"함수 다이로그 항등식(functional dilogarithm identity)"의 두 판 사이의 차이
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Pythagoras0 (토론 | 기여) |
Pythagoras0 (토론 | 기여) |
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| (같은 사용자의 중간 판 12개는 보이지 않습니다) | |||
| 1번째 줄: | 1번째 줄: | ||
==개요== | ==개요== | ||
| − | * 로저스 다이로그 함수 (Rogers' dilogarithm)가 만족시키는 두 함수 항등식의 일반화 | + | * 로저스 다이로그 함수 (Rogers' dilogarithm)가 만족시키는 두 함수 항등식의 일반화 |
** 2항 관계식, 반사공식(오일러) <math>0\leq x \leq 1</math> 일 때, :<math>L(x)+L(1-x)=L(1)</math> | ** 2항 관계식, 반사공식(오일러) <math>0\leq x \leq 1</math> 일 때, :<math>L(x)+L(1-x)=L(1)</math> | ||
** [[5항 관계식 (5-term relation) |5항 관계식 (5-term relation)]] <math>0\leq x,y\leq 1</math> 일 때, | ** [[5항 관계식 (5-term relation) |5항 관계식 (5-term relation)]] <math>0\leq x,y\leq 1</math> 일 때, | ||
:<math>L(x)+L(1-xy)+L(y)+L\left(\frac{1-y}{1-xy}\right)+L\left(\frac{1-x}{1-xy} \right)=3L(1)</math> | :<math>L(x)+L(1-xy)+L(y)+L\left(\frac{1-y}{1-xy}\right)+L\left(\frac{1-x}{1-xy} \right)=3L(1)</math> | ||
* 클러스터 대수(cluster algebra) 를 이용하여 일반화됨 | * 클러스터 대수(cluster algebra) 를 이용하여 일반화됨 | ||
| − | * n 변수로 구성된 <math>(n^2+3n)/2</math> 항 관계식을 찾을 수 있음 | + | * 가령 <math>A_n</math> 딘킨 다이어그램으로부터, n 변수로 구성된 <math>(n^2+3n)/2</math> 항 관계식을 찾을 수 있음 |
| + | * <math>2, 5, 9, 14, 20, 27, 35, 44, 54, 65,\cdots</math> | ||
| 24번째 줄: | 25번째 줄: | ||
==9항 관계식== | ==9항 관계식== | ||
| − | * | + | * <math>S</math>를 다음과 같이 두자 |
| − | :<math>S=\left\{x,y,z,\frac{ | + | :<math> |
| + | S=\left\{x,y,z,\frac{(x+1) (z+1)}{y},\frac{(x+y+1) (y+z+1)}{x y z},\\ | ||
| + | \frac{x z+x+y+z+1}{x y},\frac{x z+x+y+z+1}{y z},\frac{y+1}{x},\frac{y+1}{z}\right\}</math> | ||
* 다이로그 함수에 대하여 다음이 성립한다 | * 다이로그 함수에 대하여 다음이 성립한다 | ||
:<math>\sum_{a\in S}L(\frac{1}{1+a})=3L(1)</math> | :<math>\sum_{a\in S}L(\frac{1}{1+a})=3L(1)</math> | ||
| 32번째 줄: | 35번째 줄: | ||
==14항 관계식== | ==14항 관계식== | ||
| − | :<math>\left\{x,z,\frac{(x+1) (z+1)}{y},\frac{z+1}{w},\frac{x z+x+y+z+1}{x y},\frac{(w+z+1) (x z+x+y+z+1)}{w y z},\frac{(y+z+1) (w (x+y+1)+x z+x+y+z+1)}{w x y z}, \frac{w (x+y+1)+x z+x+y+z+1}{y z},\frac{w y+w+y+z+1}{w z},\frac{(x+y+1) (w y+w+y+z+1)}{x y z},\frac{(w+1) (y+1)}{z},\frac{y+1}{x},w,y\right\}</math> | + | :<math> |
| + | \left\{x,z,\frac{(x+1) (z+1)}{y},\frac{z+1}{w},\frac{x z+x+y+z+1}{x y}, | ||
| + | \frac{(w+z+1) (x z+x+y+z+1)}{w y z}\\,\frac{(y+z+1) (w (x+y+1)+x z+x+y+z+1)}{w x y z}, \frac{w (x+y+1)+x z+x+y+z+1}{y z},\\ | ||
| + | \frac{w y+w+y+z+1}{w z},\frac{(x+y+1) (w y+w+y+z+1)}{x y z}, | ||
| + | \frac{(w+1) (y+1)}{z},\frac{y+1}{x},w,y\right\} | ||
| + | </math> | ||
:<math>\sum_{a\in S}L(\frac{1}{1+a})=4L(1)</math> | :<math>\sum_{a\in S}L(\frac{1}{1+a})=4L(1)</math> | ||
| + | |||
| + | ==메모== | ||
