"Bootstrap percolation"의 두 판 사이의 차이
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imported>Pythagoras0 잔글 (찾아 바꾸기 – “</h5>” 문자열을 “==” 문자열로) |
imported>Pythagoras0 잔글 (찾아 바꾸기 – “<h5 (.*)">” 문자열을 “==” 문자열로) |
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| 1번째 줄: | 1번째 줄: | ||
| − | + | ==introduction== | |
* one of important question in 2d percolation is the calculation of power-law exponent for boostrap percolation<br> | * one of important question in 2d percolation is the calculation of power-law exponent for boostrap percolation<br> | ||
| 6번째 줄: | 6번째 줄: | ||
| − | + | ==bootstrap percolation== | |
* growth rule<br> | * growth rule<br> | ||
| 17번째 줄: | 17번째 줄: | ||
| − | + | ==partitions without k-gaps== | |
* partitions without k-gaps (or k-sequences)<br> | * partitions without k-gaps (or k-sequences)<br> | ||
| 33번째 줄: | 33번째 줄: | ||
| − | + | ==q-series from percolation== | |
* definition<br><math>P_k(q)=(q;q)_{\infty}G_k(q)</math><br> | * definition<br><math>P_k(q)=(q;q)_{\infty}G_k(q)</math><br> | ||
| 43번째 줄: | 43번째 줄: | ||
| − | + | ==Andrews' conjecture on asymptotics== | |
* asymptotics of P_2(q) is known <br><math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<br><math>P_2(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{18t})</math><br> | * asymptotics of P_2(q) is known <br><math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<br><math>P_2(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{18t})</math><br> | ||
| 52번째 줄: | 52번째 줄: | ||
| − | + | ==tricky integrals== | |
* Henrik Eriksson: [http://www.math.ubc.ca/~holroyd/integral.pdf A Tricky Integral]<br><math>f_1(x)=1-x</math><br><math>f_2(x)=\frac{1-x+\sqrt{(1-x)(1+3x)}}{2}</math><br> | * Henrik Eriksson: [http://www.math.ubc.ca/~holroyd/integral.pdf A Tricky Integral]<br><math>f_1(x)=1-x</math><br><math>f_2(x)=\frac{1-x+\sqrt{(1-x)(1+3x)}}{2}</math><br> | ||
| 62번째 줄: | 62번째 줄: | ||
| − | + | ==relevance to dedekind eta function== | |
* Dedekind eta function ([http://pythagoras0.springnote.com/pages/3325777 데데킨트 에타함수])<br><math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<br><math>\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})</math><br> more generally, <math>q=\exp(\frac{2\pi ih}{k})e^{-t}</math> and <math>t\to 0</math> implies<br><math>\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}</math><br> | * Dedekind eta function ([http://pythagoras0.springnote.com/pages/3325777 데데킨트 에타함수])<br><math>q=e^{-t}</math> 으로 두면 <math>t\sim 0</math> 일 때,<br><math>\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})</math><br> more generally, <math>q=\exp(\frac{2\pi ih}{k})e^{-t}</math> and <math>t\to 0</math> implies<br><math>\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}</math><br> | ||
| 70번째 줄: | 70번째 줄: | ||
| − | + | ==history== | |
* http://www.google.com/search?hl=en&tbs=tl:1&q= | * http://www.google.com/search?hl=en&tbs=tl:1&q= | ||
| 78번째 줄: | 78번째 줄: | ||
| − | + | ==related items== | |
* [[asymptotic analysis of basic hypergeometric series]]<br> | * [[asymptotic analysis of basic hypergeometric series]]<br> | ||
| 87번째 줄: | 87번째 줄: | ||
| − | + | ==encyclopedia== | |
* http://ko.wikipedia.org/wiki/ | * http://ko.wikipedia.org/wiki/ | ||
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| − | + | ==books== | |
| 112번째 줄: | 112번째 줄: | ||
| − | + | ==articles== | |
* [http://arxiv.org/abs/1002.3881 A sharper threshold for bootstrap percolation in two dimensions]<br> | * [http://arxiv.org/abs/1002.3881 A sharper threshold for bootstrap percolation in two dimensions]<br> | ||
| 141번째 줄: | 141번째 줄: | ||
| − | + | ==question and answers(Math Overflow)== | |
* http://mathoverflow.net/search?q= | * http://mathoverflow.net/search?q= | ||
| 150번째 줄: | 150번째 줄: | ||
| − | + | ==blogs== | |
* 구글 블로그 검색<br> | * 구글 블로그 검색<br> | ||
| 160번째 줄: | 160번째 줄: | ||
| − | + | ==experts on the field== | |
* [http://www.math.ubc.ca/%7Eholroyd/ http://www.math.ubc.ca/~holroyd/] | * [http://www.math.ubc.ca/%7Eholroyd/ http://www.math.ubc.ca/~holroyd/] | ||
| 169번째 줄: | 169번째 줄: | ||
| − | + | ==TeX == | |
* [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | * [http://detexify.kirelabs.org/classify.html Detexify2 - LaTeX symbol classifier] | ||
2012년 10월 28일 (일) 17:07 판
introduction
- one of important question in 2d percolation is the calculation of power-law exponent for boostrap percolation
- this is related to the theory of partitions without k-gaps
bootstrap percolation
- growth rule
- http://www.math.ubc.ca/~holroyd/boot/
partitions without k-gaps
- partitions without k-gaps (or k-sequences)
- p_k(n) is the number of partitions of n that do not contain any sequence of consecutive integers of length k. p_2 (7) = 8.
