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QUIZ Which of these convection patterns is non-Boussinesq?

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Homological Characterization Of Convection Patterns Kapilanjan Krishan Marcio Gameiro Michael Schatz Konstantin Mischaikow School of Physics School of Mathematics Georgia Institute of Technology Supported by: DOE, DARPA, NSF

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Patterns and Drug Delivery Caffeine in Polyurethane Matrix D. Saylor et al., (U.S. Food and Drug Administration)

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Patterns and Strength of Materials Maximal Principal Stresses in Alumina E. Fuller et al., (NIST)

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Patterns and Convection Camera Light Source Reduced Rayleigh number =(T-T c )/ T c =0.125 Convection cell

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Spiral Defect Chaos

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Homology Using algebra to determine topology Simplicial Cubical Representations

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Elementary Cubes and Chains 0-cube 1-cube 2-cube 1-chain 2-chain 0-chain

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Boundary Operator f e1e1 e2e2 e3e3 e4e4 e5e5 e6e6 e7e7 e8e8 dimension # of Loops enclosing holes = of homology group H 1

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Homology Summary Patterns are described by Dimension of =, the ith Betti number Homology: Computable topology

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Reduction to Binary Representation Hot flow Cold flowExperiment image

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Number of Components Zeroth Betti number = 34

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Hot flows vs. Cold flows Hot flowCold flow

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Spiral Defect Chaos

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Time ~ 10 3 Number of distinct components Hot flow vs. Cold flow

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Number of holes First Betti number = 13

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Time ~ 10 3 Number of distinct holes Hot flow vs. Cold flow

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Betti numbers vs Epsilon Hot flow and Cold flow Asymmetry between hot and cold regions Non-Boussinesq effects ? Betti numbers

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Which of these convection patterns is non-Boussinesq?

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Simulations (SF 6 ) BoussinesqNon-Boussinesq (Madruga and Riecke) Q=4.5

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Boussinesq Simulations (SF 6 ) Time Series ComponentsHoles

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Non-Boussinesq Simulations (SF 6 ) ComponentsHoles Time Series

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Simulations (CO 2 ) at Experimental Conditions ComponentsHoles Q=0.7

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Boundary Influence

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Time ~ 10 3 Number of connected components Hot flow vs. Cold flow

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Time ~ 10 3 Percentage of connected components Hot flow vs. Cold flow

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Convergence to Attractor Frequency of occurrence cold 0 : Number of cold flow components ( ~1 )

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Entropy Joint Probability P( hot 0, cold 0, hot 1, cold 1 ) Entropy (- P i log P i ) Bifurcations?

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Entropy vs epsilon Entropy=8.3Entropy=7.9 Entropy=8.9

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Space-Time Topology 1-D Gray-Scott model Space Time Time Series—First Betti number Exhbits Chaos

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Summary Homology characterizes complex patterns Underlying symmetries detected in data Alternative measure of boundary effects Detects transitions between complex states Space-time topology may reveal new insights Homology source codes available at: http://www.math.gatech.edu/~chomp

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