A model problem for conformal parameterizations of the Einstein constraint equations

We investigate the possibility that the conformal and conformal thin sandwich (CTS) methods can be used to parameterize the set of solutions of the vacuum Einstein constraint equations. To this end we develop a model problem obtained by taking the quotient of certain symmetric data on conformally flat tori. Specializing the model problem to a three-parameter family of conformal data we observe a number of new phenomena for the conformal and CTS methods. Within this family, we obtain a general existence theorem so long as the mean curvature does not change sign. When the mean curvature changes sign, we find that for certain data solutions exist if and only if the transverse-traceless tensor is sufficiently small. When such solutions exist, there are generically more than one. Moreover, the theory for mean curvatures changing sign is shown to be extremely sensitive with respect to the value of a coupling constant in the Einstein constraint equations.Comment: 40 pages, 4 figure

Sensitivity of solar off-limb line profiles to electron density stratification and the velocity distribution anisotropy

The effect of the electron density stratification on the intensity profiles of the H I Ly-$\alpha$ line and the O VI and Mg X doublets formed in solar coronal holes is investigated. We employ an analytical 2-D model of the large scale coronal magnetic field that provides a good representation of the corona at the minimum of solar activity. We use the mass-flux conservation equation to determine the outflow speed of the solar wind at any location in the solar corona and take into account the integration along the line of sight (LOS). The main assumption we make is that no anisotropy in the kinetic temperature of the coronal species is considered. We find that at distances greater than 1 Rsun from the solar surface the widths of the emitted lines of O VI and Mg X are sensitive to the details of the adopted electron density stratification. However, Ly-$\alpha$, which is a pure radiative line, is hardly affected. The calculated total intensities of Ly-$\alpha$ and the O VI doublet depend to a lesser degree on the density stratification and are comparable to the observed ones for most of the considered density models. The widths of the observed profiles of Ly-$\alpha$ and Mg X are well reproduced by most of the considered electron density stratifications, while for the O VI doublet only few stratifications give satisfying results. The densities deduced from SOHO data result in O VI profiles whose widths and intensity ratio are relatively close to the values observed by UVCS although only isotropic velocity distributions are employed. These density profiles also reproduce the other considered observables with good accuracy. Thus the need for a strong anisotropy of the velocity distribution (i.e. a temperature anisotropy) is not so clear cut as previous investigations of UVCS data suggested. ...Comment: 11 pages; 11 figure

Precompactness of solutions to the Ricci flow in the absence of injectivity radius estimates

Consider a sequence of pointed n-dimensional complete Riemannian manifolds {(M_i,g_i(t), O_i)} such that t in [0,T] are solutions to the Ricci flow and g_i(t) have uniformly bounded curvatures and derivatives of curvatures. Richard Hamilton showed that if the initial injectivity radii are uniformly bounded below then there is a subsequence which converges to an n-dimensional solution to the Ricci flow. We prove a generalization of this theorem where the initial metrics may collapse. Without injectivity radius bounds we must allow for convergence in the Gromov-Hausdorff sense to a space which is not a manifold but only a metric space. We then look at the local geometry of the limit to understand how it relates to the Ricci flow.Comment: Published by Geometry and Topology at http://www.maths.warwick.ac.uk/gt/GTVol7/paper14.abs.htm

High velocity spikes in Gowdy spacetimes

We study the behavior of spiky features in Gowdy spacetimes. Spikes with velocity initially high are, generally, driven to low velocity. Let n be any integer greater than or equal to 1. If the initial velocity of an upward pointing spike is between 4n-3 and 4n-1 the spike persists with final velocity between 1 and 2, while if the initial velocity is between 4n-1 and 4n+1, the spiky feature eventually disappears. For downward pointing spikes the analogous rule is that spikes with initial velocity between 4n-4 and 4n-2 persist with final velocity between 0 and 1, while spikes with initial velocity between 4n-2 and 4n eventually disappear.Comment: discussion of constraints added. Accepted for publication in Phys. Rev.

