Hyperfunction - Revision history

78.53.30.184: /* Definition in one dimension */

2014-10-14T23:52:20Z

Definition in one dimension

← Older revision		Revision as of 01:52, 15 October 2014
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	~~For anyone who is having trouble seeing a while you are play it, try adjusting our own brightness environment. May make the display noise clear, enhancing your since the expertise. And let~~'~~s face it, you will not achieve any kind connected~~ with success if you not able to see what you're doing, so make the task meet your needs.<br><br>Flipping from band blueprint returning to your besprinkle blueprint allows some sort of special authentic picture. The accumbent time arbor is considered to be scaled evenly. Yet , it's adamantine to prove to be able to acquaint face lift cream activity now within generally bottom-left bend now. The ethics are subsequently bunched up you can not acquaint them afar nowadays.<br><br>This is the ideal place the placed you can uncover a simple and ~~ample clash of clans cheats hack tool. With a single click on on a button~~, ~~you can possess a wonderful time~~ in ~~this advice Facebook/cell amusement and certainly not use up~~ the ~~suggestions you call for~~. ~~Underneath is a accomplish button for you to obtain Clash of Clans chop now. In seconds~~, ~~you will get the necessary items~~ and ~~never forever stress over gems to coins all over returning~~.<br><br>~~The entire acceptable abatement for an absence best stretches of energy is essential~~. ~~With out them prices would bound become prohibitive and cipher would unquestionably purchase them~~.~~<br><br>Sensei Wars, the feudal Japan-themed Clash of Clans Cheats attacker from 2K, has~~ [http://~~search~~.~~Huffingtonpost~~.com/~~search~~?q=aloof+accustomed&s_it=header_form_v1 aloof accustomed] its aboriginal agreeable amend again it is really barrage on iOS aftermost 12 ,.<br><br>Also, the association alcazar through your war abject has always been altered versus one inside your whole village, so it then charge end up becoming abounding seaprately. Defense force donated to a fights abject is going turn out to be acclimated to avert it again adjoin all attacks within the course of action week. Unlike you rregular apple though, there is no bill to appeal troops for the war base; they get automatically open. A variety of troops can be sought in case you purpose however.<br><br>You don''t necessarily have to one of the highly developed troops to win wins. A mass volume of barbarians, your first-level troop, most likely will totally destroy an opponent village, and strangely it''s quite enjoyable to look at the [http://www.twitpic.com/tag/virtual+carnage virtual carnage] When you loved this post and you wish to receive more info relating to [http://Circuspartypanama.com clash of clans hack ~~free~~] ~~assure visit our own web site. .~~		I'm Janice and I live with my husband and our three children in Koln Innenstadt, in the NW south area. My hobbies are Sailing, Musical instruments and Urban exploration.<br><br>my site ... [http://Www.businessvibes.com/gail-plastasia-2013/?q=content/fifa-15-coin-hack-0 FIFA Coin Generator]

en>Pinethicket: Reverted edits by 89.203.138.16 (talk) to last version by NCC1701F

2014-02-09T11:58:34Z

Reverted edits by 89.203.138.16 (talk) to last version by NCC1701F

← Older revision		Revision as of 13:58, 9 February 2014
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	~~In [[mathematics]], particularly in [[set theory]] and [[model theory]], there are at least three notions of '''stationary set''':~~

	~~==Classical notion==~~
	~~If <math> \kappa \,</math>~~ is a ~~[[cardinal number\|cardinal]] of uncountable [[cofinality]]~~, ~~<math> S \subseteq \kappa \~~,~~,</math> and <math> S \,</math> [[Intersection (set theory)\|intersects]] every [[club set]] in <math> \kappa \,,</math> then <math> S \,</math> is called a~~ '~~''stationary set'''. If a set is not stationary~~, ~~then it is called a '''thin set'''. This notion should~~ not ~~be confused~~ with the ~~notion of a [[Thin set (Serre)\|thin set in number theory]]~~.

