"Quantum dilogarithm"의 두 판 사이의 차이
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| 20번째 줄: | 20번째 줄: | ||
* <math>0<q<1</math>에 대하여 다음과 같이 정의<br><math>\int_0^a f(x) d_q x = a(1-q)\sum_{k=0}^{\infty}q^k f(aq^k )</math><br><math>\int_0^{\infty} f(x) d_q x =(1-q)\sum_{k=-\infty}^{\infty}q^k f(aq^k )</math><br> | * <math>0<q<1</math>에 대하여 다음과 같이 정의<br><math>\int_0^a f(x) d_q x = a(1-q)\sum_{k=0}^{\infty}q^k f(aq^k )</math><br><math>\int_0^{\infty} f(x) d_q x =(1-q)\sum_{k=-\infty}^{\infty}q^k f(aq^k )</math><br> | ||
* <math>q\to 1</math> 이면, <math>\int_0^a f(x) d_q x \to \int_0^a f(x) dx </math><br> | * <math>q\to 1</math> 이면, <math>\int_0^a f(x) d_q x \to \int_0^a f(x) dx </math><br> | ||
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<math>\Psi(z)=\exp(\frac{\operatorname{Li}_{2,q}(z)}{q-1})</math> | <math>\Psi(z)=\exp(\frac{\operatorname{Li}_{2,q}(z)}{q-1})</math> | ||
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<math>\operatorname{Li}_{2,q}(z) = -\int_0^z{{\ln (1-t)}\over t} d_{q}t </math> | <math>\operatorname{Li}_{2,q}(z) = -\int_0^z{{\ln (1-t)}\over t} d_{q}t </math> | ||
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* In Weyl algebra, the following identity holds<br><math>(v)_{\infty}(u)_{\infty}=(u)_{\infty}(-vu)_{\infty}(v)_{\infty}</math><br> | * In Weyl algebra, the following identity holds<br><math>(v)_{\infty}(u)_{\infty}=(u)_{\infty}(-vu)_{\infty}(v)_{\infty}</math><br> | ||
* [[1 manufacturing matrices from lower ranks|manufacturing matrices from lower ranks]]<br> | * [[1 manufacturing matrices from lower ranks|manufacturing matrices from lower ranks]]<br> | ||
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* [http://dx.doi.org/10.1088/0305-4470/28/8/014 Remarks on the quantum dilogarithm]<br> | * [http://dx.doi.org/10.1088/0305-4470/28/8/014 Remarks on the quantum dilogarithm]<br> | ||
** V V Bazhanov and N Yu Reshetikhin, 1995 J. Phys. A: Math. Gen. 28 2217 | ** V V Bazhanov and N Yu Reshetikhin, 1995 J. Phys. A: Math. Gen. 28 2217 | ||
| + | * [http://dx.doi.org/10.1142/S0217732395001526 A link invariant from quantum dilogarithm]<br> | ||
| + | ** Kashaev, R. M., Modern Phys. Lett. A 10 (1995), 1409–1418 | ||
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* [http://dx.doi.org/10.1142/S0217732394003610 Quantum Dilogarithm as a 6j-Symbol]<br> | * [http://dx.doi.org/10.1142/S0217732394003610 Quantum Dilogarithm as a 6j-Symbol]<br> | ||
** R. M. Kashaev, MPLA [http://www.worldscinet.com/mpla/mkt/archive.shtml?1994&9 Volume: 9, ][http://www.worldscinet.com/mpla/09/0940/S02177323940940.html Issue: 40](1994) pp. 3757-3768<br> | ** R. M. Kashaev, MPLA [http://www.worldscinet.com/mpla/mkt/archive.shtml?1994&9 Volume: 9, ][http://www.worldscinet.com/mpla/09/0940/S02177323940940.html Issue: 40](1994) pp. 3757-3768<br> | ||
2010년 7월 30일 (금) 22:12 판
introduction
quantum plane
- also called the Weyl algebra
- noncommutative geometry
- \(uv=qvu\)
q-integral (Jackson integral)
- \(0<q<1\)에 대하여 다음과 같이 정의
\(\int_0^a f(x) d_q x = a(1-q)\sum_{k=0}^{\infty}q^k f(aq^k )\)
\(\int_0^{\infty} f(x) d_q x =(1-q)\sum_{k=-\infty}^{\infty}q^k f(aq^k )\) - \(q\to 1\) 이면, \(\int_0^a f(x) d_q x \to \int_0^a f(x) dx \)
quantum dilogarithm
\(\Psi(z)=\prod_{n=0}^{\infty}(1-zq^n)=\sum_{n\geq 0}\frac{(-1)^nq^{n(n-1)/2}}{(1-q)(1-q^2)\cdots(1-q^n)} z^n\)
\(\Psi(z)=\exp(\frac{\operatorname{Li}_{2,q}(z)}{q-1})\)
\(\operatorname{Li}_{2,q}(z) = -\int_0^z{{\ln (1-t)}\over t} d_{q}t \)
\(\operatorname{Li}_2(z) = -\int_0^z{{\ln (1-t)}\over t} dt \)
asymptotics
- \(q=e^{-t}\) and as the t goes 0 (i.e. as q goes to 1)
\(\sum_{n=0}^{\infty}\frac{q^{\frac{A}{2}n^2+cn}}{(q)_n}\sim\exp(\frac{C}{t})\)
where C= sum of Rogers dilogarithms
quantum 5-term relation
- In Weyl algebra, the following identity holds
\((v)_{\infty}(u)_{\infty}=(u)_{\infty}(-vu)_{\infty}(v)_{\infty}\) - manufacturing matrices from lower ranks
history
- Boson and Fermion summation form
- asymptotic analysis of basic hypergeometric series
- Quantum groups
- Kashaev's volume Conjecture
encyclopedia
- q-적분
- 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=
[[4909919|]]
articles
-
- Quantum dilogarithm.
- Zudilin, W., Preprint, Bonn and Moscow (2006), 8 pages
- The hyperbolic volume of knots from quantum dilogarithm
- R. M. Kashaev, 1996
- Remarks on the quantum dilogarithm
- V V Bazhanov and N Yu Reshetikhin, 1995 J. Phys. A: Math. Gen. 28 2217
- A link invariant from quantum dilogarithm
- Kashaev, R. M., Modern Phys. Lett. A 10 (1995), 1409–1418
- Quantum Dilogarithm as a 6j-Symbol
- R. M. Kashaev, MPLA Volume: 9, Issue: 40(1994) pp. 3757-3768
- R. M. Kashaev, MPLA Volume: 9, Issue: 40(1994) pp. 3757-3768
- Quantum Dilogarithm
- L.D.Fadeev and R.M.Kashaev, Mod. Phys. Lett. A. 9 (1994) p.427–434
- http://ncatlab.org/nlab/show/quantum+dilogarithm
- 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[1]
- 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/10.1023/A:1007364912784
question and answers(Math Overflow)
blogs
experts on the field