7,403 research outputs found
Isoperimetry in two-dimensional percolation
We consider the unique infinite connected component of supercritical bond
percolation on the square lattice and study the geometric properties of
isoperimetric sets, i.e., sets with minimal boundary for a given volume. For
almost every realization of the infinite connected component we prove that, as
the volume of the isoperimetric set tends to infinity, its asymptotic shape can
be characterized by an isoperimetric problem in the plane with respect to a
particular norm. As an application we then show that the anchored isoperimetric
profile with respect to a given point as well as the Cheeger constant of the
giant component in finite boxes scale to deterministic quantities. This settles
a conjecture of Itai Benjamini for the plane.Comment: 40 pages, 2 figs; version to appear in Commun. Pure Appl. Mat
Wissenschaft und biologisch-dynamische Forschung
Einblicke in das Empfinden, Pflanzen nach Belieben genetisch manipulieren zu können, sowie die der materialistisch gesinnten Wissenschaft gegenüberstehenden biologisch-dynamischen Herangehensweisen, werden in diesem Essay gegeben. Ergänzend wird die Notwendigkeit aufgezeigt, Methoden zu entwickeln, welche die Objektivität biologisch-dynamischer Forschungsergebnisse belegen und eine Nachvollziehbarkeit zuwege bringen
Automated Mapping of Hydrographic Systems from Satellite Imagery Using Self-Organizing Maps and Principal Curves
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