"Rank of partition and mock theta conjecture"의 두 판 사이의 차이

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imported>Pythagoras0
 
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2번째 줄: 2번째 줄:
  
 
* [[3rd order mock theta functions]] :<math>f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+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>
 
* [[3rd order mock theta functions]] :<math>f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+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>
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*  coefficients<br> 1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244<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>
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*  coefficients 1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244
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* use mathematica '[[integer partitions]]' or the following
 
  
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# Series[(1 + 4 Sum[(-1)^n q^(n (3 n + 1)/2)/(1 + q^n), {n, 1, 10}])/<br> Sum[(-1)^n q^(n (3 n + 1)/2), {n, -8, 8}], {q, 0, 100}]
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==Andrews-Dragonette==
 
==Andrews-Dragonette==
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* '''[Dragonette1952]''' and '''[Andrews1966]'''
 
* '''[Dragonette1952]''' and '''[Andrews1966]'''
 
* concerns the question of partitions with even rank and odd rank
 
* concerns the question of partitions with even rank and odd rank
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*  rank of partition =  largest part - number of parts<br> 9의 분할인 {7,1,1}의 경우, rank=7-3=4<br> 9의 분할인 {4,3,1,1}의 경우, rank=4-4=0<br>
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*  rank of partition = largest part - number of parts 9의 분할인 {7,1,1}의 경우, rank=7-3=4 9의 분할인 {4,3,1,1}의 경우, rank=4-4=0
 
* <math>N_e(n), N_o(n)</math> number of partition with even rank and odd rank
 
* <math>N_e(n), N_o(n)</math> number of partition with even rank and odd rank
 
* <math>p(n)=N_e(n)+N_o(n)</math>
 
* <math>p(n)=N_e(n)+N_o(n)</math>
 
* <math>\alpha(n)=N_e(n)-N_o(n)</math>
 
* <math>\alpha(n)=N_e(n)-N_o(n)</math>
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*  this is in fact the coefficient of mock theta function<br><math>f(q) = \sum_{n\ge 0} \alpha(n)q^n</math><br>
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*  this is in fact the coefficient of the [[3rd order mock theta functions]]
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:<math>f(q) = \sum_{n\ge 0} \alpha(n)q^n</math>
 
* thus we need modularity of f(q) to get exact formula for <math>\alpha(n)</math> as <math>p(n)</math> was obtained by the circle method
 
* thus we need modularity of f(q) to get exact formula for <math>\alpha(n)</math> as <math>p(n)</math> was obtained by the circle method
  
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==harmonic Maass form of weight 1/2==
 
==harmonic Maass form of weight 1/2==
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* Zweger's completion
 
* Zweger's completion
  
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==construction of the Maass-Poincare series==
 
==construction of the Maass-Poincare series==
  
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==generalization==
 
==generalization==
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* crank
 
* crank
  
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==history==
 
==history==
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* http://www.google.com/search?hl=en&tbs=tl:1&q=
 
* http://www.google.com/search?hl=en&tbs=tl:1&q=
  
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==related items==
 
==related items==
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* [[3rd order mock theta functions]]
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* [[Dyson rank generating function]]
  
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==books==
 
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* [[4909919|찾아볼 수학책]]<br>
 
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* http://gigapedia.info/1/mock+theta
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://gigapedia.info/1/
 
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* http://www.amazon.com/s/ref=nb_ss_gw?url=search-alias%3Dstripbooks&field-keywords=
 
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==encyclopedia==
 
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* http://ko.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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* http://en.wikipedia.org/wiki/
 
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==question and answers(Math Overflow)==
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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* http://mathoverflow.net/search?q=
 
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==blogs==
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==computational resource==
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* https://docs.google.com/file/d/0B8XXo8Tve1cxSzZNSVVMNFZIUzg/edit
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* http://oeis.org/A000025
  
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* [http://www.maa.org/news/030807puzzlesolved.html Puzzle Solved: Ramanujan's Mock Theta Conjectures]<br> 구글 블로그 검색<br>
 
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** http://blogsearch.google.com/blogsearch?q=
 
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** http://blogsearch.google.com/blogsearch?q=
 
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** http://blogsearch.google.com/blogsearch?q=
 
