Sections of Point Sets

Fuzzy sets form A meta-system-theoretic point of view

Level Sets of Functions and Symmetry Sets of Surface Sections

We prove that the level sets of a real C function of two variables near a non-degenerate critical point are of class C [s/2] and apply this to the study of planar sections of surfaces close to the singular section by the tangent plane at an elliptic or hyperbolic point, and in particular at an umbilic point. We go on to use the results to study symmetry sets of the planar sections. We also anal...

Complete Poincar E Sections and Tangent Sets

Trying to extend a local deenition of a surface of section and the corresponding Poincar e map to a global one, one can encounter severe diiculties. We show that global transverse sections often do not exist for Hamiltonian systems with two degrees of freedom. As a consequence we present a method to generate so called W-section, which by construction will be intersected by (almost) all orbits. ...

Prize-collecting Point Sets

Given a set of points P in the plane and profits (or prizes) π : P → R≥0 we want to select a maximum profit set X ⊆ P which maximizes P p∈X π(p) − μ(X) for some particular criterion μ(X). In this paper we consider four such criteria, namely the perimeter and the area of the smallest axis-parallel rectangle containing X, and the perimeter and the area of the convex hull conv(X) of X. Our key res...

Connecting colored point sets

We study the following Ramsey-type problem. Let S = B ∪ R be a two-colored set of n points in the plane. We show how to construct, in O(n log n) time, a crossing-free spanning tree T (B) for B, and a crossing-free spanning tree T (R) for R, such that both the number of crossings between T (B) and T (R) and the diameters of T (B) and T (R) are kept small. The algorithm is conceptually simple and...