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QUIZ

QUIZ. Which of these convection patterns is non-Boussinesq?. Homological Characterization Of Convection Patterns. Kapilanjan Krishan Marcio Gameiro Michael Schatz Konstantin Mischaikow School of Physics School of Mathematics

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QUIZ

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  1. QUIZ Which of these convection patterns is non-Boussinesq?

  2. 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

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

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

  5. Patterns and Convection Light Source Camera Reduced Rayleigh number e=(T-Tc)/ Tc Convection cell e=0.125

  6. Spiral Defect Chaos

  7. Homology Using algebra to determine topology Representations Simplicial Cubical

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

  9. Boundary Operator e5 e1 e6 f e2 e8 e4 e7 e3 dimension # of Loops enclosing holes = of homology group H1

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

  11. Reduction to Binary Representation Experiment image Cold flow Hot flow

  12. Number of Components Zeroth Betti number = 34

  13. Hot flows vs. Cold flows Hot flow Cold flow

  14. Spiral Defect Chaos e~2 e~1

  15. Number of distinct components Hot flow vs. Cold flow Time ~ 103 tn Time ~ 103 tn e~1 e~2

  16. Number of holes First Betti number = 13

  17. Number of distinct holes Hot flow vs. Cold flow Time ~ 103 tn Time ~ 103 tn e~1 e~2

  18. Betti numbers vs Epsilon Hot flow and Cold flow b0 Betti numbers b1 b0 b1 e Asymmetry between hot and cold regions Non-Boussinesq effects  ?

  19. Which of these convection patterns is non-Boussinesq?

  20. Simulations (SF6 ) (Madruga and Riecke) Boussinesq Non-Boussinesq e=1.4 e=1.4, Q=4.5

  21. Boussinesq Simulations (SF6 ) Time Series Components Holes b0 b0 b1 b1

  22. Non-Boussinesq Simulations (SF6 ) Time Series Components Holes b0 b1 b0 b1

  23. Simulations (CO2 ) at Experimental Conditions e=2, Q=0.7 Components Holes b0 b1 b0 b1

  24. Boundary Influence

  25. Number of connected components Hot flow vs. Cold flow Time ~ 103 tn Time ~ 103 tn e~1 e~2

  26. Percentage of connected components Hot flow vs. Cold flow Time ~ 103 tn Time ~ 103 tn e~1 e~2

  27. Convergence to Attractor Frequency of occurrence bcold0: Number of cold flow components ( e~1 )

  28. Bifurcations? Joint Probability P(bhot0 ,bcold0 ,bhot1 , bcold1) Entropy Entropy (-S Pi log Pi) e

  29. Entropy vs epsilon e=1.00 e=1.13 e=1.25 Entropy=8.3 Entropy=7.9 Entropy=8.9

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

  31. 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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