Properties of Information Carrying Waves in Cosmology

Recently we studied the effects of information carrying waves propagating through isotropic cosmologies. By information carrying we mean that the waves have an arbitrary dependence on a function. We found that the waves introduce shear and anisotropic stress into the universe. We then constructed explicit examples of pure gravity wave perturbations for which the presence of this anisotropic stress is essential and the null hypersurfaces playing the role of the histories of the wave-fronts in the background space-time are shear-free. Motivated by this result we now prove that these two properties are true for all information carrying waves in isotropic cosmologies.Comment: 15 pages, Latex File, accepted for publication in Physical Review

Anatomy of Soft Tissues of the Spinal Canal

Background and Objectives. Important issues regarding the spread of solutions in the epidural space and the anatomy of the site of action of spinal and epidural injections are unresolved. However, the detailed anatomy of the spinal canal has been incompletely determined. We therefore examined the microscopic anatomy of the spinal canal soft tissues, including relationships to the canal walls. Methods. Whole mounts were prepared of decalcified vertebral columns with undisturbed contents from three adult humans. Similar material was prepared from a macaque and baboon immediately on death to control for artifact of tissue change after death. Other tissues examined included nerve root and proximal spinal nerve complex and dorsal epidural fat obtained during surgery. Slides were examined by light microscopy at magnifications of 10-40×. Results. There is no fibrous tissue in the epidural space. The epidural fat is composed of uniform cells enclosed in a fine membrane. The dorsal fat is only attached to the canal wall in the dorsal midline and is often tenuously attached to the dura. The dura is joined to the canal wall only ventrally at the discs. Veins are evident predominantly in the ventral epidural space. Nerve roots are composed of multiple fascicles which disperse as they approach the dorsal root ganglion. An envelope of arachnoid encloses the roots near the site of exit from the dura. Conclusions. These features of the fat explain its semifluid consistency. Lack of substantial attachments to the dura facilitate movement of the dura relative to the canal wall and allow distribution of injected solution. Fibrous barriers are an unlikely explanation for asymmetric epidural anesthesia, but the midline fat could impede solution spread. Details of nerve-root structure and their envelope of pia-arachnoid membrane may be relevant to anesthetic action

Shear-Free Gravitational Waves in an Anisotropic Universe

We study gravitational waves propagating through an anisotropic Bianchi I dust-filled universe (containing the Einstein-de-Sitter universe as a special case). The waves are modeled as small perturbations of this background cosmological model and we choose a family of null hypersurfaces in this space-time to act as the histories of the wavefronts of the radiation. We find that the perturbations we generate can describe pure gravitational radiation if and only if the null hypersurfaces are shear-free. We calculate the gauge-invariant small perturbations explicitly in this case. How these differ from the corresponding perturbations when the background space-time is isotropic is clearly exhibited.Comment: 32 pages, accepted for publication in Physical Review

Gravitational Wave Propagation in Isotropic Cosmologies

We study the propagation of gravitational waves carrying arbitrary information through isotropic cosmologies. The waves are modelled as small perturbations of the background Robertson-Walker geometry. The perfect fluid matter distribution of the isotropic background is, in general, modified by small anisotropic stresses. For pure gravity waves, in which the perturbed Weyl tensor is radiative (i.e. type N in the Petrov classification), we construct explicit examples for which the presence of the anisotropic stress is shown to be essential and the histories of the wave-fronts in the background Robertson-Walker geometry are shear-free null hypersurfaces. The examples derived in this case are analogous to the Bateman waves of electromagnetic theory.Comment: 27 pages, accepted for publication in Phys.Rev.

