Papyrus 14 - Revision history

en>Joncampo2011: Changed the date from 5th century to 6th century, as per information found here: http://ntvmr.uni-muenster.de/liste/?ObjID=10014

2014-07-28T20:49:27Z

Changed the date from 5th century to 6th century, as per information found here: http://ntvmr.uni-muenster.de/liste/?ObjID=10014

← Older revision		Revision as of 22:49, 28 July 2014
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	~~In mathematics~~, the '~~''Dyson conjecture''' {{harvs \|first=Freeman \|last=Dyson \|year=1962 \|authorlink=Freeman Dyson}} is a conjecture~~ about ~~the constant term of certain~~ [~~[Laurent polynomial]~~]s, ~~proved by~~ [~~[Kenneth G~~. ~~Wilson\|Wilson~~]~~] and Gunson~~. ~~Andrews generalized it to the '''q~~-~~Dyson conjecture''', proved by Zeilberger and Bressoud and sometimes called the '''Zeilberger–Bressoud theorem'''~~. ~~Macdonald generalized it further to more general root systems with the '''Macdonald constant term conjecture''', proved~~ by ~~Cherednik~~.		I'm a 33 years old, married and study at the high school (Educational Policy Studies).<br>In my spare time I teach myself German. I've been twicethere and look forward to returning anytime soon. I love to read, preferably on my kindle. I really love to watch The Big Bang Theory and 2 Broke Girls as well as docus about anything geological. I like Stone collecting.<br>xunjie 国内の人々に影響を与える。
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			デザイナーのクリストフ·ルメールチップの材料にさらに注意を比較した。 [http://www.tobler-verlag.ch/flash/ja/li/top/gaga/ ��ߥ�� rӋ ��] バングラデシュの衣服の輸出は130億ドルのターゲットバングラデシュの4,000人以上の縫製工場が、
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			外の美しさの気質のバージョンは、
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			財布やiPhoneケースを含む単一の特殊な製品の彼の英国風の創作を販売しています。 [http://belpars.by/Files/Config/tp/new/best/toms.html �ȥॺ ��<br><br> ��å�]

	~~==Dyson conjecture==~~		my website; [http://www.steadfast-hawaii.org/img/p/hot/chloe.php クロエ財布]

	~~The Dyson conjecture states that the [[Laurent polynomial]]~~

	~~:<math>\prod _{1\le i\ne j\le n}(1-t_i/t_j)^{a_i}</math>~~

	~~has constant term~~

	~~:<math>\frac{(a_1+a_2+\cdots+a_n)!}{a_1!a_2!\cdots a_n!}.</math>~~

	The conjecture was first proved independently by {{harvtxt\|Wilson\|1962}} and {{harvtxt\|Gunson\|1962}}. {{harvtxt\|Good\|1970}} later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations

	~~:<math>F(a_1,\dots,a_n) = \sum_{i=1}^nF(a_1,\dots,a_i-1,\dots,a_n).</math>~~

	~~The case ''n''&nbsp~~;~~= 3 of Dyson's conjecture follows from the~~ [~~[Dixon identity]].~~

	~~{{harvtxt\|Sills\|Zeilberger\|2006}} and {{harv\|Sills\|2006}} used a computer to find expressions for non-constant coefficients of~~
	~~Dyson's Laurent polynomial.~~

	~~==Dyson integral==~~
	~~When all the values ''a''<sub>''i''</sub> are equal to β/2, the constant term in Dyson's conjecture is the value of '''Dyson's integral'''~~

	:~~<math>\frac{1}{(2\pi)^n}\int_0^{2\pi}\cdots\int_0^{2\pi}\prod_{1\le j<k\le n}\|e^{i\theta_j}-e^{i\theta_k}\|^\beta \, d\theta_1\cdots d\theta_n.<~~/~~math>~~

	~~Dyson's integral is a special case of [[Selberg's integral]] after a change of variable and has value~~

	~~:<math>\frac{\Gamma(1+\beta n~~/~~2)}{\Gamma(1+\beta/2)^n}</math>~~

	~~which gives another proof of Dyson's conjecture in this special case~~.

