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Articles 1 - 4 of 4
Full-Text Articles in Mathematics
As Flat As Possible, Jon T. Jacobsen
As Flat As Possible, Jon T. Jacobsen
All HMC Faculty Publications and Research
How does one determine a surface which is as flat as possible, such as those created by soap film surfaces? What does it mean to be as flat as possible? In this paper we address this question from two distinct points of view, one local and one global in nature. Continuing with this theme, we put a temporal twist on the question and ask how to evolve a surface so as to flatten it as efficiently as possible. This elementary discussion provides a platform to introduce a wide range of advanced topics in partial differential equations and helps students …
Nonlinear Dynamics In Combinatorial Games: Renormalizing Chomp, Eric J. Friedman, Adam S. Landsberg
Nonlinear Dynamics In Combinatorial Games: Renormalizing Chomp, Eric J. Friedman, Adam S. Landsberg
WM Keck Science Faculty Papers
We develop a new approach to combinatorial games that reveals connections between such games and some of the central ideas of nonlinear dynamics: scaling behaviors, complex dynamics and chaos, universality, and aggregation processes. We take as our model system the combinatorial game Chomp, which is one of the simplest in a class of "unsolved" combinatorial games that includes Chess, Checkers, and Go. We discover that the game possesses an underlying geometric structure that "grows" (reminiscent of crystal growth), and show how this growth can be analyzed using a renormalization procedure adapted from physics. In effect, this methodology allows one to …
Approximations Of Continuous Newton's Method: An Extension Of Cayley's Problem, Jon T. Jacobsen, Owen Lewis '05, Bradley Tennis '06
Approximations Of Continuous Newton's Method: An Extension Of Cayley's Problem, Jon T. Jacobsen, Owen Lewis '05, Bradley Tennis '06
All HMC Faculty Publications and Research
Continuous Newton's Method refers to a certain dynamical system whose associated flow generically tends to the roots of a given polynomial. An Euler approximation of this system, with step size h=1, yields the discrete Newton's method algorithm for finding roots. In this note we contrast Euler approximations with several different approximations of the continuous ODE system and, using computer experiments, consider their impact on the associated fractal basin boundaries of the roots
Turing Patterns On Growing Spheres: The Exponential Case, Julijana Gjorgjieva, Jon T. Jacobsen
Turing Patterns On Growing Spheres: The Exponential Case, Julijana Gjorgjieva, Jon T. Jacobsen
All HMC Faculty Publications and Research
We consider Turing patterns for reaction-diffusion systems on the surface of a growing sphere. In particular, we are interested in the effect of dynamic growth on the pattern formation. We consider exponential isotropic growth of the sphere and perform a linear stability analysis and compare the results with numerical simulations.