"Rank of partition and mock theta conjecture"의 두 판 사이의 차이
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| 1번째 줄: | 1번째 줄: | ||
<h5>order 3 Ramanujan mock theta function</h5> | <h5>order 3 Ramanujan mock theta function</h5> | ||
| − | * <math>f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} | + | * <math>f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} {q^{n^2}\over (-q;q)_n^2}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}</math><br>[http://www.research.att.com/%7Enjas/sequences/A000025 http://www.research.att.com/~njas/sequences/A000025]<br>[http://www.research.att.com/%7Enjas/sequences/b000025.txt http://www.research.att.com/~njas/sequences/b000025.txt]<br> |
| 23번째 줄: | 23번째 줄: | ||
<h5>harmonic Maass form of weight 1/2</h5> | <h5>harmonic Maass form of weight 1/2</h5> | ||
| − | + | * Zweger's completion | |
| 29번째 줄: | 29번째 줄: | ||
| − | <h5>Maass-Poincare series</h5> | + | <h5>construction of the Maass-Poincare series</h5> |
| 103번째 줄: | 103번째 줄: | ||
<h5>articles[http://www.maa.org/news/030807puzzlesolved.html ]</h5> | <h5>articles[http://www.maa.org/news/030807puzzlesolved.html ]</h5> | ||
| − | * | + | * [http://www.springerlink.com/content/5524655155350464/ The f(q) mock theta function conjecture and partition ranks]<br> |
** Inventiones Mathematicae, 2006 | ** Inventiones Mathematicae, 2006 | ||
* [http://www.ingentaconnect.com/content/klu/rama/2003/00000007/F0030001/05142410 Partitions : at the interface of q-series and modular forms]<br> | * [http://www.ingentaconnect.com/content/klu/rama/2003/00000007/F0030001/05142410 Partitions : at the interface of q-series and modular forms]<br> | ||
** Andrews, George E., 2003<br> | ** Andrews, George E., 2003<br> | ||
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* '''[Dragonette1952]'''[http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan]<br> | * '''[Dragonette1952]'''[http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan]<br> | ||
2010년 3월 3일 (수) 18:35 판
order 3 Ramanujan mock theta function
- \(f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} {q^{n^2}\over (-q;q)_n^2}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\)
http://www.research.att.com/~njas/sequences/A000025
http://www.research.att.com/~njas/sequences/b000025.txt
Andrews-Dragonette
- [Dragonette1952] and [Andrews1966]
- rank of partition
분할의 rank = 분할에서 가장 큰 수 - 분할의 크기
9의 분할인 {7,1,1}의 경우, rank=7-3=4
9의 분할인 {4,3,1,1}의 경우, rank=4-4=0 - \(N_e(n), N_o(n)\) number of partition with even rank and odd rank
- \(p(n)=N_e(n)+N_o(n)\)
- \(\alpha(n)=N_e(n)-N_o(n)\)
- this is in fact the coefficient of mock theta function
\(f(q) = \sum_{n\ge 0} \alpha(n)q^n\) - thus we need modularity of f(q) to get exact formula for \(\alpha(n)\) as \(p(n)\) obtained by the circle method
harmonic Maass form of weight 1/2
- Zweger's completion
construction of the Maass-Poincare series
generalization
history
books
- 찾아볼 수학책
- http://gigapedia.info/1/mock+theta
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://gigapedia.info/1/
- http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
encyclopedia
- http://ko.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- http://en.wikipedia.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
question and answers(Math Overflow)
- http://mathoverflow.net/search?q=
- http://mathoverflow.net/search?q=
- http://mathoverflow.net/search?q=
blogs
articles[1]
- The f(q) mock theta function conjecture and partition ranks
- Inventiones Mathematicae, 2006
- Partitions : at the interface of q-series and modular forms
- Andrews, George E., 2003
- Andrews, George E., 2003
- [Dragonette1952]Some asymptotic formulae for the mock theta series of Ramanujan
- Dragonette, Leila A. (1952), Transactions of the American Mathematical Society 72: 474–500
-
[Andrews1966]On the theorems of Watson and Dragonette for Ramanujan's mock theta functions
- Andrews, George E. (1966), American Journal of Mathematics 88: 454–490
- 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
- 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/
experts on the field