1,639 research outputs found
A population model based on a Poisson line tessellation
International audienceIn this paper, we introduce a new population model. Taking the geometry of cities into account by adding roads, we build a Cox process driven by a Poisson line tessellation. We perform several shot-noise computations according to various generalizations of our original process. This allows us to derive analytical formulas for the uplink coverage probability in each case
Three-dimensional random Voronoi tessellations: From cubic crystal lattices to Poisson point processes
We perturb the SC, BCC, and FCC crystal structures with a spatial Gaussian noise whose adimensional strength is controlled by the parameter a, and analyze the topological and metrical properties of the resulting Voronoi Tessellations (VT). The topological properties of the VT of the SC and FCC crystals are unstable with respect to the introduction of noise, because the corresponding polyhedra are geometrically degenerate, whereas the tessellation of the BCC crystal is topologically stable even against noise of small but finite intensity. For weak noise, the mean area of the perturbed BCC and FCC crystals VT increases quadratically with a. In the case of perturbed SCC crystals, there is an optimal amount of noise that minimizes the mean area of the cells. Already for a moderate noise (a>0.5), the properties of the three perturbed VT are indistinguishable, and for intense noise (a>2), results converge to the Poisson-VT limit. Notably, 2-parameter gamma distributions are an excellent model for the empirical of of all considered properties. The VT of the perturbed BCC and FCC structures are local maxima for the isoperimetric quotient, which measures the degre of sphericity of the cells, among space filling VT. In the BCC case, this suggests a weaker form of the recentluy disproved Kelvin conjecture. Due to the fluctuations of the shape of the cells, anomalous scalings with exponents >3/2 is observed between the area and the volumes of the cells, and, except for the FCC case, also for a->0. In the Poisson-VT limit, the exponent is about 1.67. As the number of faces is positively correlated with the sphericity of the cells, the anomalous scaling is heavily reduced when we perform powerlaw fits separately on cells with a specific number of faces
Voronoi Tessellations and the Cosmic Web: Spatial Patterns and Clustering across the Universe
The spatial cosmic matter distribution on scales of a few up to more than a
hundred Megaparsec displays a salient and pervasive foamlike pattern. Voronoi
tessellations are a versatile and flexible mathematical model for such weblike
spatial patterns. They would be the natural asymptotic result of an evolution
in which low-density expanding void regions dictate the spatial organization of
the Megaparsec Universe, while matter assembles in high-density filamentary and
wall-like interstices between the voids. We describe the results of ongoing
investigations of a variety of aspects of cosmologically relevant spatial
distributions and statistics within the framework of Voronoi tessellations.
Particularly enticing is the finding of a profound scaling of both clustering
strength and clustering extent for the distribution of tessellation nodes,
suggestive for the clustering properties of galaxy clusters. Cellular patterns
may be the source of an intrinsic ``geometrically biased'' clustering.Comment: 10 pages, 9 figures, accepted for publication as long paper in
proceedings Fourth International Symposium on Voronoi Diagrams in Science and
Engineering (ISVD 2007), ed. C. Gold, IEEE Computer Society, July 2007. For
high-res version see
http://www.astro.rug.nl/~weygaert/tim1publication/vorwey.isvd07.pd
Structure in the 3D Galaxy Distribution: I. Methods and Example Results
Three methods for detecting and characterizing structure in point data, such
as that generated by redshift surveys, are described: classification using
self-organizing maps, segmentation using Bayesian blocks, and density
estimation using adaptive kernels. The first two methods are new, and allow
detection and characterization of structures of arbitrary shape and at a wide
range of spatial scales. These methods should elucidate not only clusters, but
also the more distributed, wide-ranging filaments and sheets, and further allow
the possibility of detecting and characterizing an even broader class of
shapes. The methods are demonstrated and compared in application to three data
sets: a carefully selected volume-limited sample from the Sloan Digital Sky
Survey redshift data, a similarly selected sample from the Millennium
Simulation, and a set of points independently drawn from a uniform probability
distribution -- a so-called Poisson distribution. We demonstrate a few of the
many ways in which these methods elucidate large scale structure in the
distribution of galaxies in the nearby Universe.Comment: Re-posted after referee corrections along with partially re-written
introduction. 80 pages, 31 figures, ApJ in Press. For full sized figures
please download from: http://astrophysics.arc.nasa.gov/~mway/lss1.pd
ZOBOV: a parameter-free void-finding algorithm
ZOBOV (ZOnes Bordering On Voidness) is an algorithm that finds density
depressions in a set of points, without any free parameters, or assumptions
about shape. It uses the Voronoi tessellation to estimate densities, which it
uses to find both voids and subvoids. It also measures probabilities that each
void or subvoid arises from Poisson fluctuations. This paper describes the
ZOBOV algorithm, and the results from its application to the dark-matter
particles in a region of the Millennium Simulation. Additionally, the paper
points out an interesting high-density peak in the probability distribution of
dark-matter particle densities.Comment: 10 pages, 8 figures, MNRAS, accepted. Added explanatory figures, and
better edge-detection methods. ZOBOV code available at
http://www.ifa.hawaii.edu/~neyrinck/vobo
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