1,089 research outputs found
Geometric approach to sampling and communication
Relationships that exist between the classical, Shannon-type, and
geometric-based approaches to sampling are investigated. Some aspects of coding
and communication through a Gaussian channel are considered. In particular, a
constructive method to determine the quantizing dimension in Zador's theorem is
provided. A geometric version of Shannon's Second Theorem is introduced.
Applications to Pulse Code Modulation and Vector Quantization of Images are
addressed.Comment: 19 pages, submitted for publicatio
Curvature based triangulation of metric measure spaces
We prove that a Ricci curvature based method of triangulation of compact
Riemannian manifolds, due to Grove and Petersen, extends to the context of
weighted Riemannian manifolds and more general metric measure spaces. In both
cases the role of the lower bound on Ricci curvature is replaced by the
curvature-dimension condition . We show also that for weighted
Riemannian manifolds the triangulation can be improved to become a thick one
and that, in consequence, such manifolds admit weight-sensitive
quasimeromorphic mappings. An application of this last result to information
manifolds is considered.
Further more, we extend to weak spaces the results of Kanai
regarding the discretization of manifolds, and show that the volume growth of
such a space is the same as that of any of its discretizations.Comment: 24 pages, submitted for publicatio
Courbure discrète : théorie et applications
International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor
Unveiling the Dynamics of the Universe
We explore the dynamics and evolution of the Universe at early and late
times, focusing on both dark energy and extended gravity models and their
astrophysical and cosmological consequences. Modified theories of gravity not
only provide an alternative explanation for the recent expansion history of the
universe, but they also offer a paradigm fundamentally distinct from the
simplest dark energy models of cosmic acceleration. In this review, we perform
a detailed theoretical and phenomenological analysis of different modified
gravity models and investigate their consistency. We also consider the
cosmological implications of well motivated physical models of the early
universe with a particular emphasis on inflation and topological defects.
Astrophysical and cosmological tests over a wide range of scales, from the
solar system to the observable horizon, severely restrict the allowed models of
the Universe. Here, we review several observational probes -- including
gravitational lensing, galaxy clusters, cosmic microwave background temperature
and polarization, supernova and baryon acoustic oscillations measurements --
and their relevance in constraining our cosmological description of the
Universe.Comment: 94 pages, 14 figures. Review paper accepted for publication in a
Special Issue of Symmetry. "Symmetry: Feature Papers 2016". V2: Matches
published version, now 79 pages (new format
- …