Fakultät für Mathematik
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Fakultät für Mathematik
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Institute
Institut für Stochastik
AG Stochastische Geometrie

AG Stochastische Geometrie (Wintersemester 2012/13)

Forschungsseminar der Arbeitsgruppe Räumliche Stochastik und Stochastische Geometrie

Termine
Seminar: Freitag 9:45-11:15 Seminarraum K2
Lehrende
Seminarleitung Prof. Dr. Günter Last
Sprechstunde: nach Vereinbarung.
Zimmer 2.001, Sekretariat 2.056 Kollegiengebäude Mathematik (20.30)
Email: guenter.last@kit.edu
Seminarleitung Prof. Dr. Daniel Hug
Sprechstunde: Nach Vereinbarung.
Zimmer 2.051 Kollegiengebäude Mathematik (20.30)
Email: daniel.hug@kit.edu

Vorträge

Wenn nicht explizit anders angegeben, finden alle Vorträge im Raum K2 (Kronenstr. 32, Eingang ca. 10m links neben der Kaffeebar Schiller) statt.

Freitag, 19.10.12

  • 9.45 Uhr: Vorbesprechung

Dienstag, 30.10.12 (statt Fr. 26.10.)

  • 15.45 Uhr, Raum 1C-04 Daniel Hug: Grenzwertsätze für Boolesche Modelle

Freitag, 9.11.12

  • 9.45 Uhr Günter Last: Log-Sobolev- und Konzentrationsungleichungen für Poissonsche Funktionale

Freitag, 16.11.12, Sitzungszimmer (Raum 5C-01, Allianzgebäude)

  • 9.45 Uhr Franz Schuster (TU Wien): Eine Charakterisierung der Blaschke Addition Abstract

Freitag, 23.11.12, Sitzungszimmer (Raum 5C-01, Allianzgebäude)

  • 9.30 Uhr Zbynek Pawlas (U Prag): Nonparametric estimation of the radius distribution in the Boolean model of spheres
{\bf Abstract:} \text{ We consider a stationary } d \text{-dimensional Boolean model with} spherical grains. The main aim of the statistical inference is to retrieve information on intensity and typical grain distribution based on a realization of the model observed in a bounded window. We propose a family of nonparametric estimators and study their asymptotic properties. The central limit theorem is shown under increasing domain asymptotics. The proof is based on the truncation argument and approximation by $m$-dependent random fields.

  • 11.00 Uhr Joachim Ohser (U Darmstadt): Die bildanalytische Bestimmung des Integrals der mittleren Krümmung einer Menge auf der Grundlage einer diskreten Version der Crofton-Formel
{\bf Abstract:} \text{ Das Sampling einer Menge aus dem Konvexring auf einem } homogenen Punktgitter kann als Vordergrund eines Bin\"arbildes aufgefasst werden. Dieses Sampling ist mit einem Informationsverlust verbunden. Die Sch\"atzung des Integrals der mittleren Kr\"ummung der Menge aus dem Sampling basiert auf einer diskreten Version der entsprechenden Croftonschen Schnittformel, wobei die Diskretisierung durch das Gitter impliziert wird. Im Vortrag werden zwei Ans\"atze f\"ur die Diskretisierung der Crofton-Formel und die daraus resultierenden Sch\"atzer f\"ur das Integral der mittleren Kr\"ummung miteinander verglichen. Als Anwendung wird die Bestimmung der spezifischen Faserl\"ange von 3-dimensionalen zuf\"alligen Fasersystemen auf der Grundlage tomographischer Aufnahmen betrachtet, wobei die spezifische Faserl\"ange n\"aherungsweise proportional zum Integral der mittleren Krümmung pro Volumeneinheit ist.

