61,866 research outputs found
The geodesic problem in quasimetric spaces
In this article, we study the geodesic problem in a generalized metric space,
in which the distance function satisfies a relaxed triangle inequality
for some constant , rather
than the usual triangle inequality. Such a space is called a quasimetric space.
We show that many well-known results in metric spaces (e.g. Ascoli-Arzel\`{a}
theorem) still hold in quasimetric spaces. Moreover, we explore conditions
under which a quasimetric will induce an intrinsic metric. As an example, we
introduce a family of quasimetrics on the space of atomic probability measures.
The associated intrinsic metrics induced by these quasimetrics coincide with
the metric studied early in the study of branching structures
arisen in ramified optimal transportation. An optimal transport path between
two atomic probability measures typically has a "tree shaped" branching
structure. Here, we show that these optimal transport paths turn out to be
geodesics in these intrinsic metric spaces.Comment: 21 pages, 5 figures, published versio
Aspects of noncommutative Lorentzian geometry for globally hyperbolic spacetimes
Connes' functional formula of the Riemannian distance is generalized to the
Lorentzian case using the so-called Lorentzian distance, the d'Alembert
operator and the causal functions of a globally hyperbolic spacetime. As a step
of the presented machinery, a proof of the almost-everywhere smoothness of the
Lorentzian distance considered as a function of one of the two arguments is
given. Afterwards, using a -algebra approach, the spacetime causal
structure and the Lorentzian distance are generalized into noncommutative
structures giving rise to a Lorentzian version of part of Connes'
noncommutative geometry. The generalized noncommutative spacetime consists of a
direct set of Hilbert spaces and a related class of -algebras of
operators. In each algebra a convex cone made of self-adjoint elements is
selected which generalizes the class of causal functions. The generalized
events, called {\em loci}, are realized as the elements of the inductive limit
of the spaces of the algebraic states on the -algebras. A partial-ordering
relation between pairs of loci generalizes the causal order relation in
spacetime. A generalized Lorentz distance of loci is defined by means of a
class of densely-defined operators which play the r\^ole of a Lorentzian
metric. Specializing back the formalism to the usual globally hyperbolic
spacetime, it is found that compactly-supported probability measures give rise
to a non-pointwise extension of the concept of events.Comment: 43 pages, structure of the paper changed and presentation strongly
improved, references added, minor typos corrected, title changed, accepted
for publication in Reviews in Mathematical Physic
Robustness in Metric Spaces over Continuous Quantales and the Hausdorff-Smyth Monad
Generalized metric spaces are obtained by weakening the requirements (e.g.,
symmetry) on the distance function and by allowing it to take values in
structures (e.g., quantales) that are more general than the set of non-negative
real numbers. Quantale-valued metric spaces have gained prominence due to their
use in quantitative reasoning on programs/systems, and for defining various
notions of behavioral metrics.
We investigate imprecision and robustness in the framework of quantale-valued
metric spaces, when the quantale is continuous. In particular, we study the
relation between the robust topology, which captures robustness of analyses,
and the Hausdorff-Smyth hemi-metric. To this end, we define a preorder-enriched
monad , called the Hausdorff-Smyth monad, and when is a
continuous quantale and is a -metric space, we relate the topology
induced by the metric on with the robust topology on the
powerset defined in terms of the metric on .Comment: 19 pages, 1 figur
Tameness in generalized metric structures
We broaden the framework of metric abstract elementary classes (mAECs) in
several essential ways, chiefly by allowing the metric to take values in a
well-behaved quantale. As a proof of concept we show that the result of Boney
and Zambrano on (metric) tameness under a large cardinal assumption holds in
this more general context. We briefly consider a further generalization to
partial metric spaces, and hint at connections to classes of fuzzy structures,
and structures on sheaves
Generalized Sums over Histories for Quantum Gravity I. Smooth Conifolds
This paper proposes to generalize the histories included in Euclidean
functional integrals from manifolds to a more general set of compact
topological spaces. This new set of spaces, called conifolds, includes
nonmanifold stationary points that arise naturally in a semiclasssical
evaluation of such integrals; additionally, it can be proven that sequences of
approximately Einstein manifolds and sequences of approximately Einstein
conifolds both converge to Einstein conifolds. Consequently, generalized
Euclidean functional integrals based on these conifold histories yield
semiclassical amplitudes for sequences of both manifold and conifold histories
that approach a stationary point of the Einstein action. Therefore sums over
conifold histories provide a useful and self-consistent starting point for
further study of topological effects in quantum gravity. Postscript figures
available via anonymous ftp at black-hole.physics.ubc.ca (137.82.43.40) in file
gen1.ps.Comment: 81pp., plain TeX, To appear in Nucl. Phys.
On Fenchel-Nielsen coordinates on Teichm\"uller spaces of surfaces of infinite type
We introduce Fenchel-Nielsen coordinates on Teicm\"uller spaces of surfaces
of infinite type. The definition is relative to a given pair of pants
decomposition of the surface. We start by establishing conditions under which
any pair of pants decomposition on a hyperbolic surface of infinite type can be
turned into a geometric decomposition, that is, a decomposition into hyperbolic
pairs of pants. This is expressed in terms of a condition we introduce and
which we call Nielsen convexity. This condition is related to Nielsen cores of
Fuchsian groups. We use this to define the Fenchel-Nielsen Teichm\"uller space
associated to a geometric pair of pants decomposition. We study a metric on
such a Teichm\"uller space, and we compare it to the quasiconformal
Teichm\"uller space, equipped with the Teichm\"uller metric. We study
conditions under which there is an equality between these Teichm\"uller spaces
and we study topological and metric properties of the identity map when this
map exists
Sharp measure contraction property for generalized H-type Carnot groups
We prove that H-type Carnot groups of rank and dimension satisfy the
if and only if and . The latter
integer coincides with the geodesic dimension of the Carnot group. The same
result holds true for the larger class of generalized H-type Carnot groups
introduced in this paper, and for which we compute explicitly the optimal
synthesis. This constitutes the largest class of Carnot groups for which the
curvature exponent coincides with the geodesic dimension. We stress that
generalized H-type Carnot groups have step 2, include all corank 1 groups and,
in general, admit abnormal minimizing curves.
As a corollary, we prove the absolute continuity of the Wasserstein geodesics
for the quadratic cost on all generalized H-type Carnot groups.Comment: 18 pages. This article extends the results of arXiv:1510.05960. v2:
revised and improved version. v3: final version, to appear in Commun.
Contemp. Mat
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