8,331 research outputs found
Dynamical Boson Stars
The idea of stable, localized bundles of energy has strong appeal as a model
for particles. In the 1950s John Wheeler envisioned such bundles as smooth
configurations of electromagnetic energy that he called {\em geons}, but none
were found. Instead, particle-like solutions were found in the late 1960s with
the addition of a scalar field, and these were given the name {\em boson
stars}. Since then, boson stars find use in a wide variety of models as sources
of dark matter, as black hole mimickers, in simple models of binary systems,
and as a tool in finding black holes in higher dimensions with only a single
killing vector. We discuss important varieties of boson stars, their dynamic
properties, and some of their uses, concentrating on recent efforts.Comment: 79 pages, 25 figures, invited review for Living Reviews in
Relativity; major revision in 201
Bregman Voronoi Diagrams: Properties, Algorithms and Applications
The Voronoi diagram of a finite set of objects is a fundamental geometric
structure that subdivides the embedding space into regions, each region
consisting of the points that are closer to a given object than to the others.
We may define many variants of Voronoi diagrams depending on the class of
objects, the distance functions and the embedding space. In this paper, we
investigate a framework for defining and building Voronoi diagrams for a broad
class of distance functions called Bregman divergences. Bregman divergences
include not only the traditional (squared) Euclidean distance but also various
divergence measures based on entropic functions. Accordingly, Bregman Voronoi
diagrams allow to define information-theoretic Voronoi diagrams in statistical
parametric spaces based on the relative entropy of distributions. We define
several types of Bregman diagrams, establish correspondences between those
diagrams (using the Legendre transformation), and show how to compute them
efficiently. We also introduce extensions of these diagrams, e.g. k-order and
k-bag Bregman Voronoi diagrams, and introduce Bregman triangulations of a set
of points and their connexion with Bregman Voronoi diagrams. We show that these
triangulations capture many of the properties of the celebrated Delaunay
triangulation. Finally, we give some applications of Bregman Voronoi diagrams
which are of interest in the context of computational geometry and machine
learning.Comment: Extend the proceedings abstract of SODA 2007 (46 pages, 15 figures
50 years of first passage percolation
We celebrate the 50th anniversary of one the most classical models in
probability theory. In this survey, we describe the main results of first
passage percolation, paying special attention to the recent burst of advances
of the past 5 years. The purpose of these notes is twofold. In the first
chapters, we give self-contained proofs of seminal results obtained in the '80s
and '90s on limit shapes and geodesics, while covering the state of the art of
these questions. Second, aside from these classical results, we discuss recent
perspectives and directions including (1) the connection between Busemann
functions and geodesics, (2) the proof of sublinear variance under 2+log
moments of passage times and (3) the role of growth and competition models. We
also provide a collection of (old and new) open questions, hoping to solve them
before the 100th birthday.Comment: 160 pages, 17 figures. This version has updated chapters 3-5, with
expanded and additional material. Small typos corrected throughou
- …