 and 2.3 t ha-1, respectively. About one fourth of the total arable land showed visible erosion damage. Maximum soil losses of about 70 t ha-1 occurred on individual farm plots. Average soil erosion patterns are widely used to underline the severity of an erosion problem (e.g. impacts on water bodies). But since severe rainfall events, wheel tracks, headlands, and other “singularities” often cause high erosion rates, analysis of extreme erosion patterns such as maximum values led to a more differentiated understanding and appropriate conclusions for planning and design of soil protection measures. The study contains an assessment of soil erosion in Switzerland, emphasizing questions about extent, frequency and severity. At the same time, the effects of different types of land management are investigated in the field, aiming at the development of meaningful impact indicators of (un-)sustainable agriculture/soil erosion risk as well as the validation of erosion models. The results illustrate that conservation agriculture including no-till, strip tillage and in-mulch seeding plays an essential role in reducing soil loss as compared to conventional tillage.

hp-dGFEM for Second-Order Elliptic Problems in Plyhedra I: Stability and Quasioptimality on Geometric Meshes

We introduce and analyze hp-version discontinuous Galerkin (dG) finite element methods for the numerical approximation of linear second-order elliptic boundary-value problems in three-dimensional polyhedral domains. To resolve possible corner-, edge- and corner-edge singularities, we consider hexahedral meshes that are geometrically and anisotropically refined toward the corresponding neighborhoods. Similarly, the local polynomial degrees are increased linearly and possibly anisotropically away from singularities. We design interior penalty hp-dG methods and prove that they are well-defined for problems with singular solutions and stable under the proposed hp-refinements. We establish (abstract) error bounds that will allow us to prove exponential rates of convergence in the second part of this work.