Freitag, 7.12.12

  • 9.30 Uhr Sebastian Ziesche: Perkolation von Modellen mit positiver Korrelation
  • 11.00 Uhr Daniel Hug: Orlicz-Addition und Ungleichungen

Freitag, 21.12.12

  • 9.30 Uhr Sebastian Ziesche: Perkolation von Modellen mit positiver Korrelation II
  • 11.00 Uhr Daniel Hug: Orlicz-Addition und Ungleichungen II

Freitag, 11.1.13

  • Steffen Winter: Selbstähnliche Mosaike und Minkowski-Inhalt

Freitag, 18.1.13, 9.45 Uhr, Sitzungszimmer (Raum 5C-01, Allianzgebäude)

  • Adil Mughal (Aberystwyth University, UK): Phyllotactic Description of Hard Sphere Packings in Cylindrical Channels
{\bf Abstract:} \text{ We study the optimal packing of hard spheres in an} infinitely long cylinder. Our simulations have yielded dozens of periodic, mechanically stable, structures as the ratio of the cylinder (D) to sphere (d) diameter is varied [Mughal, 2011]. Up to D/d=2.715 the densest structures are composed entirely of spheres which are in contact with the cylinder. The density reaches a maximum at discrete values of D/d when a maximum number of contacts are established. These maximal contact packings are of the classic "phyllotactic" type, familiar in biology. However, between these points we observe another type of packing, termed line-slip. An analytic understanding of these rigid structures follows by recourse to a yet simpler problem: the packing of disks on a cylinder. We show that maximal contact packings correspond to the perfect wrapping of a honeycomb arrangement of disks around a cylindrical tube. While line-slip packings are inhomogeneous deformations of the honeycomb lattice modified to wrap around the cylinder (and have fewer contacts per sphere). Beyond D/d=2.715 the structures are more complex, since they incorporate internal spheres [Mughal, 2012], but an analysis in terms of contacts or constraints is still illuminating. We review some relevant experiments with hard spheres and small bubbles. The talk is based on joint work with Ho-Kei Chan, Aaron Meagher, Denis Weaire, and Stefan Hutzler. {\bf References}\ Mughal A., Chan H. K., and Weaire D., Physical Review Letters, 106, 115704, 2011.\ Mughal et al., Physical Review E, 85, 051305, 2012.

Freitag, 25.1.13, 9.45 Uhr, Sitzungszimmer (Raum 5C-01, Allianzgebäude)

  • Michael Klatt (U Erlangen): Morphometric analysis in gamma ray astronomy
{\bf Abstract:} \text{ H.E.S.S., an array of four imaging atmospheric Cherenkov } telescopes for gamma-rays above 100 GeV, observes an increasing number of large extended sources. To account for these additional structures compared to common point source analysis, a new analysis technique based on the morphology of the sky map is developed. The here presented morphometric data analysis is a new method to detect sources of gamma-ray emission, which is especially designed for the detection of faint extended sources. Minkowski functionals quantify the structure of the count rate map, which is then compared to the expected structure of a pure Poisson background with gamma-ray sources leading to significant deviations from background structure. The standard likelihood ratio method of Li and Ma is exclusively based on the number of excess counts and discards all further structure information of large extended sources. The morphometric data analysis incorporates this additional geometric information in an unbiased analysis, i.e. without the need of any prior knowledge about the source.

Freitag, 1.2.13, 9.45 Uhr, Sitzungszimmer (Raum 5C-01, Allianzgebäude)

  • Beatrice-Helen Vritsiou: Isotropic convex bodies and reductions of the slicing problem
{\bf Abstract:} \text{A convex body in } {\mathbb R}^n, \text{namely a compact convex set with non-} empty interior, is called isotropic if it has volume 1, its barycentre is at the origin, and if its inertia matrix is a multiple of the identity, i.e. there exists a constant $L_K$ such that \begin{equation*} \int_K \langle x, \theta\rangle^2 dx = L_K^2 \end{equation*} for every unit vector $\theta\in {\mathbb R}^n$. A well-known conjecture for this class of bodies is the hyperlane conjecture or slicing problem, that asks whether the volume of every section of an isotropic body with a hyperplane that passes through the origin can be bounded from below by a constant independent of the body and its dimension $n$. This question is equivalent to asking whether every isotropic constant $L_K$ can be bounded from above by an absolute constant. The best currently known bound is $L_K = O(\sqrt[4]{n})$ for every isotropic convex body $K$ in ${\mathbb R}^n$. In this talk, we will discuss a few recent methods that could allow one to improve the upper bounds for the isotropic constants.

Freitag, 8.2.13, 9.45 Uhr, K2

  • 9.45 Uhr Günter Last: Kovarianzstruktur innerer Volumina des isotropen Booleschen Modells