An adaptive Newton-method based on a dynamical systems approach

The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical system and thereby to cast it in the framework of an adaptive step size control procedure. In so doing, our goal is to reduce the chaotic behavior of the original method without losing its quadratic convergence property close to the roots. The performance of the modified scheme is illustrated with various examples from algebraic and differential equations.

Negotiating conservation: the construction of meaningful spaces in a world heritage debate

According to one’s personal biography, social background and the resultant degree of affectedness, a person has certain ideas about the meaning of, in our example, a World Heritage Site (WHS), what he or she can expect from it and what his or her relation to it can and should be. The handling of potentially different meaningful spaces is decisive, when it comes to the negotiation of pathways towards the sustainable development of a WHS region. Due to the fact that – in a pluralistic world – multiple realities exist, they have to be taken seriously and adequately addressed. In this article we identified the ways the Jungfrau-Aletsch- WHS was constructed by exploring the visual and verbal representations of the WHS during the decision-making process (1998-2001). The results demonstrate that in the visual representations (images), the WHS was to a large extent presented as an unspoiled natural environment similar to a touristy promotion brochure. Such a ‘picture-book’-like portrait has no direct link to the population’s daily needs, their questions and anxieties about the consequences of a WHS label. By contrast, the verbal representations (articles, letters-to-the-editor, comments) were dominated by issues concerning the economic development of the region, fears of disappropriation, and different views on nature. Whereas visual and verbal representations to a large extent differ significantly, their combination might have contributed to the final decision of the majority of people concerned to support the application for inscription of the Jungfrau-Aletsch-Bietschhorn region into the World Heritage list. The prominence of economic arguments and narratives about intergenerational responsibility in the verbal representations and their combination with the aesthetic appeal of the natural environment in the visual representations might have built a common meaningful space for one part of the population.