  
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==expositions==
 
==expositions==
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*  [http://link.springer.com/article/10.1023%2FA%3A1026224002193?LI=true Partitions : at the interface of q-series and modular forms] Andrews, George E., 2003<br>
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* [http://www.maa.org/news/030807puzzlesolved.html Puzzle Solved: Ramanujan's Mock Theta Conjectures]
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*  [http://link.springer.com/article/10.1023%2FA%3A1026224002193?LI=true Partitions : at the interface of q-series and modular forms] Andrews, George E., 2003
  
  
115번째 줄: 70번째 줄:
 
==articles==
 
==articles==
  
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* [http://dx.doi.org/10.1007/s00222-005-0493-5 The f(q) mock theta function conjecture and partition ranks] Kathrin Bringmann and Ken Ono, Inventiones Mathematicae Volume 165, Number 2, 2006<br>
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* [http://dx.doi.org/10.1007/s00222-005-0493-5 The f(q) mock theta function conjecture and partition ranks] Kathrin Bringmann and Ken Ono, Inventiones Mathematicae Volume 165, Number 2, 2006
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* '''[Dragonette1952]'''[http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan]<br>
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* George E. Andrews and F. G. Garvan, [http://dx.doi.org/10.1016/0001-8708%2889%2990070-4 Ramanujan's “Lost” Notebook VI: The mock theta conjectures] 1989
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** Dragonette, Leila A. (1952), Transactions of the American Mathematical Society 72: 474–500
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* Hickerson, Dean, <cite class="" id="CITEREFHickerson1988" style="line-height: 2em; font-style: normal;">A proof of the mock theta conjectures</cite> (1988), <cite style="line-height: 2em; font-style: normal;"><em style="line-height: 2em;">[http://en.wikipedia.org/wiki/Inventiones_Mathematicae Inventiones Mathematicae]</em> '''94''' (3): 639–660, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1007%2FBF01394279 10.1007/BF01394279], [http://en.wikipedia.org/wiki/Mathematical_Reviews MR][http://www.ams.org/mathscinet-getitem?mr=969247 969247], [http://en.wikipedia.org/wiki/International_Standard_Serial_Number ISSN] [http://worldcat.org/issn/0020-9910 0020-9910]</cite>
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* '''[Andrews1966]'''[http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions]<br>
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* '''[Andrews1966]'''[http://dx.doi.org/10.2307%2F2373202 On the theorems of Watson and Dragonette for Ramanujan's mock theta functions]
 
** Andrews, George E. (1966), American Journal of Mathematics 88: 454–490
 
** Andrews, George E. (1966), American Journal of Mathematics 88: 454–490
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* <cite class="" id="CITEREFWatson1936" style="line-height: 2em; font-style: normal;">Watson, G. N. (1936), "The Final Problem : An Account of the Mock Theta Functions", <em style="line-height: 2em;">J. London Math. Soc.</em> '''11''': 55–80, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1112%2Fjlms%2Fs1-11.1.55 10.1112/jlms/s1-11.1.55]</cite>
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* '''[Dragonette1952]'''[http://dx.doi.org/10.2307%2F1990714 Some asymptotic formulae for the mock theta series of Ramanujan]
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* <cite class="" id="CITEREFWatson1937" style="line-height: 2em; font-style: normal;">Watson, G. N. (1937), "The Mock Theta Functions (2)", <em style="line-height: 2em;">Proc. London Math. Soc.</em> '''s2-42''': 274–304, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1112%2Fplms%2Fs2-42.1.274 10.1112/plms/s2-42.1.274]</cite>
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** Dragonette, Leila A. (1952), Transactions of the American Mathematical Society 72: 474–500
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*  
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* <cite class="" id="CITEREFWatson1937" style="line-height: 2em; font-style: normal;">Watson, G. N. (1937), "The Mock Theta Functions (2)", <em style="line-height: 2em;">Proc. London Math. Soc.</em> '''s2-42''': 274–304, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1112%2Fplms%2Fs2-42.1.274 10.1112/plms/s2-42.1.274]</cite>
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*  George E. Andrews and F. G. Garvan, [http://dx.doi.org/10.1016/0001-8708%2889%2990070-4 Ramanujan's “Lost” Notebook VI: The mock theta conjectures] 1989<br>
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* <cite class="" id="CITEREFWatson1936" style="line-height: 2em; font-style: normal;">Watson, G. N. (1936), "The Final Problem : An Account of the Mock Theta Functions", <em style="line-height: 2em;">J. London Math. Soc.</em> '''11''': 55–80, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1112%2Fjlms%2Fs1-11.1.55 10.1112/jlms/s1-11.1.55]</cite>
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* Hickerson, Dean, <cite class="" id="CITEREFHickerson1988" style="line-height: 2em; font-style: normal;">A proof of the mock theta conjectures</cite> (1988), <cite style="line-height: 2em; font-style: normal;"><em style="line-height: 2em;">[http://en.wikipedia.org/wiki/Inventiones_Mathematicae Inventiones Mathematicae]</em> '''94''' (3): 639–660, [http://en.wikipedia.org/wiki/Digital_object_identifier doi]:[http://dx.doi.org/10.1007%2FBF01394279 10.1007/BF01394279], [http://en.wikipedia.org/wiki/Mathematical_Reviews MR][http://www.ams.org/mathscinet-getitem?mr=969247 969247], [http://en.wikipedia.org/wiki/International_Standard_Serial_Number ISSN] [http://worldcat.org/issn/0020-9910 0020-9910]</cite>
 
