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Applications of Circle Packing Moduli. Brock Williams Texas Tech University. Including Joint Work With. Roger W. Barnard John Calhoun Levi Johnson Kent Pearce Hamed Sari- Sarraf Arun Shammanna. Work Supported By. Grants 727944 742402 745472 1035096 0930257. Grant T365Z110259. - PowerPoint PPT Presentation
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Applications of Circle Packing Moduli
Brock WilliamsTexas Tech University
Roger W. Barnard
John Calhoun
Levi Johnson
Kent Pearce
Hamed Sari-Sarraf
Arun Shammanna
Including Joint Work With
Grants727944742402745472
10350960930257
Work Supported By
GrantT365Z110259
A circle packing is a configuration of circles with a specified pattern of tangencies.
Circle Packings
Circle Packings
Circle Packings
Realization of the Triangulation
Discrete Conformal Maps
About 900 Circles in Each Packing
Discrete Conformal Maps
About 12000 Circles in Each Packing
Modulus of Ring Domain
2mod(D)
logRr
R
r
D
1
Circle Packing Modulus
Circle Packing Modulus
Circle Packing Modulus
0.61456...
2modulus .
1l
12.9058..og
r
r
Cotton Maturity
Image from Google Maps.
Immature cotton fibers are round, thick, and hollow.
As they mature, the lumen collapses and the cellulose thins.
Cotton Fibers
Cross section of fiber
“Rounding the Fiber Out”
0.67277...
2modulus .
1l
15.8525..og
r
r
“Rounding the Fiber Out”
0.91595...
2modulus .
1l
71.5678..og
r
r
In work with John Calhoun, we’ve used random walk ideas to model disease
outbreaks.
Disease Modeling
Zombie Virus
Munz, Hudea, Imad, and Smith published a PDE model of a zombie outbreak in 2009.
They found logistic growth of the zombie population (and corresponding logistic decay of the human population).
Previous Zombie Work
Incorporates additional parameters such as◦ Geography of the City◦ Hospitals◦ Human Offensive Capability
Predicts human population survival under some circumstances
Our Model
Conformal Landscape
Conformal Landscape
0.7292...
modu 19.89lus ...65
r
Effect on the Disease
Human Population Zombie Population
A Different Conformal Landscape
A Different Conformal Landscape
0.2387...
mod 4.386..ulus .
r
Effect on the Disease
Human Population Zombie Population
Questions?
Thanks!