Numerical Bifurcation Analysis of Conformal Formulations of the Einstein Constraints

The Einstein constraint equations have been the subject of study for more than fifty years. The introduction of the conformal method in the 1970's as a parameterization of initial data for the Einstein equations led to increased interest in the development of a complete solution theory for the constraints, with the theory for constant mean curvature (CMC) spatial slices and closed manifolds completely developed by 1995. The first general non-CMC existence result was establish by Holst et al. in 2008, with extensions to rough data by Holst et al. in 2009, and to vacuum spacetimes by Maxwell in 2009. The non-CMC theory remains mostly open; moreover, recent work of Maxwell on specific symmetry models sheds light on fundamental non-uniqueness problems with the conformal method as a parameterization in non-CMC settings. In parallel with these mathematical developments, computational physicists have uncovered surprising behavior in numerical solutions to the extended conformal thin sandwich formulation of the Einstein constraints. In particular, numerical evidence suggests the existence of multiple solutions with a quadratic fold, and a recent analysis of a simplified model supports this conclusion. In this article, we examine this apparent bifurcation phenomena in a methodical way, using modern techniques in bifurcation theory and in numerical homotopy methods. We first review the evidence for the presence of bifurcation in the Hamiltonian constraint in the time-symmetric case. We give a brief introduction to the mathematical framework for analyzing bifurcation phenomena, and then develop the main ideas behind the construction of numerical homotopy, or path-following, methods in the analysis of bifurcation phenomena. We then apply the continuation software package AUTO to this problem, and verify the presence of the fold with homotopy-based numerical methods.Comment: 13 pages, 4 figures. Final revision for publication, added material on physical implication

Far-from-constant mean curvature solutions of Einstein's constraint equations with positive Yamabe metrics

In this article we develop some new existence results for the Einstein constraint equations using the Lichnerowicz-York conformal rescaling method. The mean extrinsic curvature is taken to be an arbitrary smooth function without restrictions on the size of its spatial derivatives, so that it can be arbitrarily far from constant. The rescaled background metric belongs to the positive Yamabe class, and the freely specifiable part of the data given by the traceless-transverse part of the rescaled extrinsic curvature and the matter fields are taken to be sufficiently small, with the matter energy density not identically zero. Using topological fixed-point arguments and global barrier constructions, we then establish existence of solutions to the constraints. Two recent advances in the analysis of the Einstein constraint equations make this result possible: A new type of topological fixed-point argument without smallness conditions on spatial derivatives of the mean extrinsic curvature, and a new construction of global super-solutions for the Hamiltonian constraint that is similarly free of such conditions on the mean extrinsic curvature. For clarity, we present our results only for strong solutions on closed manifolds. However, our results also hold for weak solutions and for other cases such as compact manifolds with boundary; these generalizations will appear elsewhere. The existence results presented here for the Einstein constraints are apparently the first such results that do not require smallness conditions on spatial derivatives of the mean extrinsic curvature.Comment: 4 pages, no figures, accepted for publication in Physical Review Letters. (Abstract shortenned and other minor changes reflecting v4 version of arXiv:0712.0798

Nonconvolution nonlinear integral Volterra equations with monotone operators

Some results about existence, uniqueness, and attractive behaviour of solutions for nonlinear Volterra integral equations with non-convolution kernels are presented in this paper. These results are based on similar ones about nonlinear Volterra integral equations with convolution kernels and some comparison techniques. Therefore, this paper is devoted to find a wide class of nonconvolution Volterra integral equations where their solutions behave like those of Volterra equations with convolution kernels.Comment: 13 page

Entwicklung eines Inkubationssystems für ein inverses Mikroskop zur Langzeitbeobachtung von Zellkulturen in gekammerten Objektträgern

Trifunctional bispecific antibodies open up new immunological possibilities in tumour treatment. Prior to clinical application, comprehensive investigations using animal models and in vitro examinations need to be done. To investigate long-term interactions between Various immunologically active blood cells and individual tumour cells in the presence of antibodies, we developed an incubation system for experimental cell cultures on an inverted microscope. The system consists of a perspex box with a central moisture chamber with integrated water reservoir, external air circulation heating, and a CO2 supply. The sterile cell cultures are located in the wells of a slide positioned within a depression in the water reservoir. The newly developed incubation system enables continuous observation over the long term of experiments under optimal cell cultures conditions in combination with modern video techniques

Design of a fault tolerant airborne digital computer. Volume 2: Computational requirements and technology

This final report summarizes the work on the design of a fault tolerant digital computer for aircraft. Volume 2 is composed of two parts. Part 1 is concerned with the computational requirements associated with an advanced commercial aircraft. Part 2 reviews the technology that will be available for the implementation of the computer in the 1975-1985 period. With regard to the computation task 26 computations have been categorized according to computational load, memory requirements, criticality, permitted down-time, and the need to save data in order to effect a roll-back. The technology part stresses the impact of large scale integration (LSI) on the realization of logic and memory. Also considered was module interconnection possibilities so as to minimize fault propagation

Persistent clinical efficacy and safety of anti-tumour necrosis factor \textgreeka therapy with infliximab in patients with ankylosing spondylitis over 5 years: evidence for different types of response

Background: There is insufficient evidence for the long-term efficacy and safety of anti-tumour necrosis factor therapy in patients with ankylosing spondylitis (AS). This is the first report on the treatment with infliximab over 5 years.Methods: As part of a multicentre randomised trial, 69 patients with active AS at baseline (BL) have been continuously treated with infliximab (5 mg/kg i.v. every 6 weeks)---except for a short discontinuation after 3 years (FU1). The primary outcome of this extension was remission according to the ASsessment in Ankylosing Spondylitis (ASAS) criteria at the end of year 5 of the study (FU2).Results: Of the 43 patients who completed year 3, 42 agreed to continue, 38 of which (90.5%) finished year 5 (55% of 69 initially). Partial clinical remission was achieved in 13 of 38 patients (34.2%) at FU1 and FU2. At FU2, the mean Bath Ankylosing Spondylitis Disease Activity Index (BASDAI) was 2.5±1.9 (BL:6.4, FU1:2.5). BASDAI values <4 were seen in 79% of patients at both, FU1 and FU2. ASAS 20% and 40% responses were seen in 32 (84%) and 24 (63%) patients at FU2, respectively. Most patients classified as non-responders at FU2 were part-time responders, as all but one patient achieved an ASAS 20% response at least once within the last 2 years. Three types of responders were identified. No major side effects occurred during years 4 and 5 of infliximab therapy.Conclusions: Infliximab is safe and efficacious in AS patients over 5 years. The majority of the patients remained on treatment and had rather persistent levels of low disease activity. Different response types could be identified

The Carbon2Chem_® Laboratory in Oberhausen - A Workplace for Lab-Scale Setups within the Cross-Industrial Project

Within the Carbon2Chem® network, basic research is mandatory for a successful implementation and realization of sustainable technologies for CO2 emission reduction. For this purpose, the exchange of knowledge between the project partners in the individual subareas is as essential as obtaining precise data on the fundamental parameters on a laboratory scale in order to transfer them later to large-scale plants. Therefore, the Carbon2Chem® laboratory offers a platform to gain detailed insights into the individual sub-processes and to then apply these findings at the technical center in Duisburg