	~~==''q''~~-~~Dyson conjecture==~~
	~~{{harvtxt\|Andrews\|1975}} found a [[q-analog]] of Dyson's conjecture, stating that the constant term of~~
	~~:<math>\prod_{1\le i<j\le n}\left(\frac{x_i}{x_j};q\right)_{a_i}\left(\frac{qx_j}{x_i};q\right)_{a_j}</math>~~
	is
	~~:<math>\frac{(q;q)_{a_1+\cdots+a_n}}{(q;q)_{a_1}\cdots(q;q)_{a_n}}.</math>~~
	~~Here (''a'';''q'')<sub>''n''</sub> is the [[q-Pochhammer symbol]].~~
	~~This conjecture reduces to Dyson's conjecture for ''q''=1, and was proved by {{harvtxt\|Zeilberger\|Bressoud\|1985}}.~~

	~~==Macdonald conjectures==~~
	~~{{harvtxt\|Macdonald\|1982}} extended the conjecture to arbitrary finite or affine [[root system]]s, with Dyson's original conjecture corresponding to~~
	the case of the ''A''<sub>''n''−1</sub> root system and Andrews's conjecture corresponding to the affine ''A''<sub>''n''−1</sub> root system. Macdonald reformulated these conjectures as conjectures about the norms of [[Macdonald polynomial]]s. Macdonald's conjectures were proved by {{harv\|Cherednik\|1995}} using doubly affine Hecke algebras.

	~~[[Ian G. Macdonald\|Macdonald]]'s form of Dyson's conjecture for root systems of type BC is closely related to [[Selberg's integral]].~~

	~~==References==~~
	*{{Citation \| authorlink=George Andrews (mathematician) \| last1=Andrews \| first1=George E. \| title=Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) \| publisher=[[Academic Press]] \| location=Boston, MA \| mr=0399528 \| year=1975 \| chapter=Problems and prospects for basic hypergeometric functions \| pages=191–224}}
	*{{citation \|jstor=2118632 \|pages=191–216 \|last1=Cherednik \|first1=I. \|title=Double Affine Hecke Algebras and Macdonald's Conjectures \|volume=141 \|issue=1 \|journal=The Annals of Mathematics \|year=1995 \|doi=10.2307/2118632}}
	*{{Citation \| last1=Dyson \| first1=Freeman J. \| title=Statistical theory of the energy levels of complex systems. I \| doi=10.1063/1.1703773 \| mr=0143556 \| year=1962 \| journal=[[Journal of Mathematical Physics]] \| issn=0022-2488 \| volume=3 \| pages=140–156}}
	*{{Citation \| last1=Good \| first1=I. J. \| author1-link=I. J. Good \| title=Short proof of a conjecture by Dyson \| doi=10.1063/1.1665339 \| mr=0258644 \| year=1970 \| journal=[[Journal of Mathematical Physics]] \| issn=0022-2488 \| volume=11 \| pages=1884 \| issue=6}}
	*{{Citation \| last1=Gunson \| first1=J. \| title=Proof of a conjecture by Dyson in the statistical theory of energy levels \| doi=10.1063/1.1724277 \| mr=0148401 \| year=1962 \| journal=[[Journal of Mathematical Physics]] \| issn=0022-2488 \| volume=3 \| pages=752–753 \| issue=4}}
	*{{Citation \| last1=Macdonald \| first1=I. G. \| title=Some conjectures for root systems \| doi=10.1137/0513070 \| mr=674768 \| year=1982 \| journal=SIAM Journal on Mathematical Analysis \| issn=0036-1410 \| volume=13 \| issue=6 \| pages=988–1007}}
	*{{Citation \| last1=Sills \| first1=Andrew V. \| title=Disturbing the Dyson conjecture, in a generally GOOD way \| doi=10.1016/j.jcta.2005.12.005 \| mr=2259066 \| year=2006 \| journal=Journal of Combinatorial Theory. Series A \| issn=1096-0899 \| volume=113 \| issue=7 \| pages=1368–1380}}
	*{{Citation \| last1=Sills \| first1=Andrew V. \| last2=Zeilberger \| first2=Doron \| author2-link=Doron Zeilberger \| title=Disturbing the Dyson conjecture (in a GOOD way) \| url=http://projecteuclid.org/~~euclid.em~~/~~1175789739 \| mr=2253005 \| year=2006 \| journal=Experimental Mathematics \| issn=1058-6458 \| volume=15 \| issue=2 \| pages=187–191 \| doi=10.1080~~/~~10586458.2006.10128959}}~~
	*{{Citation \| last1=Wilson \| first1=Kenneth G. \| author1-link = Kenneth G. Wilson \| title=Proof of a conjecture by Dyson \| doi=10.1063/1.~~1724291 \| mr=0144627 \| year=1962 \| journal=[[Journal of Mathematical Physics]] \| issn=0022-2488 \| volume=3 \| pages=1040–1043 \| issue=5}}~~
	*{{Citation \| last1=Zeilberger \| first1=Doron \| author1-link=Doron Zeilberger \| last2=Bressoud \| first2=David M. \|author2-link=David Bressoud \| title=A proof of Andrews' q-Dyson conjecture \| doi=10.1016/0012-365X(85)90081-0 \| mr=791661 \| year=1985 \| journal=[[Discrete Mathematics (journal)\|Discrete Mathematics]] \| issn=0012-365X \| volume=54 \| issue=2 \| pages=201–224}}

