614 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
Baire and weakly Namioka spaces
Recall that a Hausdorff space is said to be Namioka if for every compact
(Hausdorff) space and every metric space , every separately continuous
function is continuous on for some
dense subset of . It is well known that in the class of all
metrizable spaces, Namioka and Baire spaces coincide (Saint-Raymond, 1983).
Further it is known that every completely regular Namioka space is Baire and
that every separable Baire space is Namioka (Saint-Raymond, 1983).
In our paper we study spaces , we call them weakly Namioka, for which the
conclusion of the theorem for Namioka spaces holds provided that the assumption
of compactness of is replaced by second countability of . We will prove
that in the class of all completely regular separable spaces and in the class
of all perfectly normal spaces, is Baire if and only if it is weakly
Namioka.Comment: 11 page
Morse homology for the heat flow
We use the heat flow on the loop space of a closed Riemannian manifold to
construct an algebraic chain complex. The chain groups are generated by
perturbed closed geodesics. The boundary operator is defined in the spirit of
Floer theory by counting, modulo time shift, heat flow trajectories that
converge asymptotically to nondegenerate closed geodesics of Morse index
difference one.Comment: 89 pages, 3 figure
SRB Measures for A Class of Partially Hyperbolic Attractors in Hilbert spaces
In this paper, we study the existence of SRB measures and their properties
for infinite dimensional dynamical systems in a Hilbert space. We show several
results including (i) if the system has a partially hyperbolic attractor with
nontrivial finite dimensional unstable directions, then it has at least one SRB
measure; (ii) if the attractor is uniformly hyperbolic and the system is
topological mixing and the splitting is H\"older continuous, then there exists
a unique SRB measure which is mixing; (iii) if the attractor is uniformly
hyperbolic and the system is non-wondering and and the splitting is H\"older
continuous, then there exists at most finitely many SRB measures; (iv) for a
given hyperbolic measure, there exist at most countably many ergodic components
whose basin contains an observable set
Ergodic solenoids and generalized currents
We introduce the concept of solenoid as an abstract laminated space. We do a
thorough study of solenoids, leading to the notion of ergodic and uniquely
ergodic solenoids. We define generalized currents associated with immersions of
oriented solenoids with a transversal measure into smooth manifolds,
generalizing Ruelle-Sullivan currents.Comment: 28 pages, 2 figures. Fully revised presentation of the paper.
Accepted in Revista Matematica Complutense. The final publication is
available at http://www.springerlink.co
The Residual Method for Regularizing Ill-Posed Problems
Although the \emph{residual method}, or \emph{constrained regularization}, is
frequently used in applications, a detailed study of its properties is still
missing. This sharply contrasts the progress of the theory of Tikhonov
regularization, where a series of new results for regularization in Banach
spaces has been published in the recent years. The present paper intends to
bridge the gap between the existing theories as far as possible. We develop a
stability and convergence theory for the residual method in general topological
spaces. In addition, we prove convergence rates in terms of (generalized)
Bregman distances, which can also be applied to non-convex regularization
functionals. We provide three examples that show the applicability of our
theory. The first example is the regularized solution of linear operator
equations on -spaces, where we show that the results of Tikhonov
regularization generalize unchanged to the residual method. As a second
example, we consider the problem of density estimation from a finite number of
sampling points, using the Wasserstein distance as a fidelity term and an
entropy measure as regularization term. It is shown that the densities obtained
in this way depend continuously on the location of the sampled points and that
the underlying density can be recovered as the number of sampling points tends
to infinity. Finally, we apply our theory to compressed sensing. Here, we show
the well-posedness of the method and derive convergence rates both for convex
and non-convex regularization under rather weak conditions.Comment: 29 pages, one figur
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