"Bootstrap percolation"의 두 판 사이의 차이

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4번째 줄: 4번째 줄:
  
 
*  calculation of power-law exponent for boostrap percolation<br>
 
*  calculation of power-law exponent for boostrap percolation<br>
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*  growth rule<br>
  
 
* http://mathworld.wolfram.com/BootstrapPercolation.html<br>
 
* http://mathworld.wolfram.com/BootstrapPercolation.html<br>
9번째 줄: 10번째 줄:
 
 
 
 
  
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*  Andrews' conjecture<br>
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">tricky integrals</h5>
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*  Henrik Eriksson: [http://www.math.ubc.ca/~holroyd/integral.pdf A Tricky Integral]<br>
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* <math>\lambda_k=\frac{\pi^2}{3k(k+1)}</math><br>
  
 
 
 
 
  
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">integrals</h5>
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">q-series</h5>
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*  definition<br><math>P_k(q)</math><br>
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*  asymptotics of P_2(q)<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})=\sqrt{\frac{2\pi}{t}}\exp(-\frac{(2\pi)^2}{24t})</math><br>
  
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<math>\lambda_k=\frac{\pi^2}{3k(k+1)}</math>
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Henrik Eriksson: [http://www.math.ubc.ca/~holroyd/integral.pdf A Tricky Integral]
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<h5 style="line-height: 2em; margin-top: 0px; margin-right: 0px; margin-bottom: 0px; margin-left: 0px;">Andrews' conjecture</h5>
  
 
 
 
 
32번째 줄: 46번째 줄:
  
 
*  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})=\sqrt{\frac{2\pi}{t}}\exp(-\frac{(2\pi)^2}{24t})</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})=\sqrt{\frac{2\pi}{t}}\exp(-\frac{(2\pi)^2}{24t})</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>
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91번째 줄: 107번째 줄:
 
**  Alexander E. Holroyd, 2003<br>
 
**  Alexander E. Holroyd, 2003<br>
 
*  Holroyd, Liggett and Romik<br>
 
*  Holroyd, Liggett and Romik<br>
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* [[2010년 books and articles|논문정리]]
 
* [[2010년 books and articles|논문정리]]

2010년 3월 15일 (월) 13:23 판

introduction

 

  • calculation of power-law exponent for boostrap percolation
  • growth rule

 

 

 

tricky integrals
  • \(\lambda_k=\frac{\pi^2}{3k(k+1)}\)

 

 

q-series
  • definition
    \(P_k(q)\)
  • asymptotics of P_2(q)
    \(\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})=\sqrt{\frac{2\pi}{t}}\exp(-\frac{(2\pi)^2}{24t})\)

 

 

 

Andrews' conjecture

 

 

relevance to dedekind eta function

 

  • Dedekind eta function (데데킨트 에타함수)
    \(q=e^{-t}\) 으로 두면 \(t\sim 0\) 일 때,
    \(\prod_{n=1}^{\infty}(1-q^n)=1+\sum_{n\geq 1}^{\infty}\frac{(-1)^nq^{n(n+1)/2}}{(q)_n}\sim \sqrt\frac{2\pi}{t}\exp(-\frac{\pi^2}{6t})=\sqrt{\frac{2\pi}{t}}\exp(-\frac{(2\pi)^2}{24t})\)
    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

 

 

related items

 

 

encyclopedia

 

 

books

 

[[4909919|]]

 

 

articles

 

 

 

 

question and answers(Math Overflow)

 

 

blogs

 

 

experts on the field

 

 

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