  
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* [[2010년 books and articles|논문정리]]
 
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* http://www.ams.org/mathscinet
 
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* http://www.zentralblatt-math.org/zmath/en/
 
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* http://pythagoras0.springnote.com/
 
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* [http://math.berkeley.edu/%7Ereb/papers/index.html http://math.berkeley.edu/~reb/papers/index.html]
 
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* http://front.math.ucdavis.edu/search?a=&t=&c=&n=40&s=Listings&q=
 
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* 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=
 
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* http://dx.doi.org/10.1007/s00222-005-0493-5
 
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==experts on the field==
 
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* http://arxiv.org/
 
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==TeX ==[[분류:개인노트]]
 
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[[분류:math and physics]]
 
 
[[분류:math and physics]]
 
[[분류:math and physics]]
 
[[분류:mock modular forms]]
 
[[분류:mock modular forms]]
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==메타데이터==
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===위키데이터===
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* ID :  [https://www.wikidata.org/wiki/Q7293214 Q7293214]
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===Spacy 패턴 목록===
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* [{'LOWER': 'rank'}, {'LOWER': 'of'}, {'LOWER': 'a'}, {'LEMMA': 'partition'}]

2021년 2월 17일 (수) 02:26 기준 최신판

order 3 Ramanujan mock theta function

  • 3rd order mock theta functions \[f(q) = \sum_{n\ge 0} {q^{n^2}\over (-q;q)_n^2} =1+\sum_{n\ge 1} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots{(1+q^{n})^2}}={2\over \prod_{n>0}(1-q^n)}\sum_{n\in Z}{(-1)^nq^{3n^2/2+n/2}\over 1+q^n}\]
  • coefficients 1, 1, -2, 3, -3, 3, -5, 7, -6, 6, -10, 12, -11, 13, -17, 20, -21, 21, -27, 34, -33, 36, -46, 51, -53, 58, -68, 78, -82, 89, -104, 118, -123, 131, -154, 171, -179, 197, -221, 245, -262, 279, -314, 349, -369, 398, -446, 486, -515, 557, -614, 671, -715, 767, -845, 920, -977, 1046, -1148, 1244


Andrews-Dragonette

  • [Dragonette1952] and [Andrews1966]
  • concerns the question of partitions with even rank and odd rank
  • rank of partition = largest part - number of parts 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 the 3rd order mock theta functions

\[f(q) = \sum_{n\ge 0} \alpha(n)q^n\]

  • thus we need modularity of f(q) to get exact formula for \(\alpha(n)\) as \(p(n)\) was obtained by the circle method



harmonic Maass form of weight 1/2

  • Zweger's completion



construction of the Maass-Poincare series

generalization

  • crank



history



related items


computational resource


expositions


articles

메타데이터

위키데이터

Spacy 패턴 목록

  • [{'LOWER': 'rank'}, {'LOWER': 'of'}, {'LOWER': 'a'}, {'LEMMA': 'partition'}]
원본 주소 "https://wiki.mathnt.net/index.php?title=Rank_of_partition_and_mock_theta_conjecture&oldid=51746"