| + | * http://perso.univ-rennes1.fr/luc.pirio/SELECTApirio.pdf | ||
==역사== | ==역사== | ||
| 54번째 줄: | 65번째 줄: | ||
==관련논문== | ==관련논문== | ||
| + | * Tomoki Nakanishi, Rogers dilogarithms of higher degree and generalized cluster algebras, arXiv:1605.04777 [math.QA], May 16 2016, http://arxiv.org/abs/1605.04777 | ||
| + | * Soudères, Ismaël. “Functional Equations for Rogers Dilogarithm.” arXiv:1509.02869 [math], September 9, 2015. http://arxiv.org/abs/1509.02869. | ||
| + | * Kerr, Matt, James D. Lewis, and Patrick Lopatto. “Simplicial Abel-Jacobi Maps and Reciprocity Laws.” arXiv:1502.05459 [math], February 18, 2015. http://arxiv.org/abs/1502.05459. | ||
| + | * Herbert Gangl, Functional equations and ladders for polylogarithms http://www.maths.dur.ac.uk/~dma0hg/ladders2_ams.pdf | ||
* Nakanishi, Tomoki. 2011. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” Nagoya Mathematical Journal 202 (June): 23–43. doi:10.1215/00277630-1260432. | * Nakanishi, Tomoki. 2011. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” Nagoya Mathematical Journal 202 (June): 23–43. doi:10.1215/00277630-1260432. | ||
| − | * Chapoton, Frédéric. 2005. “Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems.” <em>Bulletin of the London Mathematical Society</em> 37 (5) (October 1): 755 -760. doi:10.1112/S0024609305004510. | + | * Chapoton, Frédéric. 2005. “Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems.” <em>Bulletin of the London Mathematical Society</em> 37 (5) (October 1): 755 -760. doi:10.1112/S0024609305004510. |
| − | * | + | * Herbert Gangl, Functional equations of polylogarithms http://www.mathematik.hu-berlin.de/~maphy/kyoto.pdf |
| + | * Zagier, D. "Special Values and Functional Equations of Polylogarithms." Appendix A in Structural Properties of Polylogarithms (Ed. L. Lewin) http://people.mpim-bonn.mpg.de/zagier/files/tex/LewinPolylogarithms/fulltext.pdf | ||
* L.J. Rogers, On Function Sum Theorems Connected with the Series Formula Proc. London Math. Soc. (1907) s2-4(1): 169-189 doi:[http://dx.doi.org/10.1112/plms/s2-4.1.169%20 10.1112/plms/s2-4.1.169] | * L.J. Rogers, On Function Sum Theorems Connected with the Series Formula Proc. London Math. Soc. (1907) s2-4(1): 169-189 doi:[http://dx.doi.org/10.1112/plms/s2-4.1.169%20 10.1112/plms/s2-4.1.169] | ||
[[분류:다이로그]] | [[분류:다이로그]] | ||
2020년 11월 16일 (월) 04:09 기준 최신판
개요
- 로저스 다이로그 함수 (Rogers' dilogarithm)가 만족시키는 두 함수 항등식의 일반화
- 2항 관계식, 반사공식(오일러) \(0\leq x \leq 1\) 일 때, \[L(x)+L(1-x)=L(1)\]
- 5항 관계식 (5-term relation) \(0\leq x,y\leq 1\) 일 때,
\[L(x)+L(1-xy)+L(y)+L\left(\frac{1-y}{1-xy}\right)+L\left(\frac{1-x}{1-xy} \right)=3L(1)\]
- 클러스터 대수(cluster algebra) 를 이용하여 일반화됨
