applications of circle packing moduli
DESCRIPTION
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 PresentationTRANSCRIPT
![Page 1: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/1.jpg)
Applications of Circle Packing Moduli
Brock WilliamsTexas Tech University
![Page 2: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/2.jpg)
Roger W. Barnard
John Calhoun
Levi Johnson
Kent Pearce
Hamed Sari-Sarraf
Arun Shammanna
Including Joint Work With
![Page 3: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/3.jpg)
Grants727944742402745472
10350960930257
Work Supported By
GrantT365Z110259
![Page 4: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/4.jpg)
A circle packing is a configuration of circles with a specified pattern of tangencies.
Circle Packings
![Page 5: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/5.jpg)
Circle Packings
![Page 6: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/6.jpg)
Circle Packings
![Page 7: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/7.jpg)
Realization of the Triangulation
![Page 8: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/8.jpg)
Discrete Conformal Maps
About 900 Circles in Each Packing
![Page 9: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/9.jpg)
Discrete Conformal Maps
About 12000 Circles in Each Packing
![Page 10: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/10.jpg)
Modulus of Ring Domain
2mod(D)
logRr
R
r
D
1
![Page 11: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/11.jpg)
Circle Packing Modulus
![Page 12: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/12.jpg)
Circle Packing Modulus
![Page 13: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/13.jpg)
Circle Packing Modulus
0.61456...
2modulus .
1l
12.9058..og
r
r
![Page 14: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/14.jpg)
Cotton Maturity
Image from Google Maps.
![Page 15: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/15.jpg)
Immature cotton fibers are round, thick, and hollow.
As they mature, the lumen collapses and the cellulose thins.
Cotton Fibers
Cross section of fiber
![Page 16: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/16.jpg)
“Rounding the Fiber Out”
0.67277...
2modulus .
1l
15.8525..og
r
r
![Page 17: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/17.jpg)
“Rounding the Fiber Out”
0.91595...
2modulus .
1l
71.5678..og
r
r
![Page 18: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/18.jpg)
In work with John Calhoun, we’ve used random walk ideas to model disease
outbreaks.
Disease Modeling
![Page 19: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/19.jpg)
Zombie Virus
![Page 20: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/20.jpg)
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
![Page 21: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/21.jpg)
Incorporates additional parameters such as◦ Geography of the City◦ Hospitals◦ Human Offensive Capability
Predicts human population survival under some circumstances
Our Model
![Page 22: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/22.jpg)
Conformal Landscape
![Page 23: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/23.jpg)
Conformal Landscape
0.7292...
modu 19.89lus ...65
r
![Page 24: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/24.jpg)
Effect on the Disease
Human Population Zombie Population
![Page 25: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/25.jpg)
A Different Conformal Landscape
![Page 26: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/26.jpg)
A Different Conformal Landscape
0.2387...
mod 4.386..ulus .
r
![Page 27: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/27.jpg)
Effect on the Disease
Human Population Zombie Population
![Page 28: Applications of Circle Packing Moduli](https://reader036.vdocuments.us/reader036/viewer/2022062305/56815081550346895dbe7f33/html5/thumbnails/28.jpg)
Questions?
Thanks!