1,683 research outputs found
Localic completion of uniform spaces
We extend the notion of localic completion of generalised metric spaces by
Steven Vickers to the setting of generalised uniform spaces. A generalised
uniform space (gus) is a set X equipped with a family of generalised metrics on
X, where a generalised metric on X is a map from the product of X to the upper
reals satisfying zero self-distance law and triangle inequality.
For a symmetric generalised uniform space, the localic completion lifts its
generalised uniform structure to a point-free generalised uniform structure.
This point-free structure induces a complete generalised uniform structure on
the set of formal points of the localic completion that gives the standard
completion of the original gus with Cauchy filters.
We extend the localic completion to a full and faithful functor from the
category of locally compact uniform spaces into that of overt locally compact
completely regular formal topologies. Moreover, we give an elementary
characterisation of the cover of the localic completion of a locally compact
uniform space that simplifies the existing characterisation for metric spaces.
These results generalise the corresponding results for metric spaces by Erik
Palmgren.
Furthermore, we show that the localic completion of a symmetric gus is
equivalent to the point-free completion of the uniform formal topology
associated with the gus.
We work in Aczel's constructive set theory CZF with the Regular Extension
Axiom. Some of our results also require Countable Choice.Comment: 39 page
From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity
This article reviews some aspects in the current relationship between
mathematical and numerical General Relativity. Focus is placed on the
description of isolated systems, with a particular emphasis on recent
developments in the study of black holes. Ideas concerning asymptotic flatness,
the initial value problem, the constraint equations, evolution formalisms,
geometric inequalities and quasi-local black hole horizons are discussed on the
light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity.
Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24
November, 2006), part of the "General Relativity Trimester" at the Institut
Henri Poincare (Fall 2006). Comments and references added. Typos corrected.
Submitted to Classical and Quantum Gravit
Multi-moment maps
We introduce a notion of moment map adapted to actions of Lie groups that
preserve a closed three-form. We show existence of our multi-moment maps in
many circumstances, including mild topological assumptions on the underlying
manifold. Such maps are also shown to exist for all groups whose second and
third Lie algebra Betti numbers are zero. We show that these form a special
class of solvable Lie groups and provide a structural characterisation. We
provide many examples of multi-moment maps for different geometries and use
them to describe manifolds with holonomy contained in G_2 preserved by a
two-torus symmetry in terms of tri-symplectic geometry of four-manifolds.Comment: 27 page
Several Remarks on Norm Attainment in Tensor Product Spaces
Funding for open access publishing: Universidad de
Granada/CBUAThe aim of this note is to obtain results about when the norm of a projective tensor product is strongly subdifferentiable. We prove that if X (circle times) over cap Y-pi is strongly subdifferentiable and either X or Y has the metric approximation property then every bounded operator from X to Y-* is compact. We also prove that (l(p)(I)(circle times) over cap (pi)l(q)(J))(*) has the w(*)-Kadec-Klee property for every non-empty sets I,J and every 2in particular that the norm of the space l(p)(I)(circle times) over cap (pi)l(q)(J) is strongly subdifferentiable. This extends several results of Dantas, Kim, Lee and Mazzitelli. We also find examples of spaces X and Y for which the set of norm-attaining tensors in X (circle times) over cap Y-pi is dense but whose complement is dense too.Universidad de Granada/CBU
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