Initial boundary value problems for Einstein's field equations and geometric uniqueness

While there exist now formulations of initial boundary value problems for Einstein's field equations which are well posed and preserve constraints and gauge conditions, the question of geometric uniqueness remains unresolved. For two different approaches we discuss how this difficulty arises under general assumptions. So far it is not known whether it can be overcome without imposing conditions on the geometry of the boundary. We point out a natural and important class of initial boundary value problems which may offer possibilities to arrive at a fully covariant formulation.Comment: 19 page

The Role of the Mitochondrial Genome in Ageing and Carcinogenesis

Mitochondrial DNA mutations and polymorphisms have been the focus of intensive investigations for well over a decade in an attempt to understand how they affect fundamental processes such as cancer and aging. Initial interest in mutations occurring in mitochondrial DNA of cancer cells diminished when most were found to be the same mutations which occurred during the evolution of human mitochondrial haplogroups. However, increasingly correlations are being found between various mitochondrial haplogroups and susceptibility to cancer or diseases in some cases and successful aging in others

Einstein equations in the null quasi-spherical gauge

The structure of the full Einstein equations in a coordinate gauge based on expanding null hypersurfaces foliated by metric 2-spheres is explored. The simple form of the resulting equations has many applications -- in the present paper we describe the structure of timelike boundary conditions; the matching problem across null hypersurfaces; and the propagation of gravitational shocks.Comment: 12 pages, LaTeX (revtex, amssymb), revision 18 pages, contains expanded discussion and explanations, updated references, to appear in CQ

Gluing Initial Data Sets for General Relativity

We establish an optimal gluing construction for general relativistic initial data sets. The construction is optimal in two distinct ways. First, it applies to generic initial data sets and the required (generically satisfied) hypotheses are geometrically and physically natural. Secondly, the construction is completely local in the sense that the initial data is left unaltered on the complement of arbitrarily small neighborhoods of the points about which the gluing takes place. Using this construction we establish the existence of cosmological, maximal globally hyperbolic, vacuum space-times with no constant mean curvature spacelike Cauchy surfaces.Comment: Final published version - PRL, 4 page

Trapped Surfaces in Vacuum Spacetimes

An earlier construction by the authors of sequences of globally regular, asymptotically flat initial data for the Einstein vacuum equations containing trapped surfaces for large values of the parameter is extended, from the time symmetric case considered previously, to the case of maximal slices. The resulting theorem shows rigorously that there exists a large class of initial configurations for non-time symmetric pure gravitational waves satisfying the assumptions of the Penrose singularity theorem and so must have a singularity to the future.Comment: 14 page

Embedding spherical spacelike slices in a Schwarzschild solution

Given a spherical spacelike three-geometry, there exists a very simple algebraic condition which tells us whether, and in which, Schwarzschild solution this geometry can be smoothly embedded. One can use this result to show that any given Schwarzschild solution covers a significant subset of spherical superspace and these subsets form a sequence of nested domains as the Schwarzschild mass increases. This also demonstrates that spherical data offer an immediate counter example to the thick sandwich `theorem'

The Yamabe invariant for axially symmetric two Kerr black holes initial data

An explicit 3-dimensional Riemannian metric is constructed which can be interpreted as the (conformal) sum of two Kerr black holes with aligned angular momentum. When the separation distance between them is large we prove that this metric has positive Ricci scalar and hence positive Yamabe invariant. This metric can be used to construct axially symmetric initial data for two Kerr black holes with large angular momentum.Comment: 14 pages, 2 figure

Mechanisms of melanoma resistance to treatment with BRAF and MEK inhibitors

Several mechanisms of resistance to inhibition of BRAF activity in melanoma cells have been described so far. Genetic studies have shown that mutations in MEK1 kinase (MAP kinase kinase), which result in constitutive activation of ERK kinase, result in resistance to treatment. Another mechanism of the acquired BRAF inhibition resistance is the accumulation of activating mutations in the NRAS oncogene, which drives the activation of CRAF. This in turn leads to a permanent activation of the signal transduction to MEK and ERK. Another important mechanism of resistance is the formation of variants of the BRAF V600E gene splicing, including variants that lack exons 4 to 8 containing the RAS-binding domain. The presence of the p61 BRAF V600E variant leads to the constitutive ERK signal, which is resistant to RAF inhibition. In addition, treatment resistance is affected by hyperactivation of tyrosine kinase receptors such as platelet-derived factor receptor β (PDFRβ), insulin-like growth factor 1 receptor (IGF-1R) and erythropoietin-producing hepatocellular receptors (EPH) – leading to the induction of the 3-phosphoinositol kinase pathway (PI3K) in patients treated with BRAF or MEK inhibitors. Another interesting path of BRAFi/MEKi resistance is over-expression of the epidermal growth factor receptor (EGFR) through ne­gative feedback in patients treated with BRAF inhibitors (BRAFi) – EGFR is not normally expressed in untreated melanomas

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