	~~[[Category:Enumerative combinatorics]]~~
	~~[[Category:Algebraic combinatorics]]~~
	~~[[Category:Factorial and binomial topics]]~~
	~~[[Category:Mathematical identities]]~~
	~~[[Category:Conjectures]]~~
	~~[[Category:Freeman Dyson]~~]

en>IRISZOOM: /* Description */ Already stated above.

2014-02-03T21:49:52Z

Description: Already stated above.

← Older revision		Revision as of 23:49, 3 February 2014
Line 1:		Line 1:
	~~It depends on~~ the ~~quality~~ of the ~~Wordpress theme but even if it~~'~~s not a professional one you will be able to average 50~~-~~60$ EACH link. It is used~~ by ~~around 25% of all new websites,~~ and ~~there are~~ more ~~than 27 thousand plugins currently available~~. ~~One really cool features about this amazing and free wp plugin is~~ that the ~~code it generates is completely portable~~. ~~If you're using Wordpress~~ and ~~want to make your blog~~ a ~~"dofollow" blog~~, ~~meaning~~ that ~~links from your blog pass on~~ the ~~benefits of Google pagerank~~, ~~you can install one of~~ the ~~many dofollow plugins available. provided by Word~~ - ~~Press Automatic Upgrade~~, ~~so whenever you need to update the new version does not~~, ~~it automatically creates no webmaster~~. <br>~~<br>Creating a website~~ from ~~scratch can be such~~ a ~~pain~~. ~~If you have any kind of concerns concerning~~ in ~~which and~~ the ~~best way to work with [http~~:~~//www~~.~~pigmag.com~~/it/~~2013~~/09/~~20/%e2%80%9cmake~~-~~every-show-like-it~~-is~~-your-last-show%e2%80%9d-ryan-gander-~~a-~~parigi/ wordpress dropbox backup~~]~~, you'll be able to call us at our web page~~. ~~While direct advertising is limited~~ to ~~few spots in your site~~ and ~~tied to fixed monthly payment~~ by the ~~advertisers~~, ~~affiliate marketing can give you unlimited income as long as you can convert your traffic~~ to ~~sales.~~ A ~~Wordpress plugin is a software that you can install into your Wordpress site. They provide many such popular products which you can buy for your baby. For a Wordpress website, you don~~'~~t need a powerful web hosting account to host your site.~~ <br><br>~~The least difficult~~ and ~~very best way~~ to ~~do this~~ is ~~by acquiring a Word - Press site~~. ~~By using Word - Press MLM websites or blogs, you and your companies presence on the internet can be created swiftly and simply~~. ~~all the necessary planning~~ and ~~steps~~ of ~~conversion is carried out in this phase~~, ~~such as splitting~~, ~~slicing, CSS code~~, ~~adding images~~, ~~header footer etc~~. ~~Every single Theme might be unique~~, ~~providing several alternatives for webpage owners to reap the benefits of in an effort to instantaneously adjust their web page appear. If you have any questions on starting a Word -~~ Press ~~food blog or any blog for that matter~~, ~~please post them below~~ and I ~~will try to answer them~~. ~~<br><br>Word - Press installation is very easy~~ and ~~hassle free. And, make no mistake,India~~'s ~~Fertility Clinics and IVF specialists are amongst~~ the ~~best in the world,and have been for some time~~. ~~Next you'll go by way~~ of ~~to your simple Word~~ - ~~Press site~~. ~~So, we have to add our social media sharing buttons in website~~. ~~Now all you have to do is log into your Word~~ - ~~Press site making use~~ of ~~the very same username and password that you initially had in your previous site~~. ~~<br><br>Every single module contains published data and guidelines, usually a lot more than~~ 1 ~~video, and when pertinent, incentive links and PDF files to assist you out~~. ~~Automated deal feed integration option to populate your blog with relevant deals~~. By the ~~time you get the Gallery Word~~ - ~~Press Themes, the first thing that you should know is~~ on ~~how to install it~~. ~~) Remote Login: With~~ the ~~process~~ of ~~PSD to Wordpress conversion comes~~ the ~~advantage of flexibility~~. ~~I have never seen a plugin with such a massive array~~ of ~~features, this does everything that platinum SEO and All In One SEO, also throws in the functionality found within SEO Smart Links and~~ a ~~number~~ of ~~other plugins it is essentially the swiss army knife~~ of ~~Word~~ - ~~Press plugins~~.		In mathematics, the '''Dyson conjecture''' {{harvs \|first=Freeman \|last=Dyson \|year=1962 \|authorlink=Freeman Dyson}} is a conjecture about the constant term of certain [[Laurent polynomial]]s, proved by [[Kenneth G. Wilson\|Wilson]] and Gunson. Andrews generalized it to the '''q-Dyson conjecture''', proved by Zeilberger and Bressoud and sometimes called the '''Zeilberger–Bressoud theorem'''. Macdonald generalized it further to more general root systems with the '''Macdonald constant term conjecture''', proved by Cherednik.

