The initial value problem and the dynamical formulation of general relativity

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Symmetry breaking in general relativity

Bifurcation theory is used to analyze the space of solutions of Einstein's equations near a spacetime with symmetries. The methods developed here allow one to describe precisely how the symmetry is broken as one branches from a highly symmetric spacetime to nearby spacetimes with fewer symmetries, and finally to a generic solution with no symmetries. This phenomenon of symmetry breaking is associated with the fact that near symmetric solutions the space of solutions of Einstein's equations does not form a smooth manifold but rather has a conical structure. The geometric picture associated with this conical structure enables one to understand the breaking of symmetries. Although the results are described for pure gravity, they may be extended to classes of fields coupled to gravity, such as gauge theories. Since most of the known solutions of Einstein's equations have Killing symmetries, the study of how these symmetries are broken by small perturbations takes on considerable theoretical significance

The structure of the space of solutions of Einstein's equations. I. One Killing field.

This paper deals with globally hyperbolic solutions of the vacuum Einstein equations in a neighborhood of spacetimes that have a compact Cauchy surface of constant mean curvature. The first part of paper deal with solutions which have a single Killing vector field. Part II of the paper will then deal with the case of several Killing fields and general applications to mechanics

Manifolds of Riemannian metrics with prescribed scalar curvature

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An Introduction to Conformal Ricci Flow

We introduce a variation of the classical Ricci flow equation that modifies the unit volume constraint of that equation to a scalar curvature constraint. The resulting equations are named the Conformal Ricci Flow Equations because of the role that conformal geometry plays in constraining the scalar curvature. These equations are analogous to the incompressible Navier-Stokes equations of fluid mechanics inasmuch as a conformal pressure arises as a Lagrange multiplier to conformally deform the metric flow so as to maintain the scalar curvature constraint. The equilibrium points are Einstein metrics with a negative Einstein constant and the conformal pressue is shown to be zero at an equilibrium point and strictly positive otherwise. The geometry of the conformal Ricci flow is discussed as well as the remarkable analytic fact that the constraint force does not lose derivatives and thus analytically the conformal Ricci equation is a bounded perturbation of the classical unnormalized Ricci equation. That the constraint force does not lose derivatives is exactly analogous to the fact that the real physical pressure force that occurs in the Navier-Stokes equations is a bounded function of the velocity. Using a nonlinear Trotter product formula, existence and uniqueness of solutions to the conformal Ricci flow equations is proven. Lastly, we discuss potential applications to Perelman's proposed implementation of Hamilton's program to prove Thurston's 3-manifold geometrization conjectures.Comment: 52 pages, 1 figur

Instances and connectors : issues for a second generation process language

This work is supported by UK EPSRC grants GR/L34433 and GR/L32699Over the past decade a variety of process languages have been defined, used and evaluated. It is now possible to consider second generation languages based on this experience. Rather than develop a second generation wish list this position paper explores two issues: instances and connectors. Instances relate to the relationship between a process model as a description and the, possibly multiple, enacting instances which are created from it. Connectors refers to the issue of concurrency control and achieving a higher level of abstraction in how parts of a model interact. We believe that these issues are key to developing systems which can effectively support business processes, and that they have not received sufficient attention within the process modelling community. Through exploring these issues we also illustrate our approach to designing a second generation process language.Postprin

Experimental Study of the Shortest Reset Word of Random Automata

In this paper we describe an approach to finding the shortest reset word of a finite synchronizing automaton by using a SAT solver. We use this approach to perform an experimental study of the length of the shortest reset word of a finite synchronizing automaton. The largest automata we considered had 100 states. The results of the experiments allow us to formulate a hypothesis that the length of the shortest reset word of a random finite automaton with $n$ states and 2 input letters with high probability is sublinear with respect to $n$ and can be estimated as $1.95 n^{0.55}.

Exploring Appropriation of Global Cultural Rituals

Adolescents, as a consequence of identification with popular culture, have been described as having homogenous consumption patterns. More recently, however, it has been recognised that ‘glocalisation’ (global practices reworked to fit local contexts) affords an opportunity for differentiation. This paper considers a recent UK phenomenon, namely that of the US high school prom, and seeks to explore the ways in which this ritual has been adopted or adapted as part of youth culture. The method employed here was mixed methods and included in-depth interviews with those who attended a prom in the last three years as well as a questionnaire distributed amongst high school pupils who were anticipating a high school prom. The findings illustrate that the high school prom in the UK is becoming increasingly integrated into the fabric of youth culture although, depending on the agentic abilities employed by the emerging adults in the sample, there is differing appropriation of this ritual event particularly in relation to attitudes towards and motivations for attending the prom. A typology of prom attendees is posited. This paper contributes to our understanding of this practice in a local context

Rgnef (p190RhoGEF) Knockout Inhibits RhoA Activity, Focal Adhesion Establishment, and Cell Motility Downstream of Integrins

Cell migration is a highly regulated process that involves the formation and turnover of cell-matrix contact sites termed focal adhesions. Rho-family GTPases are molecular switches that regulate actin and focal adhesion dynamics in cells. Guanine nucleotide exchange factors (GEFs) activate Rho-family GTPases. Rgnef (p190RhoGEF) is a ubiquitous 190 kDa GEF implicated in the control of colon carcinoma and fibroblast cell motility.Rgnef exon 24 floxed mice (Rgnef(flox)) were created and crossed with cytomegalovirus (CMV)-driven Cre recombinase transgenic mice to inactivate Rgnef expression in all tissues during early development. Heterozygous Rgnef(WT/flox) (Cre+) crosses yielded normal Mendelian ratios at embryonic day 13.5, but Rgnef(flox/flox) (Cre+) mice numbers at 3 weeks of age were significantly less than expected. Rgnef(flox/flox) (Cre+) (Rgnef-/-) embryos and primary mouse embryo fibroblasts (MEFs) were isolated and verified to lack Rgnef protein expression. When compared to wildtype (WT) littermate MEFs, loss of Rgnef significantly inhibited haptotaxis migration, wound closure motility, focal adhesion number, and RhoA GTPase activation after fibronectin-integrin stimulation. In WT MEFs, Rgnef activation occurs within 60 minutes upon fibronectin plating of cells associated with RhoA activation. Rgnef-/- MEF phenotypes were rescued by epitope-tagged Rgnef re-expression.Rgnef-/- MEF phenotypes were due to Rgnef loss and support an essential role for Rgnef in RhoA regulation downstream of integrins in control of cell migration