Computation of maximal local (un)stable manifold patches by the parameterization method

In this work we develop some automatic procedures for computing high order polynomial expansions of local (un)stable manifolds for equilibria of differential equations. Our method incorporates validated truncation error bounds, and maximizes the size of the image of the polynomial approximation relative to some specified constraints. More precisely we use that the manifold computations depend heavily on the scalings of the eigenvectors: indeed we study the precise effects of these scalings on the estimates which determine the validated error bounds. This relationship between the eigenvector scalings and the error estimates plays a central role in our automatic procedures. In order to illustrate the utility of these methods we present several applications, including visualization of invariant manifolds in the Lorenz and FitzHugh-Nagumo systems and an automatic continuation scheme for (un)stable manifolds in a suspension bridge problem. In the present work we treat explicitly the case where the eigenvalues satisfy a certain non-resonance condition.Comment: Revised version, typos corrected, references adde

COMPUTER TOOLS FOR SOLVING MATHEMATICAL PROBLEMS: A REVIEW

The rapid development of digital computer hardware and software has had a dramatic influence on mathematics, and contrary. The advanced hardware and modern sophistical software such as computer visualization, symbolic computation, computerassisted proofs, multi-precision arithmetic and powerful libraries, have provided resolving many open problems, a huge very difficult mathematical problems, and discovering new patterns and relationships, far beyond a human capability. In the first part of the paper we give a short review of some typical mathematical problems solved by computer tools. In the second part we present some new original contributions, such as intriguing consequence of the presence of roundoff errors, distribution of zeros of random polynomials, dynamic study of zero-finding methods, a new three-point family of methods for solving nonlinear equations and two algorithms for the inclusion of a simple complex zero of a polynomial

Calibrating evanescent-wave penetration depths for biological TIRF microscopy

Roughly half of a cells proteins are located at or near the plasma membrane. In this restricted space the cell senses its environment, signals to its neighbors and ex-changes cargo through exo- and endocytotic mechanisms. Ligands bind to receptors, ions flow across channel pores, and transmitters and metabolites are transported against con-centration gradients. Receptors, ion channels, pumps and transporters are the molecular substrates of these biological processes and they constitute important targets for drug discovery. Total internal reflection fluorescence microscopy suppresses background from cell deeper layers and provides contrast for selectively imaging dynamic processes near the basal membrane of live-cells. The optical sectioning of total internal reflection fluorescence is based on the excitation confinement of the evanescent wave generated at the glass-cell interface. How deep the excitation light actually penetrates the sample is difficult to know, making the quantitative interpretation of total internal reflection fluorescence data problematic. Nevertheless, many applications like super-resolution microscopy, colocalization, fluorescence recovery after photobleaching, near-membrane fluorescence recovery after photobleaching, uncaging or photo-activation-switching, as well as single-particle tracking require the quantitative interpretation of evanescent-wave excited images. Here, we review existing techniques for characterizing evanescent fields and we provide a roadmap for comparing total internal reflection fluorescence data across images, experiments, and laboratories.Comment: 18 text pages, 7 figures and one supplemental figur

Efficient Object-Based Hierarchical Radiosity Methods

The efficient generation of photorealistic images is one of the main subjects in the field of computer graphics. In contrast to simple image generation which is directly supported by standard 3D graphics hardware, photorealistic image synthesis strongly adheres to the physics describing the flow of light in a given environment. By simulating the energy flow in a 3D scene global effects like shadows and inter-reflections can be rendered accurately. The hierarchical radiosity method is one way of computing the global illumination in a scene. Due to its limitation to purely diffuse surfaces solutions computed by this method are view independent and can be examined in real-time walkthroughs. Additionally, the physically based algorithm makes it well suited for lighting design and architectural visualization. The focus of this thesis is the application of object-oriented methods to the radiosity problem. By consequently keeping and using object information throughout all stages of the algorithms several contributions to the field of radiosity rendering could be made. By introducing a new meshing scheme, it is shown how curved objects can be treated efficiently by hierarchical radiosity algorithms. Using the same paradigm the radiosity computation can be distributed in a network of computers. A parallel implementation is presented that minimizes communication costs while obtaining an efficient speedup. Radiosity solutions for very large scenes became possible by the use of clustering algorithms. Groups of objects are combined to clusters to simulate the energy exchange on a higher abstraction level. It is shown how the clustering technique can be improved without loss in image quality by applying the same data-structure for both, the visibility computations and the efficient radiosity simulation.Eines der Schwerpunktthemen in der Computergraphik ist die effiziente Erzeugung von fotorealistischen Bildern. Im Gegensatz zur einfachen Bilderzeugung, die bereits durch gaengige 3D-Grafikhardware unterstuetzt wird, gehorcht die fotorealistische Bildsynthese physikalischen Gesetzen, die die Lichtausbreitung innerhalb einer bestimmten Umgebung beschreiben. Durch die Simulation der Energieausbreitung in einer dreidimensionalen Szene koennen globale Effekte wie Schatten und mehrfache Reflektionen wirklichkeitstreu dargestellt werden. Die hierarchische Radiositymethode (Hierarchical Radiosity) ist eine Moeglichkeit, um die globale Beleuchtung innerhalb einer Szene zu berechnen. Da diese Methode auf die Verwendung von rein diffus reflektierenden Oberflaechen beschraenkt ist, sind damit errechnete Loesungen blickwinkelunabhaengig und lassen sich in Echtzeit am Bildschirm durchwandern. Zudem ist dieser Algorithmus aufgrund der verwendeten physikalischen Grundlagen sehr gut zur Beleuchtungssimulation und Architekturvisualisierung geeignet. Den Schwerpunkt dieser Doktorarbeit stellt die Anwendung objektbasierter Methoden auf das Radiosityproblem dar. Durch konsequente Ausnutzung von Objektinformationen waehrend aller Berechnungsschritte konnten verschiedene Verbesserungen im Rahmen der hierarchischen Radiositymethode erzielt werden. Gekruemmte Objekte koennen aufgrund eines neuen Flaechenunterteilungsverfahrens nun effizient durch den hierarchischen Radiosityalgorithmus dargestellt werden. Dieses Verfahren ermoeglicht ebenso eine effiziente Parallelisierung des hierarchischen Radiosityalgorithmus. Es wird ein parallele Implementierung vorgestellt, die unter Minimierung der Kommunikationskosten eine effiziente Geschwindigkeitssteigerung erzielt. Radiosityberechnungen fuer sehr grosse Szenen sind nur durch Verwendung sogenannter Clustering-Algorithmen moeglich. Dabei werden Gruppen von Objekten zu Clustern kombiniert um den Energieaustausch zwischen Oberflaechen stellvertretend auf einem hoeheren Abstraktionsniveau durchzufuehren. Durch Verwendung derselben Datenstruktur fuer Sichtbarkeitsberechnungen und fuer die Steuerung der Radiositysimulation wird gezeigt, wie das Clusteringverfahren ohne Qualitaetsverluste verbessert werden kann

Combinatorics

This is the report on the Oberwolfach workshop on Combinatorics, held 1–7 January 2006. Combinatorics is a branch of mathematics studying families of mainly, but not exclusively, finite or countable structures – discrete objects. The discrete objects considered in the workshop were graphs, set systems, discrete geometries, and matrices. The programme consisted of 15 invited lectures, 18 contributed talks, and a problem session focusing on recent developments in graph theory, coding theory, discrete geometry, extremal combinatorics, Ramsey theory, theoretical computer science, and probabilistic combinatorics