- 가령 \(A_n\) 딘킨 다이어그램으로부터, n 변수로 구성된 \((n^2+3n)/2\) 항 관계식을 찾을 수 있음
- \(2, 5, 9, 14, 20, 27, 35, 44, 54, 65,\cdots\)
2항 관계식
- \(S=\left\{x,\frac{1}{x}\right\}\)라 두면,
\[\sum_{a\in S}L(\frac{1}{1+a})=L\left(\frac{1}{\frac{1}{x}+1}\right)+L\left(\frac{1}{x+1}\right)=L(1)\]
5항 관계식
- \(S=\left\{x,y,\frac{x+1}{y},\frac{y+1}{x},\frac{x+y+1}{x y}\right\}\) 이면,
\[\sum_{a\in S}L(\frac{1}{1+a})=L\left(\frac{1}{\frac{x+1}{y}+1}\right)+L\left(\frac{1}{\frac{y+1}{x}+1}\right)+L\left(\frac{1}{\frac{x+y+1}{x y}+1}\right)+L\left(\frac{1}{x+1}\right)+L\left(\frac{1}{y+1}\right)=2L(1)\]
9항 관계식
- \(S\)를 다음과 같이 두자
\[ S=\left\{x,y,z,\frac{(x+1) (z+1)}{y},\frac{(x+y+1) (y+z+1)}{x y z},\\ \frac{x z+x+y+z+1}{x y},\frac{x z+x+y+z+1}{y z},\frac{y+1}{x},\frac{y+1}{z}\right\}\]
- 다이로그 함수에 대하여 다음이 성립한다
\[\sum_{a\in S}L(\frac{1}{1+a})=3L(1)\]
14항 관계식
\[ \left\{x,z,\frac{(x+1) (z+1)}{y},\frac{z+1}{w},\frac{x z+x+y+z+1}{x y}, \frac{(w+z+1) (x z+x+y+z+1)}{w y z}\\,\frac{(y+z+1) (w (x+y+1)+x z+x+y+z+1)}{w x y z}, \frac{w (x+y+1)+x z+x+y+z+1}{y z},\\ \frac{w y+w+y+z+1}{w z},\frac{(x+y+1) (w y+w+y+z+1)}{x y z}, \frac{(w+1) (y+1)}{z},\frac{y+1}{x},w,y\right\} \] \[\sum_{a\in S}L(\frac{1}{1+a})=4L(1)\]
메모
역사
매스매티카 파일 및 계산 리소스
관련논문
- Tomoki Nakanishi, Rogers dilogarithms of higher degree and generalized cluster algebras, arXiv:1605.04777 [math.QA], May 16 2016, http://arxiv.org/abs/1605.04777
- Soudères, Ismaël. “Functional Equations for Rogers Dilogarithm.” arXiv:1509.02869 [math], September 9, 2015. http://arxiv.org/abs/1509.02869.
- Kerr, Matt, James D. Lewis, and Patrick Lopatto. “Simplicial Abel-Jacobi Maps and Reciprocity Laws.” arXiv:1502.05459 [math], February 18, 2015. http://arxiv.org/abs/1502.05459.
- Herbert Gangl, Functional equations and ladders for polylogarithms http://www.maths.dur.ac.uk/~dma0hg/ladders2_ams.pdf
- Nakanishi, Tomoki. 2011. “Dilogarithm Identities for Conformal Field Theories and Cluster Algebras: Simply Laced Case.” Nagoya Mathematical Journal 202 (June): 23–43. doi:10.1215/00277630-1260432.
- Chapoton, Frédéric. 2005. “Functional Identities for the Rogers Dilogarithm Associated to Cluster Y-Systems.” Bulletin of the London Mathematical Society 37 (5) (October 1): 755 -760. doi:10.1112/S0024609305004510.
- Herbert Gangl, Functional equations of polylogarithms http://www.mathematik.hu-berlin.de/~maphy/kyoto.pdf
- Zagier, D. "Special Values and Functional Equations of Polylogarithms." Appendix A in Structural Properties of Polylogarithms (Ed. L. Lewin) http://people.mpim-bonn.mpg.de/zagier/files/tex/LewinPolylogarithms/fulltext.pdf
- L.J. Rogers, On Function Sum Theorems Connected with the Series Formula Proc. London Math. Soc. (1907) s2-4(1): 169-189 doi:10.1112/plms/s2-4.1.169