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140
A TetrahedraBased Stream Surface Algorithm
 Proceedings of IEEE Visualization 2001
, 2001
"... This paper presents a new algorithm for the calculation of stream surfaces for tetrahedral grids. It propagates the surface through the tetrahedra, one at a time, calculating the intersections with the tetrahedral faces. The method allows us to incorporate topological information from the cells, e.g ..."
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Cited by 22 (7 self)
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This paper presents a new algorithm for the calculation of stream surfaces for tetrahedral grids. It propagates the surface through the tetrahedra, one at a time, calculating the intersections with the tetrahedral faces. The method allows us to incorporate topological information from the cells, e.g., critical points. The calculations are based on barycentric coordinates, since this simplifies theory and algorithm. The stream surfaces are ruled surfaces inside each cell, and their construction is started with line segments on the faces. Our method supports the analysis of velocity fields resulting from computational fluid dynamics (CFD) simulations.
Piecewiselinear models of genetic regulatory networks: equilibria and . . .
 J. MATH. BIOL.
, 2005
"... A formalism based on piecewiselinear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be wellsuited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutio ..."
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Cited by 21 (13 self)
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A formalism based on piecewiselinear (PL) differential equations, originally due to Glass and Kauffman, has been shown to be wellsuited to modelling genetic regulatory networks. However, the discontinuous vector field inherent in the PL models raises some mathematical problems in defining solutions on the surfaces of discontinuity. To overcome these difficulties we use the approach of Filippov, which extends the vector field to a differential inclusion. We study the stability of equilibria (called singular equilibrium sets) that lie on the surfaces of discontinuity. We prove several theorems that characterize the stability of these singular equilibria directly from the state transition graph, which is a qualitative representation of the dynamics of the system. We also formulate a stronger conjecture on the stability of these singular equilibrium sets.
Analog Computation with Dynamical Systems
 Physica D
, 1997
"... This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete th ..."
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Cited by 21 (0 self)
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This paper presents a theory that enables to interpret natural processes as special purpose analog computers. Since physical systems are naturally described in continuous time, a definition of computational complexity for continuous time systems is required. In analogy with the classical discrete theory we develop fundamentals of computational complexity for dynamical systems, discrete or continuous in time, on the basis of an intrinsic time scale of the system. Dissipative dynamical systems are classified into the computational complexity classes P d , CoRP d , NP d
On PredatorPrey Systems and SmallGain Theorems
 J. Mathematical Biosciences and Engineering
, 2002
"... This paper deals with an almost global attractivity result for LotkaVolterra systems with predatorprey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular inputoutput properties. We us ..."
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Cited by 17 (10 self)
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This paper deals with an almost global attractivity result for LotkaVolterra systems with predatorprey interactions. These systems can be written as (negative) feedback systems. The subsystems of the feedback loop are monotone control systems, possessing particular inputoutput properties. We use a smallgain theorem, adapted to a context of systems with multiple equilibrium points to obtain the desired almost global attractivity result. It provides su#cient conditions to rule out oscillatory or more complicated behavior which is often observed in predatorprey systems.
An Unconditionally Stable OneStep Scheme for Gradient Systems
, 1997
"... A semiimplicit time discretization of gradient flows is given and analyzed. It is shown that the scheme is unconditionally gradient stable and that the discrete equations are uniquely solvable for all time steps. The key feature of the method is a separation of the contractive and expansive terms o ..."
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Cited by 17 (0 self)
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A semiimplicit time discretization of gradient flows is given and analyzed. It is shown that the scheme is unconditionally gradient stable and that the discrete equations are uniquely solvable for all time steps. The key feature of the method is a separation of the contractive and expansive terms of the equation across the time step. The CahnHilliard equation is used as an example. 1 Introduction. This paper will concentrate on a numerical method for solving the initial value problem defined by the system of real ordinary differential equations and auxillary conditions du dt = \GammarF (u); u(0) = u 0 : (1) It is assumed that u(t) 2 C 1 (R + ; R p ), F (u) 2 C 2 (R p ; R), rF (u) is the gradient of F , and 8 ? ? ? ? ? ! ? ? ? ? ? : F (u) 0 8u 2 R p F (u) !1 as kuk !1 hJ(rF )(u) u; ui 8u 2 R p (2) Department of Mathematics, University of Utah, Salt Lake City, UT 84112, eyre@math.utah.edu 1 where J(rF )(u) is the Jacobian of rF (u) and 2 R, and where h\Delta; ...
A theory of complexity for continuous time systems
 Journal of Complexity
, 2002
"... We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their ..."
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Cited by 16 (0 self)
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We present a model of computation with ordinary differential equations (ODEs) which converge to attractors that are interpreted as the output of a computation. We introduce a measure of complexity for exponentially convergent ODEs, enabling an algorithmic analysis of continuous time flows and their comparison with discrete algorithms. We define polynomial and logarithmic continuous time complexity classes and show that an ODE which solves the maximum network flow problem has polynomial time complexity. We also analyze a simple flow that solves the Maximum problem in logarithmic time. We conjecture that a subclass of the continuous P is equivalent to the classical P. 2001 Elsevier Science (USA) Key Words: theory of analog computation; dynamical systems.
Theory and Experiments in Autonomous SensorBased Motion Planning with Applications for Flight Planetary Microrovers
, 1999
"... With the success of Mars Pathfinder's Sojourner rover, a new era of planetary exploration has opened, with demand for highly capable mobile robots. These robots must be able to traverse long distances over rough, unknown terrain autonomously, under severe resource constraints. Much prior work in mob ..."
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Cited by 15 (3 self)
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With the success of Mars Pathfinder's Sojourner rover, a new era of planetary exploration has opened, with demand for highly capable mobile robots. These robots must be able to traverse long distances over rough, unknown terrain autonomously, under severe resource constraints. Much prior work in mobile robot path planning has been based on assumptions that are not truly applicable to navigation through planetary terrains. Based on the author's firsthand experience with the Mars Pathfinder mission, this work reviews issues which are critical for successful autonomous navigation of planetary rovers. No current methodology addresses all of these constraints. We next develop the sensorbased "Wedgebug" motionplanning algorithm. This algorithm is complete, correct, requires minimal memory for storage of its world model, and uses only onboard sensors, which are guided by the algorithm to e#ciently sense only the data needed for motion planning, while avoiding unnecessary robot motion. The p...
Dynamics of learning in linear featurediscovery networks
 Network: Computation in Neural Systems, 2:85—105
, 1991
"... tleencse. ogi. edu We describe the dynamics of learning in unsupervised linear featurediscovery networks that have recurrent lateral connections. Bifurcation theory provides a description of the location of multiple equilibria and limit cycles in the weightspace dynamics. ..."
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Cited by 15 (1 self)
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tleencse. ogi. edu We describe the dynamics of learning in unsupervised linear featurediscovery networks that have recurrent lateral connections. Bifurcation theory provides a description of the location of multiple equilibria and limit cycles in the weightspace dynamics.