Choptuik scaling in null coordinates

A numerical simulation is performed of the gravitational collapse of a spherically symmetric scalar field. The algorithm uses the null initial value formulation of the Einstein-scalar equations, but does {\it not} use adaptive mesh refinement. A study is made of the critical phenomena found by Choptuik in this system. In particular it is verified that the critical solution exhibits periodic self-similarity. This work thus provides a simple algorithm that gives verification of the Choptuik results.Comment: latex (revtex), 6 figures included in the fil

Exact solution for scalar field collapse

We give an exact spherically symmetric solution for the Einstein-scalar field system. The solution may be interpreted as an inhomogeneous dynamical scalar field cosmology. The spacetime has a timelike conformal Killing vector field and is asymptotically conformally flat. It also has black or white hole-like regions containing trapped surfaces. We describe the properties of the apparent horizon and comment on the relevance of the solution to the recently discovered critical behaviour in scalar field collapse.Comment: 10 pages(Latex) (2 figures available upon request), Alberta-Thy-4-9

Decay of the Maxwell field on the Schwarzschild manifold

We study solutions of the decoupled Maxwell equations in the exterior region of a Schwarzschild black hole. In stationary regions, where the Schwarzschild coordinate $r$ ranges over $2M < r_1 < r < r_2$, we obtain a decay rate of $t^{-1}$ for all components of the Maxwell field. We use vector field methods and do not require a spherical harmonic decomposition. In outgoing regions, where the Regge-Wheeler tortoise coordinate is large, $r_*>\epsilon t$, we obtain decay for the null components with rates of $|\phi_+| \sim |\alpha| < C r^{-5/2}$, $|\phi_0| \sim |\rho| + |\sigma| < C r^{-2} |t-r_*|^{-1/2}$, and $|\phi_{-1}| \sim |\underline{\alpha}| < C r^{-1} |t-r_*|^{-1}$. Along the event horizon and in ingoing regions, where $r_*<0$, and when $t+r_*1$, all components (normalized with respect to an ingoing null basis) decay at a rate of C \uout^{-1} with \uout=t+r_* in the exterior region.Comment: 37 pages, 5 figure

Self-gravitating Klein-Gordon fields in asymptotically Anti-de-Sitter spacetimes

We initiate the study of the spherically symmetric Einstein-Klein-Gordon system in the presence of a negative cosmological constant, a model appearing frequently in the context of high-energy physics. Due to the lack of global hyperbolicity of the solutions, the natural formulation of dynamics is that of an initial boundary value problem, with boundary conditions imposed at null infinity. We prove a local well-posedness statement for this system, with the time of existence of the solutions depending only on an invariant H^2-type norm measuring the size of the Klein-Gordon field on the initial data. The proof requires the introduction of a renormalized system of equations and relies crucially on r-weighted estimates for the wave equation on asymptotically AdS spacetimes. The results provide the basis for our companion paper establishing the global asymptotic stability of Schwarzschild-Anti-de-Sitter within this system.Comment: 50 pages, v2: minor changes, to appear in Annales Henri Poincar\'

STUDI: a model to simulate the impacts of new metro lines on urban development in London

Urban systems are complex and change as a result of the interactions between their main elements. In order to model urban systems effectively, the dynamics of relationships between these elements need to be considered. This thesis investigates the interactions between transport and urban development, focusing on the impacts of new metro lines. A new model is developed for this purpose: the STUDI (Simulation of Transport and Urban Development Interactions) model. The main concept underlying the STUDI model is that the impacts of new transport infrastructure on urban development are reflected in the interactions between the main agents involved in the process, which are authorities, developers, businesses and population sub-models. The development sub-model is a regression model forecasting the number of new commercial and residential premises, and the business and population sub-models are microsimulation models. The business sub-model simulates business start-ups and closures and business location and relocation decisions. The population sub-model simulates in- and out- migration, demographic and employment change, and residential location decisions. The main results include changes in the spatial distributions of development, businesses and population over time under different transport supply scenarios. The STUDI model has been developed for London in order to test the wider impacts of the new metro lines. First it has been applied to evaluate the impacts of the Jubilee Line Extension (JLE) and then it was used to forecast the impacts of a line to open in the future: the East London Line Extension. Both cases indicate the positive impact of new transport infrastructure on urban development

Self-Similar Scalar Field Collapse: Naked Singularities and Critical Behaviour

Homothetic scalar field collapse is considered in this article. By making a suitable choice of variables the equations are reduced to an autonomous system. Then using a combination of numerical and analytic techniques it is shown that there are two classes of solutions. The first consists of solutions with a non-singular origin in which the scalar field collapses and disperses again. There is a singularity at one point of these solutions, however it is not visible to observers at finite radius. The second class of solutions includes both black holes and naked singularities with a critical evolution (which is neither) interpolating between these two extremes. The properties of these solutions are discussed in detail. The paper also contains some speculation about the significance of self-similarity in recent numerical studies.Comment: 27 pages including 5 encapsulated postcript figures in separate compressed file, report NCL94-TP1

Spherically symmetric perfect fluid in area-radial coordinates

We study the spherically symmetric collapse of a perfect fluid using area-radial coordinates. We show that analytic mass functions describe a static regular centre in these coordinates. In this case, a central singularity can not be realized without an infinite discontinuity in the central density. We construct mass functions involving fluid dynamics at the centre and investigate the relationship between those and the nature of the singularities.Comment: Accepted by CQG. LaTex file, 14 pages, no figure

An automatic controller tuning algorithm.

A project report submitted to the Faculty of Engineering, University of the Witwatersrand, Johannesburg, in partial fulfillment of the requirements for 'the degree of Master of Science in Engineering. Johannesburg 1991.The report describes the design of an algorithm which can be used for automatic controller tuning purposes. It uses an on-line parameter estimator and a pole assignrnent design method. The resulting control law is formulated to approximate a proportional-integral (PI) industrial controller. The development ofthe algorithm is based on the delta-operator, Some implementation aspects such as covariance resetting, dead zone, and signal conditioning are also discussed. Robust stability and performance are two issues that govern the design approach. Additionally transient and steady state system response criteria are utilized from the time and frequency domains. The design work is substantiated with the use of simulation and real plant tests.AC201

An automatic controller tuning algorithm.

A project report submitted to the Faculty of Engineering, University of the Witwatersrand, Johannesburg, in partial fulfillment of the requirements for 'the degree of Master of Science in Engineering. Johannesburg 1991.The report describes the design of an algorithm which can be used for automatic controller tuning purposes. It uses an on-line parameter estimator and a pole assignrnent design method. The resulting control law is formulated to approximate a proportional-integral (PI) industrial controller. The development ofthe algorithm is based on the delta-operator, Some implementation aspects such as covariance resetting, dead zone, and signal conditioning are also discussed. Robust stability and performance are two issues that govern the design approach. Additionally transient and steady state system response criteria are utilized from the time and frequency domains. The design work is substantiated with the use of simulation and real plant tests.AC201