The scattering map in two coupled piecewise-smooth systems, with numerical application to rocking blocks

We consider a non-autonomous dynamical system formed by coupling two piecewise-smooth systems in \RR^2 through a non-autonomous periodic perturbation. We study the dynamics around one of the heteroclinic orbits of one of the piecewise-smooth systems. In the unperturbed case, the system possesses two $C^0$ normally hyperbolic invariant manifolds of dimension two with a couple of three dimensional heteroclinic manifolds between them. These heteroclinic manifolds are foliated by heteroclinic connections between $C^0$ tori located at the same energy levels. By means of the {\em impact map} we prove the persistence of these objects under perturbation. In addition, we provide sufficient conditions of the existence of transversal heteroclinic intersections through the existence of simple zeros of Melnikov-like functions. The heteroclinic manifolds allow us to define the {\em scattering map}, which links asymptotic dynamics in the invariant manifolds through heteroclinic connections. First order properties of this map provide sufficient conditions for the asymptotic dynamics to be located in different energy levels in the perturbed invariant manifolds. Hence we have an essential tool for the construction of a heteroclinic skeleton which, when followed, can lead to the existence of Arnol'd diffusion: trajectories that, on large time scales, destabilize the system by further accumulating energy. We validate all the theoretical results with detailed numerical computations of a mechanical system with impacts, formed by the linkage of two rocking blocks with a spring

A comparison among four different retrieval methods for ice-cloud properties using data from CloudSat, CALIPSO, and MODIS

The A-Train constellation of satellites provides a new capability to measure vertical cloud profiles that leads to more detailed information on ice-cloud microphysical properties than has been possible up to now. A variational radar–lidar ice-cloud retrieval algorithm (VarCloud) takes advantage of the complementary nature of the CloudSat radar and Cloud–Aerosol Lidar and Infrared Pathfinder Satellite Observations (CALIPSO) lidar to provide a seamless retrieval of ice water content, effective radius, and extinction coefficient from the thinnest cirrus (seen only by the lidar) to the thickest ice cloud (penetrated only by the radar). In this paper, several versions of the VarCloud retrieval are compared with the CloudSat standard ice-only retrieval of ice water content, two empirical formulas that derive ice water content from radar reflectivity and temperature, and retrievals of vertically integrated properties from the Moderate Resolution Imaging Spectroradiometer (MODIS) radiometer. The retrieved variables typically agree to within a factor of 2, on average, and most of the differences can be explained by the different microphysical assumptions. For example, the ice water content comparison illustrates the sensitivity of the retrievals to assumed ice particle shape. If ice particles are modeled as oblate spheroids rather than spheres for radar scattering then the retrieved ice water content is reduced by on average 50% in clouds with a reflectivity factor larger than 0 dBZ. VarCloud retrieves optical depths that are on average a factor-of-2 lower than those from MODIS, which can be explained by the different assumptions on particle mass and area; if VarCloud mimics the MODIS assumptions then better agreement is found in effective radius and optical depth is overestimated. MODIS predicts the mean vertically integrated ice water content to be around a factor-of-3 lower than that from VarCloud for the same retrievals, however, because the MODIS algorithm assumes that its retrieved effective radius (which is mostly representative of cloud top) is constant throughout the depth of the cloud. These comparisons highlight the need to refine microphysical assumptions in all retrieval algorithms and also for future studies to compare not only the mean values but also the full probability density function

Bifurcations of piecewise smooth flows:perspectives, methodologies and open problems

In this paper, the theory of bifurcations in piecewise smooth flows is critically surveyed. The focus is on results that hold in arbitrarily (but finitely) many dimensions, highlighting significant areas where a detailed understanding is presently lacking. The clearest results to date concern equilibria undergoing bifurcations at switching boundaries, and limit cycles undergoing grazing and sliding bifurcations. After discussing fundamental concepts, such as topological equivalence of two piecewise smooth systems, discontinuity-induced bifurcations are defined for equilibria and limit cycles. Conditions for equilibria to exist in n-dimensions are given, followed by the conditions under which they generically undergo codimension-one bifurcations. The extent of knowledge of their unfoldings is also summarized. Codimension-one bifurcations of limit cycles and boundary-intersection crossing are described together with techniques for their classification. Codimension-two bifurcations are discussed with suggestions for further study

Metric Perturbation Approach to Gravitational Waves in Isotropic Cosmologies

Gravitational waves in isotropic cosmologies were recently studied using the gauge-invariant approach of Ellis-Bruni. We now construct the linearised metric perturbations of the background Robertson-Walker space-time which reproduce the results obtained in that study. The analysis carried out here also facilitates an easy comparison with Bardeen.Comment: 29 pages, Latex file, accepted for publication in Physical Review