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38
Visualizing the Structure of the World Wide Web in 3D Hyperbolic Space
 IN PROCEEDINGS OF VRML '95
, 1995
"... 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 i ..."
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Cited by 75 (5 self)
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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.
Fluid Interaction for High Resolution WallSize Displays
, 2002
"... that I have read this dissertation and that in my opinion it is fully adequate, ..."
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Cited by 18 (2 self)
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that I have read this dissertation and that in my opinion it is fully adequate,
Interactive Topological Drawing
, 1998
"... The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional space. In topological drawing, issues such as adjacency and connectedness, which are topological in nature, take precedence over purely geometric issues ..."
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Cited by 18 (1 self)
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The research presented here examines topological drawing, a new mode of constructing and interacting with mathematical objects in threedimensional 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 threedimensional 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 selfintersections. 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
Growing 1D and quasi 2D unstable manifolds of maps
, 1998
"... 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 twodimensional unstable manifolds of maps with threedimensional state spaces. Here we present a Q2D algorithm for the special case of a quas ..."
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Cited by 16 (7 self)
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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 twodimensional unstable manifolds of maps with threedimensional 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 onedimensional manifolds of a map modeling mixing in a stirring tank. With the Q2D algorithm we compute twodimensional unstable manifolds in the quasiperiodically forced H'enon map. Mathematical Subject Classification: 65C20,...
TwoDimensional Global Manifolds of Vector Fields
 CHAOS
"... We present an algorithm for computing twodimensional stable and unstable manifolds of threedimensional 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 plane ..."
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Cited by 14 (6 self)
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We present an algorithm for computing twodimensional stable and unstable manifolds of threedimensional 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...
Computing geodesic level sets on global (un)stable manifolds of vector fields
 SIAM JOURNAL ON APPLIED DYNAMICAL SYSTEMS
, 2003
"... 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 an ..."
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Cited by 10 (6 self)
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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 saddletype. 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 kdimensional unstable manifold of an equilibrium or periodic orbit (or more general normally hyperbolic invariant manifold) of a vector field with an ndimensional 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
Vérification Et Synthèse Des Systèmes Hybrides
, 2000
"... 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 methodo ..."
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Cited by 10 (1 self)
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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 nonlinear 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.
Integration of Declarative Approaches
 In Graph Drawing (GD'96), Berkeley/CA
, 1997
"... 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 ..."
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Cited by 7 (1 self)
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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
THE GEOMETRY OF SLOW MANIFOLDS NEAR A FOLDED NODE
"... This paper is concerned with the geometry of slow manifolds of a dynamical system with two slow and one fast variable. Specifically, we study the dynamics near a folded node singularity, which is known to give rise to socalled canard solutions. Geometrically, canards are intersection curves of two ..."
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Cited by 6 (1 self)
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This paper is concerned with the geometry of slow manifolds of a dynamical system with two slow and one fast variable. Specifically, we study the dynamics near a folded node singularity, which is known to give rise to socalled canard solutions. Geometrically, canards are intersection curves of twodimensional attracting and repelling slow manifolds, and they are a key element of slowfast dynamics. For example, canard solutions are associated with mixedmode oscillations, where they organize regions with different numbers of small oscillations. We perform a numerical study of the geometry of twodimensional slow manifolds in the normal form of a folded node in R 3. Namely, we view the part of a slow manifold that is of interest as a oneparameter family of orbit segments up to a suitable crosssection. Hence, it is the solution of a twopoint boundary value problem, which we solve by numerical continuation with the package AUTO. The computed family of orbit segments is used to obtain a mesh representation of the manifold as a surface. With this approach we show how the attracting and repelling slow manifolds change in dependence on the eigenvalue ratio µ of the reduced flow. At µ = 1 two primary canards bifurcate and secondary canards are created at odd integer values of µ. We compute 24 secondary canards to investigate how they spiral more and more around one of the primary canards. The first twelve secondary canards are continued in µ to obtain a numerical bifurcation diagram.
Crossing the c=1 barrier in 2d Lorentzian quantum gravity, Phys
 Rev. D
, 2000
"... Abstract: In an extension of earlier work we investigate the behaviour of twodimensional Lorentzian quantum gravity under coupling to a conformal field theory with c> 1. This is done by analyzing numerically a system of eight Ising models (corresponding to c=4) coupled to dynamically triangulate ..."
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Cited by 5 (3 self)
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Abstract: In an extension of earlier work we investigate the behaviour of twodimensional Lorentzian quantum gravity under coupling to a conformal field theory with c> 1. This is done by analyzing numerically a system of eight Ising models (corresponding to c=4) coupled to dynamically triangulated Lorentzian geometries. It is known that a single Ising model couples weakly to Lorentzian quantum gravity, in the sense that the Hausdorff dimension of the ensemble of twogeometries is two (as in pure Lorentzian quantum gravity) and the matter behaviour is governed by the Onsager exponents. By increasing the amount of matter to 8 Ising models, we find that the geometry of the combined system has undergone a phase transition. The new phase is characterized by an anomalous scaling of spatial length relative to proper time at large distances, and as a consequence the Hausdorff dimension is now three. In spite of this qualitative change in the geometric sector, and a very strong interaction between matter and geometry, the critical exponents of the Ising model retain their Onsager values. This provides evidence for the conjecture that the KPZ values of the critical exponents in 2d Euclidean quantum gravity are entirely due to the presence of baby universes. Lastly, we summarize the lessons learned so far from 2d Lorentzian quantum gravity. Contents