Computer Algebra Applications for Numerical Relativity

We discuss the application of computer algebra to problems commonly arising in numerical relativity, such as the derivation of 3+1-splits, manipulation of evolution equations and automatic code generation. Particular emphasis is put on working with abstract index tensor quantities as much as possible.Comment: 5 pages, no figures. To appear in the Proceedings of the Spanish Relativity Meeting (ERE 2002), Mao, Menorca, Spain, 22-24 Sept 200

From Tensor Equations to Numerical Code -- Computer Algebra Tools for Numerical Relativity

In this paper we present our recent work in developing a computer-algebra tool for systems of partial differential equations (PDEs), termed "Kranc". Our work is motivated by the problem of finding solutions of the Einstein equations through numerical simulations. Kranc consists of Mathematica based computer-algebra packages, that facilitate the task of dealing with symbolic tensorial calculations and realize the conversion of systems of partial differential evolution equations into parallelized C or Fortran code.Comment: LaTeX llncs style, 9 pages, 1 figure, to appaer in the proceedings of "SYNASC 2004 - 6th International Symposium on Symbolic and Numeric Algorithms for Scientific Computing", Timisoara, Romania, September 26-30 200

Fast jets from bubbles close to solid objects: examples from pillars in water to infinite planes in different liquids

The dynamics of a single, laser-induced cavitation bubble on top of a solid cylinder and right at a plane solid boundary is studiedboth experimentally and numerically. The most intriguing phenomenon that occurs for a millimeter sized bubble right at a flatsolid boundary in water is the formation of a fast jet that is directed towards the solid with a speed of the order of 1000 m/s.Paradoxically, in this setting, fast jet formation causally is related to the viscosity of the liquid.Thus, results from numericalsimulations with varying liquid viscosity and bubble size are presented. Bubble dynamics and jet formation mechanisms arediscussed. It is shown, that fast jet formation persists for a wide range of liquid viscosities, including e.g. 50 cSt silicone oil. Forbubbles generated close to the flat top of a long, thin cylinder the parameter space of initial distance to the cylinder, bubble size andcylinder radius is scanned numerically and partly compared to experiments. When the maximum radius of the bubble exceeds theone of the cylinder the bubble collapses in the form of a mushroom or can resemble a trophy, depending on the values of thegeometry parameters. Complex patterns of jet formation with jet speeds ranging from the order of a few hundred m/s to severalthousand m/s are found.The results represent a contribution to understand the behavior of bubbles collapsing close to solid surfaces,in particular, how thin, fast jets are generated

Kranc: a Mathematica application to generate numerical codes for tensorial evolution equations

We present a suite of Mathematica-based computer-algebra packages, termed "Kranc", which comprise a toolbox to convert (tensorial) systems of partial differential evolution equations to parallelized C or Fortran code. Kranc can be used as a "rapid prototyping" system for physicists or mathematicians handling very complicated systems of partial differential equations, but through integration into the Cactus computational toolkit we can also produce efficient parallelized production codes. Our work is motivated by the field of numerical relativity, where Kranc is used as a research tool by the authors. In this paper we describe the design and implementation of both the Mathematica packages and the resulting code, we discuss some example applications, and provide results on the performance of an example numerical code for the Einstein equations.Comment: 24 pages, 1 figure. Corresponds to journal versio

Combining anatomical, diffusion, and resting state functional magnetic resonance imaging for individual classification of mild and moderate Alzheimer's disease

AbstractMagnetic resonance imaging (MRI) is sensitive to structural and functional changes in the brain caused by Alzheimer's disease (AD), and can therefore be used to help in diagnosing the disease. Improving classification of AD patients based on MRI scans might help to identify AD earlier in the disease's progress, which may be key in developing treatments for AD. In this study we used an elastic net classifier based on several measures derived from the MRI scans of mild to moderate AD patients (N=77) from the prospective registry on dementia study and controls (N=173) from the Austrian Stroke Prevention Family Study. We based our classification on measures from anatomical MRI, diffusion weighted MRI and resting state functional MRI. Our unimodal classification performance ranged from an area under the curve (AUC) of 0.760 (full correlations between functional networks) to 0.909 (grey matter density). When combining measures from multiple modalities in a stepwise manner, the classification performance improved to an AUC of 0.952. This optimal combination consisted of grey matter density, white matter density, fractional anisotropy, mean diffusivity, and sparse partial correlations between functional networks. Classification performance for mild AD as well as moderate AD also improved when using this multimodal combination. We conclude that different MRI modalities provide complementary information for classifying AD. Moreover, combining multiple modalities can substantially improve classification performance over unimodal classification

GRB Light Curves in the Relativistic Turbulence Model

Randomly oriented relativistic emitters in a relativistically expanding shell provides an alternative to internal shocks as a mechanism for producing GRBs' variable light curves with efficient conversion of energy to radiation. In this model the relativistic outflow is broken into small emitters moving relativistically in the outflow's rest frame. Variability arises because an observer sees an emitter only when its velocity points towards him so that only a small fraction of the emitters are seen by a given observer. Models with significant relativistic random motions require converting and maintaining a large fraction of the overall energy into these motions. While it is not clear how this is achieved, we explore here, using two toy models, the constraints on parameters required to produce light curves comparable to the observations. We find that a tight relation between the size of the emitters and the bulk and random Lorentz factors is needed and that the random Lorentz factor determines the variability. While both models successfully produce the observed variability there are several inconsistencies with other properties of the light curves. Most of which, but not all, might be resolved if the central engine is active for a long time producing a number of shells, resembling to some extent the internal shocks model.Comment: Significantly revised with a discussion of additional models. Accepted for publication in APJ

Type II Critical Collapse of a Self-Gravitating Nonlinear σ\sigma-Model

We report on the existence and phenomenology of type II critical collapse within the one-parameter family of SU(2) $\sigma$-models coupled to gravity. Numerical investigations in spherical symmetry show discretely self-similar (DSS) behavior at the threshold of black hole formation for values of the dimensionless coupling constant \ccbeta ranging from 0.2 to 100; at 0.18 we see small deviations from DSS. While the echoing period $\Delta$ of the critical solution rises sharply towards the lower limit of this range, the characteristic mass scaling has a critical exponent $\gamma$ which is almost independent of \ccbeta, asymptoting to $0.1185 \pm 0.0005$ at large \ccbeta. We also find critical scaling of the scalar curvature for near-critical initial data. Our numerical results are based on an outgoing-null-cone formulation of the Einstein-matter equations, specialized to spherical symmetry. Our numerically computed initial-data critical parameters $p^*$ show 2nd order convergence with the grid resolution, and after compensating for this variation in $p^*$, our individual evolutions are uniformly 2nd order convergent even very close to criticality.Comment: 23 pages, includes 10 postscript figure files, uses REVTeX, epsf, psfrag, and AMS math fonts (amstex + amssymb); to appear in PRD15. Summary of revisions from v2: fix wrong formula in figure 6 caption and y-axis label, also minor wording changes and update publication status of refs 5-