122,267 research outputs found
The nature of spacetime singularities
Present knowledge about the nature of spacetime singularities in the context
of classical general relativity is surveyed. The status of the BKL picture of
cosmological singularities and its relevance to the cosmic censorship
hypothesis are discussed. It is shown how insights on cosmic censorship also
arise in connection with the idea of weak null singularities inside black
holes. Other topics covered include matter singularities and critical collapse.
Remarks are made on possible future directions in research on spacetime
singularities.Comment: Submitted to 100 Years of Relativity - Space-Time Structure: Einstein
and Beyond, A. Ashtekar (ed.
Kirchhoff equations in generalized Gevrey spaces: local existence, global existence, uniqueness
In this note we present some recent results for Kirchhoff equations in
generalized Gevrey spaces. We show that these spaces are the natural framework
where classical results can be unified and extended. In particular we focus on
existence and uniqueness results for initial data whose regularity depends on
the continuity modulus of the nonlinear term, both in the strictly hyperbolic
case, and in the degenerate hyperbolic case.Comment: 20 pages, 4 tables, conference paper (7th ISAAC congress, London
2009
Aberrations in shift-invariant linear optical imaging systems using partially coherent fields
Here the role and influence of aberrations in optical imaging systems
employing partially coherent complex scalar fields is studied. Imaging systems
require aberrations to yield contrast in the output image. For linear
shift-invariant optical systems, we develop an expression for the output
cross-spectral density under the space-frequency formulation of statistically
stationary partially coherentfields. We also develop expressions for the output
cross{spectral density and associated spectral density for weak-phase,
weak-phase-amplitude, and single-material objects in one transverse spatial
dimension
Limit operators, collective compactness, and the spectral theory of infinite matrices
In the first half of this memoir we explore the interrelationships between the abstract theory of limit operators (see e.g. the recent monographs of Rabinovich, Roch and Silbermann (2004) and Lindner (2006)) and the concepts and results of the generalised collectively compact operator theory introduced by Chandler-Wilde and Zhang (2002). We build up to results obtained by applying this generalised collectively compact operator theory to the set of limit operators of an operator (its operator spectrum). In the second half of this memoir we study bounded linear operators on the generalised sequence space , where and is some complex Banach space. We make what seems to be a more complete study than hitherto of the connections between Fredholmness, invertibility, invertibility at infinity, and invertibility or injectivity of the set of limit operators, with some emphasis on the case when the operator is a locally compact perturbation of the identity. Especially, we obtain stronger results than previously known for the subtle limiting cases of and . Our tools in this study are the results from the first half of the memoir and an exploitation of the partial duality between and and its implications for bounded linear operators which are also continuous with respect to the weaker topology (the strict topology) introduced in the first half of the memoir. Results in this second half of the memoir include a new proof that injectivity of all limit operators (the classic Favard condition) implies invertibility for a general class of almost periodic operators, and characterisations of invertibility at infinity and Fredholmness for operators in the so-called Wiener algebra. In two final chapters our results are illustrated by and applied to concrete examples. Firstly, we study the spectra and essential spectra of discrete Schrödinger operators (both self-adjoint and non-self-adjoint), including operators with almost periodic and random potentials. In the final chapter we apply our results to integral operators on
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