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Abstract. The first chapter deals with singularities occurring in quadratic planar vector fields. We make distinction between singularities which as a general system are of finite codimension and singularities which are of infinite codimension in the sense that they are nonisolated, or Hamiltonian, or integrable, or that they have an axis of symmetry after a linear coordinate change or that they can be approximated by centers. In the second chapter we provide quadratic models for all the known versal fc-parameter unfoldings with k = 1, 2, 3 , except for the nilpotent focus which cannot occur as a quadratic system. We finally show that a certain type of elliptic points of codimension 4 does not have a quadratic versal unfolding.

In this paper we show how both geometry-driven diffusion and optimization of the Mumford-Shah functional can be used to develop a type of curve-evolution that is able to preserve salient features of closed curves (such as corners and straight line segments), while simultaneously suppressing noise and irrelevant details. The idea is to characterize the curve by means of its angle-function (i.e. the angle between the tangent and a fixed axis) and to apply the appropriate dynamics to this one-dimensional representation. We show how constrained evolution equations can be used to keep the corresponding curve closed at all times.

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