Webrelaunch 2020

AG Stochastische Geometrie am 19.6.2009

Klaus Mecke: Integral Geometry and Density Functional Theory for Liquids

Die Moleküle eines nematischen Flüssigkristalls lassen sich durch ein elektrisches Feld orientieren und so z.B. als elektrische Anzeige (LCD) nutzen. Die Fähigkeit sich spontan auszurichten hängt von der Form der Moleküle ab, d.h. stäbchenförmige Moleküle bilden bei hohen Dichten nematische Phasen, sphärische dagegen nicht. Bereits 1949 zeigte Lars Onsager, dass das Phänomen allein aus entropischen Gründen
auftritt und bereits bei harten Kolloidteilchen mit rein repulsiver Wechselwirkungen zu beobachten ist. 1989 stellte Y. Rosenfeld eine Dichtefunktionaltheorie vor, die die Eigenschaften inhomogener Flüssigkeiten von harten Kugeln quantitativ genau erklären kann, allerdings keine nematische Phasen.
Jetzt ist es basierend auf integralgeometrischen Methoden gelungen, die Form der Moleküle durch Minkowski-Tensoren in einer Dichtefunktionaltheorie zu berücksichtigen und die Ausrichtung der Moleküle in Übereinstimmung mit Simulationen zu beschreiben.

Eva Vedel-Jensen: Lévy-based spatio-temporal modelling - with a view to growth

In this talk, I will give a review of a modelling framework for spatio-temporal processes, based on Lévy theory.
I will exemplify the potential of this approach in stochastic geometry and spatial statistics by studying Lévy-based growth modelling of planar objects. The growth models considered are spatio-temporal stochastic processes on the circle. As a by product, non-separable flexible models for space-time covariance functions on the circle are provided.

Markus Kiderlen: Summary characteristics of point processes based on the medial axis

The spherical contact distribution function H(X,.) of a stationary point process X in n-dimensional space describes the distribution of the distance from the origin to its nearest neighbour in X. Equivalently, H(X,r), r>0, can be described as the specific volume covered by the random closed set X_r, obtained as the union of all balls with radii equal to r and midpoints in X. Generalizing this, K. Mecke and D. Stoyan suggested to replace the specific volume by other specific Minkowski functionals, thus obtaining n+1 morphological functions, which can be used as summary characteristics of X. We extend this idea further by extracting certain shape information from X_r. We consider the medial axis M(X_r) (also called Blum axis or inner skeleton) of X_r consisting of all points in X_r whose nearest neigbour in the boundary of X_r is not unique. The set M(X_r) is a locally finite union of lower-dimensional polytopes and thus characteristics like the mean number of k-faces per unit volume, the mean number of j-faces incident with a typical k-face and similar quantities can be considered. We will discuss the case where X is a Poisson process in detail, as this can serve as a reference model.