H\"older estimates for parabolic operators on domains with rough boundary

We investigate linear parabolic, second-order boundary value problems with mixed boundary conditions on rough domains. Assuming only boundedness and ellipticity on the coefficient function and very mild conditions on the geometry of the domain, including a very weak compatibility condition between the Dirichlet boundary part and its complement, we prove H\"older continuity of the solution in space and time.Comment: 1 figur

Communicating Mental Illness in the Black American Community

Human-human interactions are of central relevance for the success in professional and occupational environments, which also substantially influence quality of life. This is especially true in the case of individuals with high-functioning autism (HFA), who experience deficits in social cognition that often lead to social exclusion and unemployment. Despite good education and high motivation, individuals with HFA do not reach employment rates that are substantially higher than 50 %. This is an alarmingly high rate of unemployment considering that the United Nations have recently emphasized the inclusion of handicapped persons as a mandatory human right. To date, the specific needs of autistic persons with respect to their working environment are largely unexplored. It remains moreover an open question how support systems and activities, including newly developed communication devices for professional environments of individuals with HFA, should look like. The German health and social care systems are not adequately prepared for the proper support of this population. This leads us to suggest that supported employment programs should be developed for adults with HFA that specifically address their needs and requirements. Such programs should comprise (1) the adequate assessment of HFA, including a neuropsychological profile and an individual matching of persons' preferences with requirements of the working place, (2) on-the-job coaching activities that include systematic communication and interaction training, and (3) instruction of non-autistic peers, including colleagues and supervisors, about weaknesses and strengths of HFA

Local freedom in the gravitational field

In a cosmological context, the electric and magnetic parts of the Weyl tensor, E_{ab} and H_{ab}, represent the locally free curvature - i.e. they are not pointwise determined by the matter fields. By performing a complete covariant decomposition of the derivatives of E_{ab} and H_{ab}, we show that the parts of the derivative of the curvature which are locally free (i.e. not pointwise determined by the matter via the Bianchi identities) are exactly the symmetrised trace-free spatial derivatives of E_{ab} and H_{ab} together with their spatial curls. These parts of the derivatives are shown to be crucial for the existence of gravitational waves.Comment: New results on gravitational waves included; new references added; revised version (IOP style) to appear Class. Quantum Gra

Dynamical systems approach to G2 cosmology

In this paper we present a new approach for studying the dynamics of spatially inhomogeneous cosmological models with one spatial degree of freedom. By introducing suitable scale-invariant dependent variables we write the evolution equations of the Einstein field equations as a system of autonomous partial differential equations in first-order symmetric hyperbolic format, whose explicit form depends on the choice of gauge. As a first application, we show that the asymptotic behaviour near the cosmological initial singularity can be given a simple geometrical description in terms of the local past attractor on the boundary of the scale-invariant dynamical state space. The analysis suggests the name ``asymptotic silence'' to describe the evolution of the gravitational field near the cosmological initial singularity.Comment: 28 pages, 3 tables, 1 *.eps figure, LaTeX2e (10pt), matches version accepted for publication by Classical and Quantum Gravit

A dynamical system approach to higher order gravity

The dynamical system approach has recently acquired great importance in the investigation on higher order theories of gravity. In this talk I review the main results and I give brief comments on the perspectives for further developments.Comment: 6 pages, 1 figure, 2 tables, talk given at IRGAC 2006, July 200

On the propagation of jump discontinuities in relativistic cosmology

A recent dynamical formulation at derivative level \ptl^{3}g for fluid spacetime geometries $({\cal M}, {\bf g}, {\bf u})$, that employs the concept of evolution systems in first-order symmetric hyperbolic format, implies the existence in the Weyl curvature branch of a set of timelike characteristic 3-surfaces associated with propagation speed |v| = \sfrac{1}{2} relative to fluid-comoving observers. We show it is the physical role of the constraint equations to prevent realisation of jump discontinuities in the derivatives of the related initial data so that Weyl curvature modes propagating along these 3-surfaces cannot be activated. In addition we introduce a new, illustrative first-order symmetric hyperbolic evolution system at derivative level \ptl^{2}g for baryotropic perfect fluid cosmological models that are invariant under the transformations of an Abelian $G_{2}$ isometry group.Comment: 19 pages, 1 table, REVTeX v3.1 (10pt), submitted for publication to Physical Review D; added Report-No, corrected typo

Isotropic singularity in inhomogeneous brane cosmological models

We discuss the asymptotic dynamical evolution of spatially inhomogeneous brane-world cosmological models close to the initial singularity. By introducing suitable scale-invariant dependent variables and a suitable gauge, we write the evolution equations of the spatially inhomogeneous $G_{2}$ brane cosmological models with one spatial degree of freedom as a system of autonomous first-order partial differential equations. We study the system numerically, and we find that there always exists an initial singularity, which is characterized by the fact that spatial derivatives are dynamically negligible. More importantly, from the numerical analysis we conclude that there is an initial isotropic singularity in all of these spatially inhomogeneous brane cosmologies for a range of parameter values which include the physically important cases of radiation and a scalar field source. The numerical results are supported by a qualitative dynamical analysis and a calculation of the past asymptotic decay rates. Although the analysis is local in nature, the numerics indicates that the singularity is isotropic for all relevant initial conditions. Therefore this analysis, and a preliminary investigation of general inhomogeneous ($G_0$) models, indicates that it is plausible that the initial singularity is isotropic in spatially inhomogeneous brane-world cosmological models and consequently that brane cosmology naturally gives rise to a set of initial data that provide the conditions for inflation to subsequently take place.Comment: 32 pages with 8 pictures. submitted to Class. Quant. Gra

Causal propagation of geometrical fields in relativistic cosmology

We employ the extended 1+3 orthonormal frame formalism for fluid spacetime geometries $({\cal M}, {\bf g}, {\bf u})$, which contains the Bianchi field equations for the Weyl curvature, to derive a 44-D evolution system of first-order symmetric hyperbolic form for a set of geometrically defined dynamical field variables. Describing the matter source fields phenomenologically in terms of a barotropic perfect fluid, the propagation velocities $v$ (with respect to matter-comoving observers that Fermi-propagate their spatial reference frames) of disturbances in the matter and the gravitational field, represented as wavefronts by the characteristic 3-surfaces of the system, are obtained. In particular, the Weyl curvature is found to account for two (non-Lorentz-invariant) Coulomb-like characteristic eigenfields propagating with $v = 0$ and four transverse characteristic eigenfields propagating with $|v| = 1$, which are well known, and four (non-Lorentz-invariant) longitudinal characteristic eigenfields propagating with |v| = \sfrac{1}{2}. The implications of this result are discussed in some detail and a parallel is drawn to the propagation of irregularities in the matter distribution. In a worked example, we specialise the equations to cosmological models in locally rotationally symmetric class II and include the constraints into the set of causally propagating dynamical variables.Comment: 25 pages, RevTeX (10pt), accepted for publication by Physical Review