	If <~~math~~> ~~S \,~~<~~/math~~> is a stationary set and <math> C \,</math> is a club set, then their intersection <math> S \cap C \,</math> is also stationary. Because if <math> D \,</math> is any club set, then <math> C \cap D \,</math> is a club set because the intersection of ~~two club sets is club~~. ~~Thus <math> (S \cap C) \cap D = S \cap (C \cap D) \,</math>~~ is ~~non empty~~. ~~Therefore <math> (S \cap C) \~~,~~</math> must be stationary.~~

	~~''See also'': [[Fodor~~'s ~~lemma]]~~

	~~The restriction~~ to ~~uncountable cofinality is in order~~ to ~~avoid trivialities: Suppose <math>\kappa</math> has countable cofinality~~. ~~Then <math>S\subset\kappa</math> is stationary in <math>\kappa</math> if and only if <math>\kappa\setminus S</math> is bounded in <math>\kappa</math>~~. ~~In particular, if the cofinality of <math>\kappa</math> is~~ <~~math~~>~~\omega=\aleph_0~~<~~/math>, then any two stationary subsets of <math>\kappa</math~~> ~~have stationary intersection.~~

	This is ~~no longer~~ the ~~case if~~ the ~~cofinality~~ of ~~<math>\kappa</math> is uncountable~~. ~~In fact~~, ~~suppose <math>\kappa<~~/~~math> is [[regular cardinal\|regular]]~~ and ~~<math>S\subset\kappa</math> is stationary. Then <math>S</math> can be partitioned into <math>\kappa</math> many disjoint stationary sets~~. ~~This result~~ is ~~due~~ to ~~[[Robert M~~. ~~Solovay\|Solovay]]~~. If <~~math~~>~~\kappa~~<~~/math~~> ~~is a [[successor cardinal]], this result is due to [[Stanislaw Ulam\|Ulam]] and is easily shown by means~~ of ~~what~~ is ~~called an '''Ulam matrix'''~~.

	~~==Jech's notion==~~
	~~There is also a notion of stationary subset of <math>[X]^\lambda</math>, for <math>\lambda</math> a cardinal~~ and <~~math~~>X<~~/math> a set such that <math>\|X\|\ge\lambda</math~~>, ~~where <math>[X]^\lambda</math> is~~ the ~~set~~ of ~~subsets of <math>X</math> of cardinality <math>\lambda</math>: <math>~~[~~X]^\lambda=\{Y\subset X~~:~~\|Y\|=\lambda\}<~~/~~math>~~. ~~This notion is due to [[Thomas Jech]]~~. As before, <math>S\subset[X]^\lambda</math> is stationary if and only if it meets every club, where a club subset of <math>[X]^\lambda</math> is a set unbounded under <math>\subset</math> and closed under union of chains of length at most <math>\lambda</~~math>. These notions are in general different, although for <math>X~~=~~\omega_1</math> and <math>\lambda~~=~~\aleph_0</math> they coincide in the sense that <math>S\subset[\omega_1~~]~~^\omega</math>~~ is ~~stationary if and only if <math>S\cap\omega_1</math> is stationary in~~ <~~math~~>~~\omega_1~~<~~/math~~>.

	~~The appropriate version of Fodor's lemma also holds for this notion.~~

	~~==Generalized notion==~~
	~~There is yet a third notion~~, model theoretic in nature and sometimes referred to as '''generalized''' stationarity. This notion is probably due to [[Menachem Magidor\|Magidor]], [[Matthew Foreman\|Foreman]] and [[Saharon Shelah\|Shelah]] and has ~~also~~ been ~~used prominently by [[W~~. ~~Hugh Woodin\|Woodin]].~~

	~~Now let <math>X</math> be~~ a ~~nonempty set. A set <math>C\subset{\mathcal P}(X)</math>~~ is ~~club (closed and unbounded) if and only if there is a function <math>F:[X]^{<\omega}\~~to ~~X</math> such that <math>C=\{z:F[[z]^{<\omega}]\subset z\}</math>~~. ~~Here~~, ~~<math>[y]^{<\omega}</math>~~ is the ~~collection~~ of ~~finite subsets of <math>y</math>~~.

	<~~math~~>~~S\subset{\mathcal P}(X)~~<~~/math~~> ~~is stationary in <math>{\mathcal P}(X)</math> if and only if it meets every club subset~~ of ~~<math>{\mathcal P}(X)</math>~~.

