فرم ماک ماژولار: تفاوت میان نسخهها
جز Mojtabakd صفحهٔ فرم شبه مدولار را به فرم ماک ماژولار منتقل کرد: اصلاح، «شبه» بر اساس چه منبعی معادل ماک هست؟ |
#1Lib1Ref & #1Lib1RefIran |
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خط ۱: | خط ۱: | ||
در ریاضیات، '''فرم ماک ماژولار''' {{انگلیسی|Mock Modular Form}}، بخش هولومورفیک از یک فرم ماس (Maass Form) ضعیف هارمونیک بوده و یک '''تابع تتای ماک''' اساساً فرم ماژولار ماکی با وزن {{sfrac|1|2}} است. اولین مثالها از توابع تتای ماک توسط رامانوجان سرینیواسا در آخرین نامهاش در سال ۱۹۲۰ به گ.هـ. هاردی و در دفترچهٔ گمشدهاش توصیف شدهاست. سندر زوگر کشف نمود که افزودن برخی از توابع غیر-هولومورفیک بهشان، آنها را تبدیل به فرمهای ماس ضعیف میکند.{{sfn|Zwegers|2001}}{{sfn|Zwegers|2002}} |
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{{منبع|تاریخ=مه ۲۰۲۰}} |
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در [[ریاضیات]] یک '''فرم شبه مدولار''' قسمت [[تابع تحلیلی مختلط|هولومورفیک]] از یک فرم ماس هارمونیک ضعیف است و یک '''تابع شبه تتا''' اساساً یک فرم شبه مدولار با وزن ۱/۲ است. اولین مثال ها از توابع شبه تتا توسط [[سرینیواسا رامانوجان]] در آخرین نامه اش به [[گادفری هارولد هاردی]] در سال ۱۹۲۰ و در دفترچه گمشده اش توصیف شده اند. |
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== تاریخچه == |
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{{quote box |
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| align=right |
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| width=33% |
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| quote= «فرض کنید تابعی به فرم اویلری وجود داشته و فرض کنید که تمام نقاط یا بینهایت از آنها، نقاط تکین نمایی باشند، و همچنین فرض کنید که در این نقاط، فرم مجانبی با همان ترتیب و سادگی موارد (A) و (B) نزدیک شود. حال سؤال این است که: آیا این تابع به صورت جمع دو تابعی است که یکی از آنها تابع <math>\theta</math> و دیگری تابع (بدیهی) <math>O(1)</math> در تمام نقاط <math>e^{2m\pi i/n}</math> میباشد؟ … وقتی چنین نباشد، من به این تابع، تابع <math>\theta</math>ی ماک میگویم.» |
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| source=تعریف اصلی رامانوجان از تابع تتای ماک{{sfn|Ramanujan|2000|loc=Appendix II}} |
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}} |
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نامه ۱۲ ژانویه ۱۹۲۰ میلادی رامانوجان به هاردی،{{sfn|Ramanujan|2000|loc=Appendix II}} 17 مثال از توابعی را که او به نام توابع تتای ماک مینامید، فهرست کرده، و دفترچه گمشده اش{{sfn|Ramanujan|1988}} شامل چندین مثال دیگر نیز میباشد (منظور رامانوجان از اصطلاح «تابع تتا»، چیزیست که امروزه به یک فرم ماژولار معروف است). رامانوجان اشاره کرد که این توابع دارای بسط مجانبی در کاسپها بوده، که با فرمهای ماژولاری با وزن {{sfn|1|2}} شباهت داشته، به طوری که احتمالاً دارای قطبهایی در کاسپها بوده اما نمیتوان آن]ا را برحسب توابع تتای «معمولی» نوشت. او توابعی با خواص مشابه را «توابع تتای ماک» نامید. زووگرز (Zwegers)، بعدها ارتباط تابع تتای ماک با فرمهای ماس ضعیف کشف نمود. |
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رامانوجان به توابع تتای ماکش، '''مرتبه''' (order) نسبت داد، که تعریف واضح و مشخصی نداشت. قبل از کار زووگرز، مراتب توابع تتای ماک شناخته شده شامل این موارد بودند: |
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{{وسطچین}} |
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<math>3, 5, 6, 7, 8, 10. |
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{{پایان وسطچین}} |
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مفهوم رامانوجان از مرتبه، بعدها تبدیل به کونداکتورِ کاراکتر نبنتیپوس (Nebentypus character) با فرمهای ماس هارمونیک از وزن {{sfn|1|2}} شد که توابع تتای ماک رامانوجان را به عنوان تصویرهای هولومورفیک میپذیرند. |
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== ارجاعات == |
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{{پانویس|چپچین=بله}} |
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== منابع == |
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{{چپچین}} |
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== برای مطالعه بیشتر == |
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* {{Citation| chapter = Mock theta functions, ranks and Maass forms |
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| last = Ono | first = Ken | year = 2008 |
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| author-link = Ken Ono |