New Dimensions for Wound Strings: The Modular Transformation of Geometry to Topology

We show, using a theorem of Milnor and Margulis, that string theory on compact negatively curved spaces grows new effective dimensions as the space shrinks, generalizing and contextualizing the results in hep-th/0510044. Milnor's theorem relates negative sectional curvature on a compact Riemannian manifold to exponential growth of its fundamental group, which translates in string theory to a higher effective central charge arising from winding strings. This exponential density of winding modes is related by modular invariance to the infrared small perturbation spectrum. Using self-consistent approximations valid at large radius, we analyze this correspondence explicitly in a broad set of time-dependent solutions, finding precise agreement between the effective central charge and the corresponding infrared small perturbation spectrum. This indicates a basic relation between geometry, topology, and dimensionality in string theory.Comment: 28 pages, harvmac big. v2: references and KITP preprint number added, minor change

Solutions of the Einstein Constraint Equations with Apparent Horizon Boundaries

We construct asymptotically Euclidean solutions of the vacuum Einstein constraint equations with an apparent horizon boundary condition. Specifically, we give sufficient conditions for the constant mean curvature conformal method to generate such solutions. The method of proof is based on the barrier method used by Isenberg for compact manifolds without boundary, suitably extended to accommodate semilinear boundary conditions and low regularity metrics. As a consequence of our results for manifolds with boundary, we also obtain improvements to the theory of the constraint equations on asymptotically Euclidean manifolds without boundary.Comment: 27 pages, 1 figure, TeX, v3. Final version to appear in CMP. Exposition has been extensively tightened and the proof of Proposition 3.5 has been simplifie

Phenomenology of the Gowdy Universe on T3×RT^3 \times R

Numerical studies of the plane symmetric, vacuum Gowdy universe on $T^3 \times R$ yield strong support for the conjectured asymptotically velocity term dominated (AVTD) behavior of its evolution toward the singularity except, perhaps, at isolated spatial points. A generic solution is characterized by spiky features and apparent ``discontinuities'' in the wave amplitudes. It is shown that the nonlinear terms in the wave equations drive the system generically to the ``small velocity'' AVTD regime and that the spiky features are caused by the absence of these terms at isolated spatial points.Comment: 19 pages, 21 figures, uses Revtex, psfi

Anti-factor Xa antibodies in patients with antiphospholipid syndrome and their effects upon coagulation assays

- Introduction: The aim of this study was to examine the prevalence and functional effects of antibodies directed against Factor (F)Xa and other serine proteases (SP) in patients with antiphospholipid syndrome (APS). - Methods: Serum from patients with APS (n = 59), systemic lupus erythematosus (SLE; n = 106), other autoimmune rheumatic disease (ARD; n = 63) and 40 healthy controls (HC) were tested for IgG activity against thrombin (Thr), FXa, FVIIa, phosphatidylserine (PS)/FXa and antithrombin (AT)-III by enzyme-linked immunosorbent assay (ELISA). Anti-FXa positive IgG were purified to measure their avidity by chaotropic ELISA and functional effects upon clotting time (FXa-ACT) and FXa enzymatic activity (± AT-III). - Results: Anti-FXa IgG were found in patients with SLE (49.1%) and APS (33.9%) (P <0.05) but not in ARD controls and HC. In contrast, anti-Thr and anti-PS/FXa IgG were identified in other ARD and anti-FVIIa IgG were low in all groups. The avidity of APS-IgG to FXa was significantly higher than SLE-IgG (P <0.05). Greatest prolongation of FXa-ACT was observed with APS-IgG and greatest inhibitory effect upon FXa enzymatic activity was found with APS-IgG followed by SLE-IgG compared to HC-IgG. ATIII inhibition of FXa was significantly reduced by APS-IgG compared with HC and SLE (P <0.05) and did not correlate with binding to AT-III. - Conclusion: APS anti-FXa IgG have higher avidity to FXa and greater effects upon the enzymatic and coagulant activity of FXa compared with SLE anti-FXa IgG. Further studies of anti-FXa antibodies in APS, SLE and other non-autoimmune thrombotic disease cohorts are now required to evaluate whether targeting FXa with selective inhibitors in patients bearing anti-FXa antibodies may be an effective treatment strategy