Probability amplitude - Revision history

198.13.228.32: /* Mathematical */

2014-12-30T04:17:20Z

Mathematical

← Older revision		Revision as of 06:17, 30 December 2014
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24.7.123.82: /* In the context of the double-slit experiment */

2014-02-17T08:51:43Z

In the context of the double-slit experiment

← Older revision		Revision as of 10:51, 17 February 2014
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	~~In [[mathematics]], the '''cyclotomic identity''' states that~~

	~~:<math>{1 \over 1-\alpha z}=\prod_{j=1}^\infty\left({1 \over 1-z^j}\right)^{M(\alpha,j)}</math>~~

	~~where ''M'' is~~ [~~[Moreau's necklace-counting function]],~~		[http://www.britannica.com/search?query=Material Material] to meet you! Some of my name is Eusebio furthermore I think it voice overs quite good when say it. As a man what I really like is behaving but I'm thinking on starting something new. I work as an get clerk. My house is so now in South Carolina in addition I don't plan directly on changing it. You will find my website here: http://prometeu.net<br><br>[http://Answers.Yahoo.com/search/search_result?p=Feel+free&submit-go=Search+Y!+Answers Feel free] to surf to my web page; [http://prometeu.net hack clash of clans ifile]

	:~~<math>M(\alpha,n)~~=~~{1\over n}\sum_{d\,\|\,n}\mu\left({n \over d}\right)\alpha^d,</math>~~

	~~and ''μ''(·) is the classic [[Möbius function]~~] of ~~[[number theory]].~~

	~~The~~ name ~~comes from the denominator, 1 − ''z''<sup> ''j''</sup>, which~~ is ~~the product of [[cyclotomic polynomial]]s~~.

	~~The left hand side of the cyclotomic identity~~ is ~~the [[generating function]] for the free associative algebra~~ on ~~α generators, and the right hand side~~ is ~~the generating function for the [[universal enveloping algebra]] of the [[free Lie algebra]]~~ on ~~α generators~~. ~~The cyclotomic identity witnesses the fact that these two algebras are isomorphic.~~

	~~There is also a symmetric generalization of the cyclotomic identity found by Strehl~~:
	:<~~math~~>~~\prod_{j=1}^\infty\left({1 \over 1-\alpha z^j}\right)^{M(\beta,j)}=\prod_{j=1}^\infty\left({1 \over 1-\beta z^j}\right)^{M(\alpha,j)}~~<~~/math~~>

	~~==References==~~

	*{{Citation \| last1=Metropolis \| first1=N. \| last2=Rota \| first2=Gian-Carlo \| author2-link=Gian-Carlo Rota \| editor1-last=Greene \| editor1-first=Curtis \| editor1-link = Curtis Greene \| title=Combinatorics and algebra (Boulder, Colo., 1983). Proceedings of the AMS-IMS-SIAM joint summer research conference held at the University of Colorado, Boulder, Colo., June 5–11, 1983. \| url=http://~~books~~.~~google~~.com/~~books~~?id=~~2axt00oBDEwC~~&pg=~~PA19 \| publisher=[[American Mathematical Society~~]~~] \| location=Providence, R.I. \| series=Contemp. Math. \| isbn=978-0-8218-5029-9 \| mr=777692 \| year=1984 \| volume=34 \| chapter=The cyclotomic identity \| pages=19–27}}~~

	[~~[Category~~:~~Mathematical identities]~~]

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2014-01-12T21:55:16Z

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← Older revision		Revision as of 23:55, 12 January 2014
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			:<math>{1 \over 1-\alpha z}=\prod_{j=1}^\infty\left({1 \over 1-z^j}\right)^{M(\alpha,j)}</math>

			where ''M'' is [[Moreau's necklace-counting function]],

			:<math>M(\alpha,n)={1\over n}\sum_{d\,\|\,n}\mu\left({n \over d}\right)\alpha^d,</math>

			and ''μ''(·) is the classic [[Möbius function]] of [[number theory]].

			The name comes from the denominator, 1 − ''z''<sup> ''j''</sup>, which is the product of [[cyclotomic polynomial]]s.

			The left hand side of the cyclotomic identity is the [[generating function]] for the free associative algebra on α generators, and the right hand side is the generating function for the [[universal enveloping algebra]] of the [[free Lie algebra]] on α generators. The cyclotomic identity witnesses the fact that these two algebras are isomorphic.

			There is also a symmetric generalization of the cyclotomic identity found by Strehl:
			:<math>\prod_{j=1}^\infty\left({1 \over 1-\alpha z^j}\right)^{M(\beta,j)}=\prod_{j=1}^\infty\left({1 \over 1-\beta z^j}\right)^{M(\alpha,j)}</math>

			==References==

			*{{Citation \| last1=Metropolis \| first1=N. \| last2=Rota \| first2=Gian-Carlo \| author2-link=Gian-Carlo Rota \| editor1-last=Greene \| editor1-first=Curtis \| editor1-link = Curtis Greene \| title=Combinatorics and algebra (Boulder, Colo., 1983). Proceedings of the AMS-IMS-SIAM joint summer research conference held at the University of Colorado, Boulder, Colo., June 5–11, 1983. \| url=http://books.google.com/books?id=2axt00oBDEwC&pg=PA19 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=Contemp. Math. \| isbn=978-0-8218-5029-9 \| mr=777692 \| year=1984 \| volume=34 \| chapter=The cyclotomic identity \| pages=19–27}}

			[[Category:Mathematical identities]]

en>Cydebot: Robot - Moving category Fundamental physics concepts to :Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

2012-07-19T00:38:32Z

Robot - Moving category Fundamental physics concepts to Category:Concepts in physics per CFD at Wikipedia:Categories for discussion/Log/2012 July 12.

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