			==Dyson conjecture==

			The Dyson conjecture states that the [[Laurent polynomial]]

			:<math>\prod _{1\le i\ne j\le n}(1-t_i/t_j)^{a_i}</math>

			has constant term

			:<math>\frac{(a_1+a_2+\cdots+a_n)!}{a_1!a_2!\cdots a_n!}.</math>

			The conjecture was first proved independently by {{harvtxt\|Wilson\|1962}} and {{harvtxt\|Gunson\|1962}}. {{harvtxt\|Good\|1970}} later found a short proof, by observing that the Laurent polynomials, and therefore their constant terms, satisfy the recursion relations

			:<math>F(a_1,\dots,a_n) = \sum_{i=1}^nF(a_1,\dots,a_i-1,\dots,a_n).</math>

			The case ''n'' = 3 of Dyson's conjecture follows from the [[Dixon identity]].

			{{harvtxt\|Sills\|Zeilberger\|2006}} and {{harv\|Sills\|2006}} used a computer to find expressions for non-constant coefficients of
			Dyson's Laurent polynomial.

			==Dyson integral==
			When all the values ''a''<sub>''i''</sub> are equal to β/2, the constant term in Dyson's conjecture is the value of '''Dyson's integral'''

			:<math>\frac{1}{(2\pi)^n}\int_0^{2\pi}\cdots\int_0^{2\pi}\prod_{1\le j<k\le n}\|e^{i\theta_j}-e^{i\theta_k}\|^\beta \, d\theta_1\cdots d\theta_n.</math>

			Dyson's integral is a special case of [[Selberg's integral]] after a change of variable and has value

			:<math>\frac{\Gamma(1+\beta n/2)}{\Gamma(1+\beta/2)^n}</math>

			which gives another proof of Dyson's conjecture in this special case.

			==''q''-Dyson conjecture==
			{{harvtxt\|Andrews\|1975}} found a [[q-analog]] of Dyson's conjecture, stating that the constant term of
			:<math>\prod_{1\le i<j\le n}\left(\frac{x_i}{x_j};q\right)_{a_i}\left(\frac{qx_j}{x_i};q\right)_{a_j}</math>
			is
			:<math>\frac{(q;q)_{a_1+\cdots+a_n}}{(q;q)_{a_1}\cdots(q;q)_{a_n}}.</math>
			Here (''a'';''q'')<sub>''n''</sub> is the [[q-Pochhammer symbol]].
			This conjecture reduces to Dyson's conjecture for ''q''=1, and was proved by {{harvtxt\|Zeilberger\|Bressoud\|1985}}.

			==Macdonald conjectures==
			{{harvtxt\|Macdonald\|1982}} extended the conjecture to arbitrary finite or affine [[root system]]s, with Dyson's original conjecture corresponding to
			the case of the ''A''<sub>''n''−1</sub> root system and Andrews's conjecture corresponding to the affine ''A''<sub>''n''−1</sub> root system. Macdonald reformulated these conjectures as conjectures about the norms of [[Macdonald polynomial]]s. Macdonald's conjectures were proved by {{harv\|Cherednik\|1995}} using doubly affine Hecke algebras.

