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Scaling limits of anisotropic Hastings–Levitov clusters

Fredrik Johansson Viklund, Alan Sola, Amanda Turner (2012)

Annales de l'I.H.P. Probabilités et statistiques

We consider a variation of the standard Hastings–Levitov model HL(0), in which growth is anisotropic. Two natural scaling limits are established and we give precise descriptions of the effects of the anisotropy. We show that the limit shapes can be realised as Loewner hulls and that the evolution of harmonic measure on the cluster boundary can be described by the solution to a deterministic ordinary differential equation related to the Loewner equation. We also characterise the stochastic fluctuations...

Shape factor extremes for prolate spheroids

Daniel Hlubinka (2006)

Kybernetika

Microscopic prolate spheroids in a given volume of an opaque material are considered. The extremes of the shape factor of the spheroids are studied. The profiles of the spheroids are observed on a random planar section and based on these observations we want to estimate the distribution of the extremal shape factor of the spheroids. We show that under a tail uniformity condition the Maximum domain of attraction is stable. We discuss the normalising constants (n.c.) for the extremes of the spheroid...

Simplices rarely contain their circumcenter in high dimensions

Jon Eivind Vatne (2017)

Applications of Mathematics

Acute triangles are defined by having all angles less than π / 2 , and are characterized as the triangles containing their circumcenter in the interior. For simplices of dimension n ≥ 3 , acuteness is defined by demanding that all dihedral angles between ( n - 1 ) -dimensional faces are smaller than π / 2 . However, there are, in a practical sense, too few acute simplices in general. This is unfortunate, since the acuteness property provides good qualitative features for finite element methods. The property of acuteness...

Size of the giant component in a random geometric graph

Ghurumuruhan Ganesan (2013)

Annales de l'I.H.P. Probabilités et statistiques

In this paper, we study the size of the giant component C G in the random geometric graph G = G ( n , r n , f ) of n nodes independently distributed each according to a certain density f ( · ) in [ 0 , 1 ] 2 satisfying inf x ∈ [ 0 , 1 ] 2 f ( x ) g t ; 0 . If c 1 n ≤ r n 2 ≤ c 2 log n n for some positive constants c 1 , c 2 and n r n 2 ⟶ ∞ as n → ∞ , we show that the giant component of G contains at least n - o ( n ) nodes with probability at least 1 - e - β n r n 2 for all n and for some positive constant β ....

Spatio-temporal modelling of a Cox point process sampled by a curve, filtering and Inference

Blažena Frcalová, Viktor Beneš (2009)

Kybernetika

The paper deals with Cox point processes in time and space with Lévy based driving intensity. Using the generating functional, formulas for theoretical characteristics are available. Because of potential applications in biology a Cox process sampled by a curve is discussed in detail. The filtering of the driving intensity based on observed point process events is developed in space and time for a parametric model with a background driving compound Poisson field delimited by special test sets. A...

Stationary map coloring

Omer Angel, Itai Benjamini, Ori Gurel-Gurevich, Tom Meyerovitch, Ron Peled (2012)

Annales de l'I.H.P. Probabilités et statistiques

We consider a planar Poisson process and its associated Voronoi map. We show that there is a proper coloring with 6 colors of the map which is a deterministic isometry-equivariant function of the Poisson process. As part of the proof we show that the 6-core of the corresponding Delaunay triangulation is empty. Generalizations, extensions and some open questions are discussed.

Stereology of dihedral angles

Vratislav Horálek (2000)

Applications of Mathematics

The paper presents a short survey of stereological problems concerning dihedral angles, their solutions and applications, and introduces a graph for determining the distribution functions of planar angles under the hypothesis that dihedral angles in ℝ 3 are of the same size and create a random field.

Stereology of grain boundary precipitates

Vratislav Horálek (1989)

Aplikace matematiky

Precipitates modelled by rotary symmetrical lens-shaped discs are situated on matrix grain boundaries and the homogeneous specimen is intersected by a plate section. The stereological model presented enables one to express all basic parameters of spatial structure and moments of the corresponding probability distributions of quantitative characteristics of precipitates in terms of planar structure parameters the values of which can be estimated from measurements carried out in the plane section....

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