We visualize the structure of sections of the World Wide Web by constructing graphical representations in 3D hyperbolic space. The felicitous property that hyperbolic space has "more room" than Euclidean space allows more information to be seen amid less clutter, and motion by hyperbolic isometries provides for mathematically elegant navigation. The 3D graphical representations, available in the WebOOGL or VRML file formats, contain link anchors which point to the original pages on the Web itself. We use the Geomview/WebOOGL 3D Web browser as an interface between the 3D representation and the actual documents on the Web. The Web is just one example of a hierarchical tree structure with links "back up the tree" i.e. a directed graph which contains cycles. Our information visualization techniques are appropriate for other types of directed graphs with cycles, such as filesystems with symbolic links.

that I have read this dissertation and that in my opinion it is fully adequate,

The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in three-dimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues. Because the domain of application is mathematics, topological drawing is also concerned with the correct representation and display of these objects on a computer. By correctness we mean that the essential topological features of objects are maintained during interaction. We have chosen to limit the scope of topological drawing to knot theory, a domain that consists essentially of one class of object (embedded circles in three-dimensional space) yet is rich enough to contain a wide variety of difficult problems of research interest. In knot theory, two embedded circles (knots) are considered equivalent if one may be smoothly deformed into the other without any cuts or self-intersections. This notion of equivalence may be thought of as the heart of knot theory. We present methods for the computer construction and interactive manipulation of a

We present a new 1D algorithm for computing the global onedimensional unstable manifold of a saddle point of a map. This method can be generalized to compute two-dimensional unstable manifolds of maps with three-dimensional state spaces. Here we present a Q2D algorithm for the special case of a quasiperiodically forced map, which allows for a substantial simplification of the general case described in [18]. The key idea is to `grow' the manifold in steps, which consist of finding a new point on the manifold at a prescribed distance from the last point. The speed of growth is determined only by the curvature of the manifold, and not by the dynamics. The performance of the 1D algorithm is demonstrated with a constructed test example, and it is then used to compute one-dimensional manifolds of a map modeling mixing in a stirring tank. With the Q2D algorithm we compute two-dimensional unstable manifolds in the quasiperiodically forced H'enon map. Mathematical Subject Classification: 65C20,...

We present an algorithm for computing two-dimensional stable and unstable manifolds of three-dimensional vector fields. The main idea is to grow the manifold in concentric (topological) circles. Each new circle is computed as a set of intersection points of the manifold with a finite number of planes perpendicular to the last circle. Together with a scheme for adding or removing such planes this guarantees the quality of the mesh representing the computed manifold. As examples we compute the stable manifold of the origin spiralling into the Lorenz atractor, and an unstable manifold in Arneodo's system converging to a limit cycle. 1 Introduction Vector fields are the mathematical models of choice in numerous applications; see for example [7, 16] and further references therein. The most basic question is to understand the dynamics of a given fixed vector field. The phase space is organized by the spine or skeleton of the dynamics, which consists of all equilibria and other invariant set...

This thesis proposes a practical framework for the verification and synthesis of hybrid systems, that is, systems combining continuous and discrete dynamics. The lack of methods for computing reachable sets of continuous dynamics has been the main obstacle towards an algorithmic verification methodology for hybrid systems. We develop two effective approximate reachability techniques for continuous systems based on an efficient representation of sets and a combination of techniques from simulation, computational geometry, optimization, and optimal control. One is specialized for linear systems and extended to systems with uncertain input, and the other can be applied for non-linear systems. Using these reachability techniques we develop a safety verification algorithm which can work for a broad class of hybrid systems with arbitrary continuous dynamics and rather general switching behavior. We next study the problem of synthesizing switching controllers for hybrid systems with respect to a safety property. We present an effective synthesis algorithm based on the calculation of the maximal invariant set and the approximate reachability techniques. Finally, we describe the experimental tool "d/dt" which provides automatic safety verification and controller synthesis for hybrid systems with linear differential inclusions. Besides numerous academic examples, we have successfully applied the tool to verify some practical systems.

Many applications give rise to dynamical systems in the form of a vector field with a phase space of moderate dimension. Examples are the Lorenz equations, mechanical and other oscillators, and models of spiking neurons. The key to understanding the global dynamics of such a system are the stable and unstable manifolds of the saddle points, of the saddle periodic orbits and, more generally, of all invariant manifolds of saddle-type. Except in very special circumstances the (un)stable manifolds are global objects that cannot be found analytically but need to be computed numerically. This is a nontrivial task when the dimension of the manifold is larger than one. In this paper we present a general algorithm to compute the k-dimensional unstable manifold of an equilibrium or periodic orbit (or more general normally hyperbolic invariant manifold) of a vector field with an n-dimensional phase space, where 1 < k < n. Stable manifolds are computed by considering the flow for negative time. The key idea is to view the unstable manifold as a purely geometric object, hence disregarging the dynamics on the manifold, and compute it as a list of approximate geodesic level sets, which are (topological) (k − 1)-spheres. Starting from a (k − 1)sphere in the linear eigenspace of the equilibrium or periodic orbit, the next geodesic level set is found in a local (and changing) coordinate system given by hyperplanes perpendicular to the last

This paper and the accompanying demo describe a strategy and a software architecture for integrating several declarative approaches. This architecture allows for the interactive specification of local criteria for each vertex and edge. The Gold

Computer graphics has proven to be a very attractive tool for investigating low-dimensional algebraic manifolds and gaining intuition about their properties [9]. In principle, a computer image of any manifold described by algebraic

this paper we describe such an application of the Evolver, and present some results. The c fl A K Peters, Ltd. 1058-6458/96 $0.50 per page 182 Experimental Mathematics, Vol. 4 (1995), No. 3 first result has already been reported [Weaire and Phelan 1994a], and has excited widespread interest [Rivier 1994]. It provides a structure for monodisperse dry foam having lower energy than that of Kelvin [Thomson 1887]. We will therefore begin by sketching the historical background and the implications of this result. Further calculations will be presented for other dry foam structures. The Evolver has also been adapted to calculate energies of wet foam structures. We will present the first such results, and discuss the topological changes that occur as the liquid fraction is varied. The relevant details of the Evolver, its application to dry foams, its adaptation to wet foams and other technical aspects of the work are described in the sidebar on page 184. As the liquid fraction is varied, the mathematics dictating the structure of foam ranges from the minimal spatial partitioning problem (space-filling cells) in the dry limit to the optimal sphere packing problem in the wet limit touching (spherical bubbles). Wet foam may give a valuable insight into the connection between these and other familiar extremisation problems, such as that of the mimimal sphere covering (filling space with equal overlapping spheres of minimum total volume). 1. SOME HISTORY