			[[Ian G. Macdonald\|Macdonald]]'s form of Dyson's conjecture for root systems of type BC is closely related to [[Selberg's integral]].

			==References==
			*{{Citation \| authorlink=George Andrews (mathematician) \| last1=Andrews \| first1=George E. \| title=Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) \| publisher=[[Academic Press]] \| location=Boston, MA \| mr=0399528 \| year=1975 \| chapter=Problems and prospects for basic hypergeometric functions \| pages=191–224}}
			*{{citation \|jstor=2118632 \|pages=191–216 \|last1=Cherednik \|first1=I. \|title=Double Affine Hecke Algebras and Macdonald's Conjectures \|volume=141 \|issue=1 \|journal=The Annals of Mathematics \|year=1995 \|doi=10.2307/2118632}}
			*{{Citation \| last1=Dyson \| first1=Freeman J. \| title=Statistical theory of the energy levels of complex systems. I \| doi=10.1063/1.1703773 \| mr=0143556 \| year=1962 \| journal=[[Journal of Mathematical Physics]] \| issn=0022-2488 \| volume=3 \| pages=140–156}}
			*{{Citation \| last1=Good \| first1=I. J. \| author1-link=I. J. Good \| title=Short proof of a conjecture by Dyson \| doi=10.1063/1.1665339 \| mr=0258644 \| year=1970 \| journal=[[Journal of Mathematical Physics]] \| issn=0022-2488 \| volume=11 \| pages=1884 \| issue=6}}
			*{{Citation \| last1=Gunson \| first1=J. \| title=Proof of a conjecture by Dyson in the statistical theory of energy levels \| doi=10.1063/1.1724277 \| mr=0148401 \| year=1962 \| journal=[[Journal of Mathematical Physics]] \| issn=0022-2488 \| volume=3 \| pages=752–753 \| issue=4}}
			*{{Citation \| last1=Macdonald \| first1=I. G. \| title=Some conjectures for root systems \| doi=10.1137/0513070 \| mr=674768 \| year=1982 \| journal=SIAM Journal on Mathematical Analysis \| issn=0036-1410 \| volume=13 \| issue=6 \| pages=988–1007}}
			*{{Citation \| last1=Sills \| first1=Andrew V. \| title=Disturbing the Dyson conjecture, in a generally GOOD way \| doi=10.1016/j.jcta.2005.12.005 \| mr=2259066 \| year=2006 \| journal=Journal of Combinatorial Theory. Series A \| issn=1096-0899 \| volume=113 \| issue=7 \| pages=1368–1380}}
			*{{Citation \| last1=Sills \| first1=Andrew V. \| last2=Zeilberger \| first2=Doron \| author2-link=Doron Zeilberger \| title=Disturbing the Dyson conjecture (in a GOOD way) \| url=http://projecteuclid.org/euclid.em/1175789739 \| mr=2253005 \| year=2006 \| journal=Experimental Mathematics \| issn=1058-6458 \| volume=15 \| issue=2 \| pages=187–191 \| doi=10.1080/10586458.2006.10128959}}
			*{{Citation \| last1=Wilson \| first1=Kenneth G. \| author1-link = Kenneth G. Wilson \| title=Proof of a conjecture by Dyson \| doi=10.1063/1.1724291 \| mr=0144627 \| year=1962 \| journal=[[Journal of Mathematical Physics]] \| issn=0022-2488 \| volume=3 \| pages=1040–1043 \| issue=5}}
			*{{Citation \| last1=Zeilberger \| first1=Doron \| author1-link=Doron Zeilberger \| last2=Bressoud \| first2=David M. \|author2-link=David Bressoud \| title=A proof of Andrews' q-Dyson conjecture \| doi=10.1016/0012-365X(85)90081-0 \| mr=791661 \| year=1985 \| journal=[[Discrete Mathematics (journal)\|Discrete Mathematics]] \| issn=0012-365X \| volume=54 \| issue=2 \| pages=201–224}}

			[[Category:Enumerative combinatorics]]
			[[Category:Algebraic combinatorics]]
			[[Category:Factorial and binomial topics]]
			[[Category:Mathematical identities]]
			[[Category:Conjectures]]
			[[Category:Freeman Dyson]]

en>Leszek Jańczuk: ref

2011-08-23T18:54:35Z

ref

New page

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