Hartmut Jürgens

Fractals for the Classroom

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Bausteine des Chaos Fraktale

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Topological Perturbations in the Numerical Study of Nonlinear Eigenvalue and Bifurcation Problems

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Strange Attractors: The Locus of Chaos

Publisher Summary This chapter discusses topological perturbations in the numerical study of nonlinear eigenvalue and bifurcation problems. The chapter reviews a new device in the context of simplicial and continuation methods that was introduced in [P-P] and to relate this device with a suitable interpretation of an idea due to M.M. Jeppson [J,A-J]. As an application their usefulness is demonstrated in some selected numerical problems—an elliptic boundary value problem with infinitely many solutions, a bifurcation problem with non-differentiable nonlinearity, and a periodicity problem given by a differential delay equation that is conjectured to have chaotic behavior. For an efficient implementation of the topological perturbation techniques it has been most important to introduce and use a new concept of triangulation. This concept, which in a sense can be understood as a “virtual” triangulation, introduces a large amount of flexibility into simplicial path following algorithms, for example, the mesh size can be modified and still the triangulation process is as simple as Kuhns triangulation.

Length, Area and Dimension: Measuring Complexity and Scaling Properties

Having discussed the phenomena of chaos and the routes leading to it in ‘simple’ one-dimensional settings, we continue with the exposition of chaos in dynamical systems of two or more dimensions. This is the relevant case for models in the natural sciences since very rarely can processes be described by only one single state variable. One of the main players in this context is the notion of strange attractors.

Limits and Self-Similarity

Geometry has always had two sides, and both together have played very important roles. There has been the analysis of patterns and forms on the one hand; and on the other, the measurement of patterns and forms. The incommensurability of the diagonal of a square was initially a problem of measuring length but soon moved to the very theoretical level of introducing irrational numbers. Attempts to compute the length of the circumference of the circle led to the discovery of the mysterious number π. Measuring the area enclosed between curves has, to a great extent, inspired the development of calculus.

Encoding Images by Simple Transformations

Dyson is referring to mathematicians, like G. Cantor, D. Hilbert, and W. Sierpinski, who have been justly credited with having helped to lead mathematics out of its crisis at the turn of the century by building marvelous abstract foundations on which modern mathematics can now flourish safely. Without question, mathematics has changed during this century. What we see is an ever increasing dominance of the algebraic approach over the geometric. In their striving for absolute truth, mathematicians have developed new standards for determining the validity of mathematical arguments. In the process, many of the previously accepted methods have been abandoned as inappropriate. Geometric or visual arguments were increasingly forced out. While Newton’s Principia Mathematica, laying the fundamentals of modern mathematics, still made use of the strength of visual arguments, the new objectivity seems to require a dismissal of this approach. From this point of view, it is ironic that some of the constructions which Cantor, Hilbert, Sierpinski and others created to perfect their extremely abstract foundations simultaneously hold the clues to understanding the patterns of nature in a visual sense. The Cantor set, Hilbert curve, and Sierpinski gasket all give testimony to the delicacy and problems of modern set theory and at the same time, as Mandelbrot has taught us, are perfect models for the complexity of nature.

Optimierte Oberflächenabtastung mit orientierten Kubusketten

So far, we have discussed two extreme ends of fractal geometry. We have explored fractal monsters, such as the Cantor set, the Koch curve, and the Sierpinski gasket; and we have argued that there are many fractals in natural structures and patterns, such as coastlines, blood vessel systems, and cauliflowers. We have discussed features, such as self-similarity, scaling properties, and fractal dimensions shared by those natural structures and the monsters; but we have not yet seen that they are close relatives in the sense that maybe a cauliflower is just a ‘mutant’ of a Sierpinski gasket, and a fern is just a Koch curve ‘let loose’. Or phrased as a question, is there a framework in which a natural structure, such as a cauliflower, and an artificial structure, such as a Sierpinski gasket, are just examples of one unifying approach; and if so, what is it? Believe it or not, there is such a theory, and this chapter is devoted to it. It goes back to Mandelbrot’s book, The Fractal Geometry of Nature, and a beautiful paper by the Australian mathematician Hutchinson.2 Barnsley and Berger have extended these ideas and advocated the point of view that they are very promising for the encoding of images.3 In fact, this will be the focus of the appendix on image compression.

ILabMed-Workstation — Eine Entwicklungsumgebung für radiologische Anwendungen

Wir stellen einen Algorithmus zur Abtastung beliebiger Oberflachen im dreidimensionalen Raum vor. Zu einem vorgegebenen Gitter von Datenpunkten (explizit gegeben aus einem Experiment oder implizit durch eine Funktion definiert), das beispielsweise Qualitaten wie Temperatur-, Druck-oder Dichteverteilung oder ein Potential reprasentiert, versucht der Algorithmus, Kuben zu bestimmen, die eine Flache konstanter Werte schneiden. Haben wir einen solchen Kubus aus einer zusammenhangenden Menge gefunden, so konnen wir zeigen, das der Algorithmus alle Elemente dieser Menge genau einmal generiert und uberhaupt nur solche bestimmt. Zur Demonstration wenden wir den Algorithmus auf dreidimensionale fraktale Mengen an.

Introduction: Causality Principle, Deterministic Laws and Chaos

Die ILabMed-Workstation ist eine Entwicklungsumgebung fur die medizinische Bildverarbeitung. Basierend auf einer grosen Anzahl an Bildverarbeitungsalgorithmen konnen verschiedenste radiologische Probleme gelost werden. Das System kann in einfacher Weise um neue Algorithmen erweitert und die individuellen Problemlosungen konnen mit einer Bedienoberflache versehen werden. Dies wird in der neu entwickelt en Programmiersprache APrIL durchgefuhrt, die durch eine interpretierte Ausfuhrung kurze Entwicklungszyklen ermoglicht. Die ILabMed-Workstation wird am Centrum fur Medizinische Diagnosesysteme und Visualisierung zur Entwicklung von radiologischen Anwendungsprojekten u.a. im Bereich der praoperativen Planung der Leberchirurgie eingesetzt.