Group with operators - Revision history

86.183.223.225 at 17:00, 23 November 2014

2014-11-23T17:00:26Z

← Older revision		Revision as of 19:00, 23 November 2014
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	~~In [[mathematics]], the '''Milliken–Taylor theorem''' in [[combinatorics]]~~ is ~~a generalization of both [[Ramsey~~'s ~~theorem]]~~ and ~~[[IP set\|Hindman's theorem]]. It~~ is ~~named after Keith Milliken and [http://www.math.union.edu/people/faculty/publications/taylora.html Alan D. Taylor].~~		Golda is what's created on my beginning certification although it is not the name on my birth certification. The favorite hobby for him and his children is to play lacross and he would by no means give it up. I am currently a journey agent. North Carolina is the place he enjoys most but now he is contemplating other options.<br><br>Have a look at my web site [http://www.skullrocker.com/blogs/post/10991 psychic readers]

	~~Let <math>\mathcal{P}_f(\mathbb{N})</math> denote the set of finite subsets of <math>\mathbb{N}</math>,~~ and ~~define a partial order on <math>\mathcal{P}_f(\mathbb{N})</math>~~ by ~~α<β [[if and only if]] max α<min β~~. ~~Given~~ a ~~sequence of integers <math>\langle a_n \rangle_{n=0}^\infty \subset \mathbb{N}</math> and {{nowrap\|''k'' > 0}}, let~~
	:<math>[FS(\langle a_n \rangle_{n=0}^\infty)]^k_< = \left \{ \left \{ \sum \alpha_1, \ldots , \sum \alpha_k \right \}: \alpha_1 ,\cdots , \alpha_k \in \mathcal{P}_f(\mathbb{N})\text{ and }\alpha_1 <\cdots < \alpha_k \right \}.~~</math>~~
	~~Let <math>[S]^k</math> denote~~ the ~~''k''-element subsets of a set ''S''~~. ~~The Milliken–Taylor theorem says that for any finite partition~~ <~~math~~>~~[\mathbb{N}]^k=C_1 \cup C_2 \cup \cdots \cup C_r~~<~~/math~~>~~, there exist some {{nowrap\|''i'' ≤ ''r''}} and~~ a ~~sequence <math>\langle a_n \rangle_{n=0}^{\infty} \subset \mathbb{N}</math> such that <math>[FS(\langle a_n \rangle_{n=0}^{\infty})]^k_< \subset C_i</math>.~~

	For each <math>\langle a_n \rangle_{n=0}^\infty \subset \mathbb{N}</math>, call <math>[FS(\langle a_n \rangle_{n=0}^\infty)]^k_< </math> an ''MT<sup>k</sup> set''. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MT<sup>''k''</sup> sets is [[partition regular]] for each ''k''.

	~~==References==~~
	~~#K. Milliken, Ramsey's Theorem with sums or unions, ''J. Comb. Theory (Series A)'' '''18''' (1975), 276–290~~ [http://dx.~~doi~~.~~org~~/~~10.1016~~/~~0097-3165(75)90039-4]~~
	~~#A. Taylor, A canonical partition relation for finite subsets of ω, ''J. Comb. Theory (Series A)'' '''21''' (1976), 137–146 [http://dx.doi.org/10.1016~~/~~0097-3165(76)90058-3]~~

	~~{{DEFAULTSORT:Milliken-Taylor theorem}}~~
	~~[[Category:Ramsey theory]]~~
	~~[[Category:Theorems in discrete mathematics]~~]


	~~{{combin-stub}}~~

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2013-03-15T07:47:03Z

Bot: Migrating 3 interwiki links, now provided by Wikidata on d:q3077823

← Older revision		Revision as of 09:47, 15 March 2013
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	~~Alyson~~ is ~~what my spouse enjoys to contact me but I don~~'~~t like when~~ people ~~use my complete title~~. ~~My working day job is an invoicing officer but I~~'~~ve currently applied for another 1~~. ~~Alaska is exactly where I~~'~~ve usually been living. I am truly fond~~ of ~~to go to karaoke but I~~'~~ve been taking on new issues recently~~.<br><br>~~Feel free to visit my website~~; [http://~~jplusfn~~.~~gaplus~~.kr/xe/~~qna~~/~~78647 certified psychics~~]		In [[mathematics]], the '''Milliken–Taylor theorem''' in [[combinatorics]] is a generalization of both [[Ramsey's theorem]] and [[IP set\|Hindman's theorem]]. It is named after Keith Milliken and [http://www.math.union.edu/people/faculty/publications/taylora.html Alan D. Taylor].

			Let <math>\mathcal{P}_f(\mathbb{N})</math> denote the set of finite subsets of <math>\mathbb{N}</math>, and define a partial order on <math>\mathcal{P}_f(\mathbb{N})</math> by α<β [[if and only if]] max α<min β. Given a sequence of integers <math>\langle a_n \rangle_{n=0}^\infty \subset \mathbb{N}</math> and {{nowrap\|''k'' > 0}}, let
			:<math>[FS(\langle a_n \rangle_{n=0}^\infty)]^k_< = \left \{ \left \{ \sum \alpha_1, \ldots , \sum \alpha_k \right \}: \alpha_1 ,\cdots , \alpha_k \in \mathcal{P}_f(\mathbb{N})\text{ and }\alpha_1 <\cdots < \alpha_k \right \}.</math>
			Let <math>[S]^k</math> denote the ''k''-element subsets of a set ''S''. The Milliken–Taylor theorem says that for any finite partition <math>[\mathbb{N}]^k=C_1 \cup C_2 \cup \cdots \cup C_r</math>, there exist some {{nowrap\|''i'' ≤ ''r''}} and a sequence <math>\langle a_n \rangle_{n=0}^{\infty} \subset \mathbb{N}</math> such that <math>[FS(\langle a_n \rangle_{n=0}^{\infty})]^k_< \subset C_i</math>.

			For each <math>\langle a_n \rangle_{n=0}^\infty \subset \mathbb{N}</math>, call <math>[FS(\langle a_n \rangle_{n=0}^\infty)]^k_< </math> an ''MT<sup>k</sup> set''. Then, alternatively, the Milliken–Taylor theorem asserts that the collection of MT<sup>''k''</sup> sets is [[partition regular]] for each ''k''.

			==References==
			#K. Milliken, Ramsey's Theorem with sums or unions, ''J. Comb. Theory (Series A)'' '''18''' (1975), 276–290 [http://dx.doi.org/10.1016/0097-3165(75)90039-4]
			#A. Taylor, A canonical partition relation for finite subsets of ω, ''J. Comb. Theory (Series A)'' '''21''' (1976), 137–146 [http://dx.doi.org/10.1016/0097-3165(76)90058-3]

			{{DEFAULTSORT:Milliken-Taylor theorem}}
			[[Category:Ramsey theory]]
			[[Category:Theorems in discrete mathematics]]


			{{combin-stub}}

en>ZéroBot: r2.7.1) (Robot: Adding fr:Groupe à opérateurs

2012-07-08T12:46:31Z

r2.7.1) (Robot: Adding fr:Groupe à opérateurs

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Alyson is what my spouse enjoys to contact me but I don't like when people use my complete title. My working day job is an invoicing officer but I've currently applied for another 1. Alaska is exactly where I've usually been living. I am truly fond of to go to karaoke but I've been taking on new issues recently.<br><br>Feel free to visit my website; [http://jplusfn.gaplus.kr/xe/qna/78647 certified psychics]