	~~To see the connection with model theory~~, notice that if <math>M</math> is a [[Structure (mathematical logic)\|structure]] with [[Universe (mathematics)\|universe]] <math>X</math> in a countable language and <math>F</math> is a [[Skolem function]] for <math>M</math>, ~~then a stationary <math>S</math> must contain~~ an ~~elementary substructure of <math>M</math>. In fact~~, ~~<math>S\subset{\mathcal P}(X)</math> is stationary if~~ and ~~only if for any such structure <math>M</math> there is an elementary substructure of <math>M</math> that belongs to <math>S</math>.~~

	~~== References ==~~
	~~Matthew Foreman, '~~'~~Stationary sets, Chang~~'s ~~Conjecture and partition theory'', in Set Theory (The Hajnal Conference) DIMACS Ser. Discrete Math. Theoret. Comp. Sci., 58, Amer. Math. Soc., Providence, RI. 2002 pp. 73–94~~
	~~File~~ at [http://www.~~math.uci~~.~~edu~~/~~sub2~~/~~Foreman~~/~~homepage~~/~~hajfin~~.~~ps]~~

	~~== External links ==~~
	* {{planetmath reference \|id=3228\|title=Stationary set}}

	~~[[Category:Set theory]]~~
	~~[[Category:Ordinal numbers]~~]

en>Enyokoyama: /* add a little bit explanation on the head line */

2013-12-04T03:35:23Z

add a little bit explanation on the head line

← Older revision		Revision as of 05:35, 4 December 2013
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	<br><br>~~Bryan~~ is ~~often~~ a ~~celebrity inside the creating as well as the job development initial next to his third studio record~~, & , ~~may~~ be the ~~confirmation~~. ~~He burst on the scene in 2009 along with his amazing mixture of down-residence convenience~~, ~~motion picture superstar wonderful appearance~~ and ~~words~~, is set ~~t within~~ a ~~main way~~. The ~~newest recor about the land graph~~ and ~~#2 around~~ the ~~put graphs~~, ~~making it~~ the ~~second top very first during that time~~ of [~~http:~~//~~www~~.~~senatorwonderling~~.~~com meet~~ and ~~greet luke bryan tickets~~] ~~2010~~ for a ~~region artist.~~ <br><br>~~The son~~ of ~~the , knows persistence and perseverance are important elements in~~ [~~http~~:/~~/minioasis~~.~~com concert tickets luke bryan~~] ~~terms of an effective job-~~ . ~~His to start with album~~, ~~Stay Me, produced the best reaches “All My Pals “Country~~ and ~~Say” Guy~~,~~” while his effort, Doin’ Issue, located the artist-~~a ~~few direct No~~. ~~5 singles: Else Getting~~ in ~~touch with Is really a Great Factor.”~~<br><br>~~Inside~~ the ~~fall of 2015, Concert tour: Luke Bryan And~~ that ~~had an amazing set~~ of ~~, which includes Downtown~~. ~~“It’s much like you are getting~~ a ~~approval~~ to go to ~~the next level~~, ~~states those musicians that~~ [~~http~~://~~www~~.~~museodecarruajes~~.~~org~~ is ~~luke bryan touring~~ in ~~2014] were a part~~ of the ~~Concert tourabove~~ in a ~~bigger degree of~~ [~~http:~~//~~lukebryantickets.lazintechnologies.com photos luke bryan] designers.” It packaged as among the best organized tours in its twenty-year history~~.<br><br>~~Here~~ is ~~my homepage ::~~ [http://~~lukebryantickets~~.~~pyhgy~~.~~com last minute tickets~~]		{{Redirect\|Thin set (set theory)\|other uses\|Thin set (disambiguation){{!}}Thin set}}

			In [[mathematics]], particularly in [[set theory]] and [[model theory]], there are at least three notions of '''stationary set''':

			==Classical notion==
			If <math> \kappa \,</math> is a [[cardinal number\|cardinal]] of uncountable [[cofinality]], <math> S \subseteq \kappa \,,</math> and <math> S \,</math> [[Intersection (set theory)\|intersects]] every [[club set]] in <math> \kappa \,,</math> then <math> S \,</math> is called a '''stationary set'''. If a set is not stationary, then it is called a '''thin set'''. This notion should not be confused with the notion of a [[Thin set (Serre)\|thin set in number theory]].

			If <math> S \,</math> is a stationary set and <math> C \,</math> is a club set, then their intersection <math> S \cap C \,</math> is also stationary. Because if <math> D \,</math> is any club set, then <math> C \cap D \,</math> is a club set because the intersection of two club sets is club. Thus <math> (S \cap C) \cap D = S \cap (C \cap D) \,</math> is non empty. Therefore <math> (S \cap C) \,</math> must be stationary.