- examples: partition of 7
{{7},{6,1},{5,2},{5,1,1},{4,3},{4,2,1},{4,1,1,1},{3,3,1},{3,2,2},{3,2,1,1},{3,1,1,1,1},{2,2,2,1},{2,2,1,1,1},{2,1,1,1,1,1},{1,1,1,1,1,1,1}}
7, 6 + 1, 5 + 2, 5 + 1 + 1, 4 + 1 + 1 + 1, 3 + 3 + 1, 3 + 1 + 1 + 1 + 1, and 1 + 1 + 1 + 1 + 1 + 1 + 1.
so there are 8 partitions without 2-gaps - Anderew's result
- generating function for partitions without k-gaps
\(G_2(q)=1+\sum_{n=1}^{\infty}\frac{q^n\prod_{j=1}^{n-1}(1-q^j+q^{2j})}{(q;q)_n}\)A116931
- generating function for partitions without k-gaps
- (*define a gap as 'b' *)
b := 2
G[b_, x_] :=
Sum[x^k*Product[1 + x^(b*j)/(1 - x^j), {j, 1, k - 1}]/(1 - x^k), {k,
1, 30}]
Series[G[b, x], {x, 0, 20}]
Table[SeriesCoefficient[%, n], {n, 0, 20}]
q-series from percolation
- definition
\(P_k(q)=(q;q)_{\infty}G_k(q)\) - Andrews and Zagier expression of \(P_k(q)\)
- result of [HLR04]
if \(q=e^{-t}\) and \(t\sim 0\)
\(P_k(q) \sim \frac{-\lambda_k}{1-q}\) as \(q \to 1\)
Andrews' conjecture on asymptotics
- asymptotics of P_2(q) is known
\(q=e^{-t}\) 으로 두면 \(t\sim 0\) 일 때,
\(P_2(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{18t})\) - conjecture
\(P_k(q) \sim \sqrt\frac{2\pi}{t}\exp(-\frac{\lambda_k}{t})\)
where \(\lambda_k=\frac{\pi^2}{3k(k+1)}\)
tricky integrals
- Henrik Eriksson: A Tricky Integral
\(f_1(x)=1-x\)
\(f_2(x)=\frac{1-x+\sqrt{(1-x)(1+3x)}}{2}\) - \(\lambda_k=\frac{\pi^2}{3k(k+1)}\)
- \(\lambda_2=\frac{\pi^2}{18}\)
relevance to dedekind eta function
- Dedekind eta function (데데킨트 에타함수)
\(q=e^{-t}\) 으로 두면 \(t\sim 0\) 일 때,
\(\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})\)
more generally, \(q=\exp(\frac{2\pi ih}{k})e^{-t}\) and \(t\to 0\) implies
\(\sqrt{\frac{t}{2\pi}}\exp({\frac{\pi^2}{6k^2t}})\eta(\frac{h}{k}+i\frac{t}{2\pi})\sim \frac{\exp\left(\pi i (\frac{h}{12k}-s(h,k)\right)}{\sqrt{k}}\)
history
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
- 2010년 books and articles
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
articles
- A sharper threshold for bootstrap percolation in two dimensions
- [1]Janko Gravner, Alexander E. Holroyd, Robert Morris, 2010
- Improved bounds on metastability thresholds and probabilities for generalized bootstrap percolation
- Kathrin Bringmann, Karl Mahlburg, 2010
- Kathrin Bringmann, Karl Mahlburg, 2010
- Integrals, partitions and MacMahon's Theorem
- George Andrewsa, Dan Romik, 2007
- George Andrewsa, Dan Romik, 2007
- Slow convergence
- Partitions with short sequences and mock theta functions
- George E. Andrews, 2005
- George E. Andrews, 2005
- [HLR04]Integrals, Partitions, and Cellular Automata
- A. E. Holroyd, T. M. Liggett & D. Romik, Transactions of the American Mathematical Society, 2004, Vol 356, 3349-3368
- A. E. Holroyd, T. M. Liggett & D. Romik, Transactions of the American Mathematical Society, 2004, Vol 356, 3349-3368
- sharp metastability threshold for two-dimensional bootstrap percolation
- Alexander E. Holroyd, 2003
- Alexander E. Holroyd, 2003
- 논문정리
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html[2]
- http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
- http://www.ams.org/mathscinet/search/publications.html?pg4=AUCN&s4=&co4=AND&pg5=TI&s5=&co5=AND&pg6=PC&s6=&co6=AND&pg7=ALLF&co7=AND&Submit=Search&dr=all&yrop=eq&arg3=&yearRangeFirst=&yearRangeSecond=&pg8=ET&s8=All&s7=
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
experts on the field