169 research outputs found
Sharp estimates on the first eigenvalue of the p-Laplacian with negative Ricci lower bound
We complete the picture of sharp eigenvalue estimates for the p-Laplacian on
a compact manifold by providing sharp estimates on the first nonzero eigenvalue
of the nonlinear operator when the Ricci curvature is bounded from
below by a negative constant. We assume that the boundary of the manifold is
convex, and put Neumann boundary conditions on it. The proof is based on a
refined gradient comparison technique and a careful analysis of the underlying
model spaces.Comment: Sign mistake fixed in the proof of the gradient comparison theorem
(theorem 5.1 pag 10), and some minor improvements aroun
On the regularity up to the boundary for certain nonlinear elliptic systems
We consider a class of nonlinear elliptic systems and we prove regularity up to the boundary for second order derivatives. In the proof we trace carefully the dependence on the various parameters of the problem, in order to establish, in a further work, results for more general systems
Quantum fields and local measurements
The process of quantum measurement is considered in the algebraic framework of quantum field theory on curved spacetimes. Measurements are carried out on one quantum field theory, the "system", using another, the "probe". The measurement process involves a dynamical coupling of "system" and "probe" within a bounded spacetime region. The resulting "coupled theory" determines a scattering map on the uncoupled combination of the "system" and "probe" by reference to natural "in" and "out" spacetime regions. No specific interaction is assumed and all constructions are local and covariant. Given any initial state of the probe in the "in" region, the scattering map determines a completely positive map from "probe" observables in the "out" region to "induced system observables", thus providing a measurement scheme for the latter. It is shown that the induced system observables may be localized in the causal hull of the interaction coupling region and are typically less sharp than the probe observable, but more sharp than the actual measurement on the coupled theory. Post-selected states conditioned on measurement outcomes are obtained using Davies-Lewis instruments that depend on the initial probe state. Composite measurements involving causally ordered coupling regions are also considered. Provided that the scattering map obeys a causal factorization property, the causally ordered composition of the individual instruments coincides with the composite instrument; in particular, the instruments may be combined in either order if the coupling regions are causally disjoint. This is the central consistency property of the proposed framework. The general concepts and results are illustrated by an example in which both "system" and "probe" are quantized linear scalar fields, coupled by a quadratic interaction term with compact spacetime support. System observables induced by simple probe observables are calculated exactly, for sufficiently weak coupling, and compared with first order perturbation theory
The Spherically Symmetric Standard Model with Gravity
Spherical reduction of generic four-dimensional theories is revisited. Three
different notions of "spherical symmetry" are defined. The following sectors
are investigated: Einstein-Cartan theory, spinors, (non-)abelian gauge fields
and scalar fields. In each sector a different formalism seems to be most
convenient: the Cartan formulation of gravity works best in the purely
gravitational sector, the Einstein formulation is convenient for the Yang-Mills
sector and for reducing scalar fields, and the Newman-Penrose formalism seems
to be the most transparent one in the fermionic sector. Combining them the
spherically reduced Standard Model of particle physics together with the
usually omitted gravity part can be presented as a two-dimensional (dilaton
gravity) theory.Comment: 58 pages, 2 eps figure
Bemerkungen zur Arbeit von U. Bischoff und H. Nolte Bupivacain 0,5%, 0,75% zur Periduralanaesthesie bei Extremitäten-Operationen
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