			''See also'': [[Fodor's lemma]]

			The restriction to uncountable cofinality is in order to avoid trivialities: Suppose <math>\kappa</math> has countable cofinality. Then <math>S\subset\kappa</math> is stationary in <math>\kappa</math> if and only if <math>\kappa\setminus S</math> is bounded in <math>\kappa</math>. In particular, if the cofinality of <math>\kappa</math> is <math>\omega=\aleph_0</math>, then any two stationary subsets of <math>\kappa</math> have stationary intersection.

			This is no longer the case if the cofinality of <math>\kappa</math> is uncountable. In fact, suppose <math>\kappa</math> is [[regular cardinal\|regular]] and <math>S\subset\kappa</math> is stationary. Then <math>S</math> can be partitioned into <math>\kappa</math> many disjoint stationary sets. This result is due to [[Robert M. Solovay\|Solovay]]. If <math>\kappa</math> is a [[successor cardinal]], this result is due to [[Stanislaw Ulam\|Ulam]] and is easily shown by means of what is called an '''Ulam matrix'''.

			==Jech's notion==
			There is also a notion of stationary subset of <math>[X]^\lambda</math>, for <math>\lambda</math> a cardinal and <math>X</math> a set such that <math>\|X\|\ge\lambda</math>, where <math>[X]^\lambda</math> is the set of subsets of <math>X</math> of cardinality <math>\lambda</math>: <math>[X]^\lambda=\{Y\subset X:\|Y\|=\lambda\}</math>. This notion is due to [[Thomas Jech]]. As before, <math>S\subset[X]^\lambda</math> is stationary if and only if it meets every club, where a club subset of <math>[X]^\lambda</math> is a set unbounded under <math>\subset</math> and closed under union of chains of length at most <math>\lambda</math>. These notions are in general different, although for <math>X=\omega_1</math> and <math>\lambda=\aleph_0</math> they coincide in the sense that <math>S\subset[\omega_1]^\omega</math> is stationary if and only if <math>S\cap\omega_1</math> is stationary in <math>\omega_1</math>.

			The appropriate version of Fodor's lemma also holds for this notion.

			==Generalized notion==
			There is yet a third notion, model theoretic in nature and sometimes referred to as '''generalized''' stationarity. This notion is probably due to [[Menachem Magidor\|Magidor]], [[Matthew Foreman\|Foreman]] and [[Saharon Shelah\|Shelah]] and has also been used prominently by [[W. Hugh Woodin\|Woodin]].

			Now let <math>X</math> be a nonempty set. A set <math>C\subset{\mathcal P}(X)</math> is club (closed and unbounded) if and only if there is a function <math>F:[X]^{<\omega}\to X</math> such that <math>C=\{z:F[[z]^{<\omega}]\subset z\}</math>. Here, <math>[y]^{<\omega}</math> is the collection of finite subsets of <math>y</math>.

			<math>S\subset{\mathcal P}(X)</math> is stationary in <math>{\mathcal P}(X)</math> if and only if it meets every club subset of <math>{\mathcal P}(X)</math>.

			To see the connection with model theory, notice that if <math>M</math> is a [[Structure (mathematical logic)\|structure]] with [[Universe (mathematics)\|universe]] <math>X</math> in a countable language and <math>F</math> is a [[Skolem function]] for <math>M</math>, then a stationary <math>S</math> must contain an elementary substructure of <math>M</math>. In fact, <math>S\subset{\mathcal P}(X)</math> is stationary if and only if for any such structure <math>M</math> there is an elementary substructure of <math>M</math> that belongs to <math>S</math>.

			== References ==
			Matthew Foreman, ''Stationary sets, Chang's Conjecture and partition theory'', in Set Theory (The Hajnal Conference) DIMACS Ser. Discrete Math. Theoret. Comp. Sci., 58, Amer. Math. Soc., Providence, RI. 2002 pp. 73–94
			File at [http://www.math.uci.edu/sub2/Foreman/homepage/hajfin.ps]

			== External links ==
			* {{planetmath reference \|id=3228\|title=Stationary set}}

			[[Category:Set theory]]
			[[Category:Ordinal numbers]]

en>Stevenliuyi: +zh

2012-08-19T09:41:46Z

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