Y-system and functional dilogarithm identities
imported>Pythagoras0님의 2012년 10월 29일 (월) 10:55 판
introduction
- 체비셰프 다항식
\(U_n(x)^2=1+U_{n-1}(x)U_{n+1}(x)\) - 정다각형의 대각선의 길이
\(r_i^2=1+r_{i-1}r_{i+1}, 1\leq i \leq n-3\) - Question : for what values of \(r_1=x\), is the recurrence \(r_i^2=1+r_{i-1}r_{i+1}\) periodic? (\(r_0=1\))
- A := RecurrenceTable[{a[n] a[n - 2] + 1 == a[n - 1]^2, a[1] == x,
a[2] == y}, a, {n, 10}]
Simplify[A]
- Laurent phenomenon is true
- total positivity is broken
- 정오각형의 경우
- \(r_i^2=1+r_{i-1}r_{i+1}\), \(r_0=1,r_3=1\)
- 3가지 점화식의 해가 존재
- \(\{1,-1,0,1\}\), \(\{1,\frac{-\sqrt{5}+1}{2},\frac{-\sqrt{5}+1}{2},1 \}\) , \(\{1,\frac{\sqrt{5}+1}{2},\frac{\sqrt{5}+1}{2},1 \}\)
- A := RecurrenceTable[{a[n] a[n - 2] + 1 == a[n - 1]^2, a[1] == 1,
a[2] == 2 y}, a, {n, 10}]
Simplify[A]
NSolve[-4 y + 8 y^3 == 1, y]
{1, 2 y, -1 + 4 y^2, -4 y + 8 y^3,
1 - 12 y^2 +
16 y^4} /. {{y -> -0.5`}, {y -> -0.30901699437494745`}, {y ->
0.8090169943749475`}} // TableForm
total positivity
- \(r_{i-1}r_{i+1}=r_i^2+1\)
- A := RecurrenceTable[{a[n] a[n - 2] - 1 == a[n - 1]^2, a[1] == x,
a[2] == y}, a, {n, 10}]
Simplify[A]
relation to 5-term relation
- 5항 관계식 (5-term relation)
\(1-x_{i}=x_{i-1}x_{i+1}\)
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/
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- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
- Dilogarithm identities in conformal field theory and cluster algebras
- Periodicities in cluster algebras and dilogarithm identities
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
- Dilogarithm identities for conformal field theories and cluster algebras: simply laced case 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
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