Degeneracy of the b-boundary in General Relativity

The b-boundary construction by B. Schmidt is a general way of providing a boundary to a manifold with connection. It has been shown to have undesirable topological properties however. C. J. S. Clarke gave a result showing that for space-times, non-Hausdorffness is to be expected in general, but the argument contains some errors. We show that under somewhat different conditions on the curvature, the b-boundary will be non-Hausdorff, and illustrate the degeneracy by applying the conditions to some well known exact solutions of general relativity.Comment: 25 pages, AMS-LaTeX v1.2, AMSFonts, submitted to Commun. Math. Phy

Planted tree fallows and their influence on soil fertility and maize production in East Africa

Soil fertility depletion is a main constraint to food production in sub-Saharan Africa. This thesis concerns the potential of N2-fixing trees to increase nitrogen inputs to agroforestry systems and accordingly to improve crop production. The suitability of five tropical tree species (including two N2-fixing species, Leucaena leucocephala and Prosopis chilensis) for tree fallows in Tanzania were evaluated by comparing their leaf chemistry, their effects on soil properties and on maize growth. After five years fallow, the per cent total soil N was higher under Prosopis compared to under other tree species. Maize biomass production was higher on soils from Leucaena or Prosopis compared to grass fallow. Prosopis contributed 11% to the total soil C over a period of 8 years. Field experiments in Kenya were performed to test a low-level 15N-tracer technique to estimate biological nitrogen fixation in Sesbania sesban over an 18-months period, and to compare the effects of short-duration tree fallows on two subsequent maize harvests with natural fallow and continuous cropping. We estimated the N derived from atmosphere by Sesbania after 18 months to between 500 and 600 kg ha-1, depending on which plant parts were used for 15N data and on the choice of reference species. We consider the 15N dilution method to be appropriate for quantifying N2 fixation in improved fallows in studies of young trees with high N2-fixing ability. In an experiment examining the effects of tree fallows on subsequent maize crops approximately 70-90% of the N in Sesbania, and 50-70% in Calliandra calothyrsus, was derived from N2-fixation. The quantity of N added by N2-fixation, 280-360 kg N ha-1 for Sesbania and 120-170 kg N ha-1 for Calliandra, resulted in a positive N balance after two cropping seasons of 170-250 kg N ha-1 and 90-140 kg N ha-1 respectively. Both the content of inorganic N in the topsoil and the quantity of N mineralised during rainy seasons were higher after the Sesbania fallows than after the other treatments. The substantial accumulation of N in planted Sesbania demonstrated its potential to increase the sustainability of crop production on N-limited soils

The Geometry of the Frame Bundle over Spacetime

One of the known mathematical descriptions of singularities in General Relativity is the b-boundary, which is a way of attaching endpoints to inextendible endless curves in a spacetime. The b-boundary of a manifold M with connection is constructed by forming the Cauchy completion of the frame bundle LM equipped with a certain Riemannian metric, the b-metric G. We study the geometry of (LM,G) as a Riemannian manifold in the case when the connection is the Levi-Civita connection of a Lorentzian metric g on M. In particular, we give expressions for the curvature and discuss the isometries and the geodesics of (LM,G) in relation to the geometry of (M,g).Comment: 14 pages, no figures, LaTeX 2e with AMSLaTeX 1.2 and AMSFonts, submitted to J. Math. Phy

On Imprisoned Curves and b-length in General Relativity

This paper is concerned with two themes: imprisoned curves and the b-length functional. In an earlier paper by the author, it was claimed that an endless incomplete curve partially imprisoned in a compact set admits an endless null geodesic cluster curve. Unfortunately, the proof was flawed. We give an outline of the problem and remedy the situation by providing a proof by different methods. Next, we obtain some results concerning the structure of b-length neighbourhoods, which gives a clue to how the geometry of a spacetime is encoded in the pseudo-orthonormal frame bundle equipped with the b-metric. We also show that a previous result by the author, proving total degeneracy of a b-boundary fibre in some cases, does not apply to imprisoned curves. Finally, we correct some results in the literature linking the b-lengths of general curves in the frame bundle with the b-length of the corresponding horizontal curves.Comment: 26 pages, 7 figures, LaTeX 2e with AMSLaTeX 1.2 and AMSFonts, submitted to J. Math. Phy

Gotzmann's persistence theorem for finite modules

We prove a generalization of Gotzmann's persistence theorem in the case of modules with constant Hilbert polynomial. As a consequence, we show that the defining equations that give the embedding of a Quot scheme of points into a Grassmannian are given by a single Fitting ideal.Comment: 13 pages; restructured the paper and clarified some proof

Rees algebras of modules and coherent functors

We show that several properties of the theory of Rees algebras of modules become more transparent using the category of coherent functors rather than working directly with modules. In particular, we show that the Rees algebra is induced by a canonical map of coherent functors.Comment: 15 pages. This is an extended part of an old version of arXiv:1402.3219 - v2: Clarified some statements and made minor notation change

A Web-based Distance Learning System in Business Administration Experiences from an Inter-Nordic Course

The paper presents a specially constructed Web-based tool, WebGPSS, developed in Sweden, for teaching business students simulation. The tool includes a Java Applet that provides a Graphical Users Interface, by which the student can build a simulation model of a system, such as an inventory system. The model is then run on a remote server, which then sends result graphs and tables over the Web to the student computer. The paper furthermore discusses the experiences made when using this tool in a distance-learning course at the Norwegian School of Economics and Business Administration (NHH) in Bergen, in 2000 – 2002.Web-based; Distance learning; Simulation; GPSS; Java