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{{پایان چپچین}} |
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== پیوند به بیرون == |
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* [https://web.archive.org/web/20081221143605/http://www.mpim-bonn.mpg.de/Events/This+Year+and+Prospect/Mock+theta+functions/ International Conference: Mock theta functions and applications 2009] |
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* [https://web.archive.org/web/20081204111537/http://www.math.psu.edu/andrews/biblio.html Papers on mock theta functions] by [[George Andrews (mathematician)|George Andrews]] |
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* [http://www.mi.uni-koeln.de/~kbringma/papers.html Papers on mock theta functions] by [[Kathrin Bringmann]] |
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* [https://web.archive.org/web/20100620082612/http://www.math.wisc.edu/~ono/reprints/index.html Papers on mock theta functions] by [[Ken Ono]] |
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* [https://web.archive.org/web/20081023234404/http://mathsci.ucd.ie/~zwegers/ Papers on mock theta functions] by [[Sander Zwegers]] |
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* {{MathWorld|urlname=MockThetaFunction|title=Mock Theta Function}} |
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[[رده:فرمهای مدولار]] |
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[[رده:Q-آنالوگها]] |
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[[رده:سرینیواسا رامانوجان]] |
[[رده:سرینیواسا رامانوجان]] |
نسخهٔ ۱۵ مهٔ ۲۰۲۱، ساعت ۱۲:۵۵
در ریاضیات، فرم ماک ماژولار (به انگلیسی: Mock Modular Form)، بخش هولومورفیک از یک فرم ماس (Maass Form) ضعیف هارمونیک بوده و یک تابع تتای ماک اساساً فرم ماژولار ماکی با وزن 1/2 است. اولین مثالها از توابع تتای ماک توسط رامانوجان سرینیواسا در آخرین نامهاش در سال ۱۹۲۰ به گ.هـ. هاردی و در دفترچهٔ گمشدهاش توصیف شدهاست. سندر زوگر کشف نمود که افزودن برخی از توابع غیر-هولومورفیک بهشان، آنها را تبدیل به فرمهای ماس ضعیف میکند.[۱][۲]
تاریخچه
«فرض کنید تابعی به فرم اویلری وجود داشته و فرض کنید که تمام نقاط یا بینهایت از آنها، نقاط تکین نمایی باشند، و همچنین فرض کنید که در این نقاط، فرم مجانبی با همان ترتیب و سادگی موارد (A) و (B) نزدیک شود. حال سؤال این است که: آیا این تابع به صورت جمع دو تابعی است که یکی از آنها تابع و دیگری تابع (بدیهی) در تمام نقاط میباشد؟ … وقتی چنین نباشد، من به این تابع، تابع ی ماک میگویم.»
تعریف اصلی رامانوجان از تابع تتای ماک[۳]
نامه ۱۲ ژانویه ۱۹۲۰ میلادی رامانوجان به هاردی،[۳] 17 مثال از توابعی را که او به نام توابع تتای ماک مینامید، فهرست کرده، و دفترچه گمشده اش[۴] شامل چندین مثال دیگر نیز میباشد (منظور رامانوجان از اصطلاح «تابع تتا»، چیزیست که امروزه به یک فرم ماژولار معروف است). رامانوجان اشاره کرد که این توابع دارای بسط مجانبی در کاسپها بوده، که با فرمهای ماژولاری با وزن [۵] شباهت داشته، به طوری که احتمالاً دارای قطبهایی در کاسپها بوده اما نمیتوان آن]ا را برحسب توابع تتای «معمولی» نوشت. او توابعی با خواص مشابه را «توابع تتای ماک» نامید. زووگرز (Zwegers)، بعدها ارتباط تابع تتای ماک با فرمهای ماس ضعیف کشف نمود.
رامانوجان به توابع تتای ماکش، مرتبه (order) نسبت داد، که تعریف واضح و مشخصی نداشت. قبل از کار زووگرز، مراتب توابع تتای ماک شناخته شده شامل این موارد بودند:
<math>3, 5, 6, 7, 8, 10.
مفهوم رامانوجان از مرتبه، بعدها تبدیل به کونداکتورِ کاراکتر نبنتیپوس (Nebentypus character) با فرمهای ماس هارمونیک از وزن [۵] شد که توابع تتای ماک رامانوجان را به عنوان تصویرهای هولومورفیک میپذیرند.
ارجاعات
- ↑ Zwegers 2001.
- ↑ Zwegers 2002.
- ↑ ۳٫۰ ۳٫۱ Ramanujan 2000, Appendix II.
- ↑ Ramanujan 1988.
- ↑ ۵٫۰ ۵٫۱ 1 & 2.
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برای مطالعه بیشتر
- Ono, Ken (2008), "Mock theta functions, ranks and Maass forms", in Alladi, Krishnaswami (ed.), Surveys in Number Theory, Developments in Mathematics, vol. 17, Springer-Verlag, pp. 119–141, ISBN 978-0-387-78509-7, Zbl 1183.11064
پیوند به بیرون
- International Conference: Mock theta functions and applications 2009
- Papers on mock theta functions by George Andrews
- Papers on mock theta functions by Kathrin Bringmann
- Papers on mock theta functions by Ken Ono
- Papers on mock theta functions by Sander Zwegers
- Weisstein, Eric W. "Mock Theta Function". MathWorld.