Y-system and functional dilogarithm identities
http://bomber0.myid.net/ (토론)님의 2011년 2월 25일 (금) 12:25 판
introduction
five-term relation of dilogarithm
- 5항 관계식 (5-term relation)
- 로저스 다이로그 함수 \(L(x)\)에 대하여 다음이 성립한다
\(0\leq x,y\leq 1\) 일 때,
\(L(x)+L(1-xy)+L(y)+L(\frac{1-y}{1-xy})+L\Left( \frac{1-x}{1-xy} )\right)=\frac{\pi^2}{2}\) - \(1-x_{i}=x_{i-1}x_{i+1}\) 를 만족시키는 다섯개의 수
- is this also an example of a cluster variable?
- asymptotic analysis of basic hypergeometric series
- f[{x_, y_, z_, w_}] := Simplify[(x - z)/(x - w)*(y - w)/(y - z)]
A := Permutations[{0, 1, w, z}]
Table[Limit[f[Ai], w -> \[Infinity]], {i, 24}]
B := Subsets[{0, x*y, 1, y, z}, {4}]
g[i_] := Table[
Limit[f[n], z -> \[Infinity]], {n, Permutations[Bi]}]
Table[f[Bi], {i, 1, 5}]
Table[g[i], {i, 5}]
rank 2 example
\(y_{m-1}y_{m+1}=y_m+1\)
Start with two variables \(y_1,y_2\).
\(y_3y_1=y_2+1\). so \(y_3=\frac{y_2+1}{y_1}\)
\(y_2y_4=y_3+1 \)implies \(y_4=\frac{y_3+1}{y_2}=\frac{y_1+y_2+1}{y_1y_2}\)
\(y_3y_5=y_4+1\) implies \(y_5=\frac{y_4+1}{y_3}= \frac{y_1+1}{y_2}\) we are getting Laurent polynomials
\(y_4y_6=y_5\) implies \(y_6=\frac{y_5+1}{y_4}= \frac{\frac{y_1+1}{y_2}+1}{\frac{y_1+y_2+1}{y_1y_2}}=\frac{y_1(y_1+1)+y_1y_2}{y_1+y_2+1}=y_1\)
history
- 3 central charge of CFT, L-values, volume of threefolds and dilogarithm
- dilogarithm and dilogarithm identities
- Bloch group, K-theory and dilogarithm
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
articles
- Periodic cluster algebras and dilogarithm identities
- Tomoki Nakanishi, 2010
- Periodicities of T and Y-systems, dilogarithm identities, and cluster algebras I: Type B_r
- Rei Inoue, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, Tomoki Nakanishi, 2010
- Rei Inoue, Osamu Iyama, Bernhard Keller, Atsuo Kuniba, Tomoki Nakanishi, 2010
- Dilogarithm identities for conformal field theories and cluster algebras: simply laced case
- Tomoki Nakanishi, 2009
- Tomoki Nakanishi, 2009
- Thermodynamic Bethe Ansatz and Dilogarithm Identities I
- Edward Frenkel, Andras Szenes, 1995
- ADE functional dilogarithm identities and integrable models
- F. Gliozzi, R. Tateo, Phys. Lett. 348B (1995) 84-88.
- Rogers Dilogarithm in Integrable Systems
- A. Kuniba, T. Nakanishi, 1992
- Spectra in Conformal Field Theories from the Rogers Dilogarithm
- Atsuo Kuniba, Tomoki Nakanishi, 1992
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
blogs
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- http://ncatlab.org/nlab/show/HomePage
experts on the field