P/poly - Revision history

en>David Eppstein: no need for the math formatting

2014-10-10T05:44:11Z

no need for the math formatting

← Older revision		Revision as of 07:44, 10 October 2014
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	~~{{no footnotes\|date=June 2013}}~~		Nothing to say about me I think.<br>I enjoy of finally being a member of this site.<br>I just hope Im useful in some way .<br><br>Feel free to surf to my homepage :: [http://scentsofheaven.com.au/activity/p/230468/ Fifa 15 Coin Generator]
	~~[[Image:Kaplan-Yorke map~~.~~png\|thumb\|A plot of 100,000 iterations of the Kaplan-Yorke map with α=0.2. The initial value (x~~<~~sub~~>~~0</sub>,y<sub>0</sub>) was (128873/350377,0.667751).]]~~
	~~The '''Kaplan–Yorke map''' is~~ a ~~[[discrete-time]] [[dynamical system]]. It is an example~~ of ~~a dynamical system that exhibits [[chaos theory\|chaotic behavior]]~~. ~~The Kaplan–Yorke [[function (mathematics)\|map]] takes a point (''x~~<~~sub~~>~~n</sub>, y<sub>n</sub> '')~~ in ~~the [[plane (mathematics)\|plane]] and [[function (mathematics)\|maps]] it to a new point given by~~

	~~:<math>x_{n+1}=2x_n\ (\textrm{mod}~1)\,</math>~~
	:<~~math~~>~~y_{n+1}=\alpha y_n+\cos(4\pi x_n)\,~~<~~/math~~>

	~~where ''mod'' is the [[modulo operation\|modulo operator]] with real arguments. The map depends on only the one [[Constant (mathematics)\|constant]] α.~~

	~~==Calculation method==~~
	~~Due~~ to ~~roundoff error, successive applications of the modulo operator will yield zero after some ten or twenty iterations when implemented as a floating point operation on a computer. It is better~~ to ~~implement the following equivalent algorithm~~:

	:~~<math>a_{n+1}=2a_n\ (\textrm{mod}~b)\,</math>~~
	:~~<math>x_{n+1}=a_n~~/~~b\,<~~/~~math>~~
	~~:<math>y_{n+1}=\alpha y_n+\cos(4\pi x_n)\,</math>~~

	~~where the <math>a_n</math> and <math>b<~~/~~math> are computational integers. It is also best to choose <math>b<~~/~~math> to be a large [[prime number]] in order to get many different values of <math>x_n<~~/~~math>.~~

	~~==References==~~
	* {{cite book \| author= J.L. Kaplan and J.A. Yorke \| title= Functional Differential Equations and Approximations of Fixed Points (Lecture notes in Mathematics 730) \|editor=H.O. Peitgen and H.O. Walther \| publisher=Springer-Verlag \| year=1979 \| isbn = 0-387-09518-7}}
	* {{cite journal \| author=P. Grassberger and I. Procaccia \| title=Measuring the strangeness of strange attractors \| journal=Physica \| year=1983 \| volume=9D \| issue=1-2\| pages=189–208 \| bibcode=1983PhyD....9..189G \| doi=10.1016/~~0167-2789(83)90298-1 }}~~

	~~{{Chaos theory}}~~

	~~{{DEFAULTSORT:Kaplan Yorke Map}}~~
	~~[[Category:Chaotic maps]~~]

	~~{{Mathapplied-stub}}~~
	~~{{physics-stub}}~~

en>Awaterl: Fix broken link to Impagliazzo paper.

2013-09-19T01:20:30Z

Fix broken link to Impagliazzo paper.

← Older revision		Revision as of 03:20, 19 September 2013
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	<br><br><br><br>In the ~~event you loved this information~~ and ~~you would want~~ to ~~[http~~://~~Support~~.~~microsoft~~.~~com~~/kb/~~896861 receive details~~] ~~relating~~ to ~~[http:~~/~~/wiki~~.~~bbshop~~.~~com~~.~~br/index~~.~~php?~~title=~~Aspects_that_alter_Medical_Technology_Advancement Crier PR wine PR] i implore you to visit~~ the ~~site~~.		{{no footnotes\|date=June 2013}}
			[[Image:Kaplan-Yorke map.png\|thumb\|A plot of 100,000 iterations of the Kaplan-Yorke map with α=0.2. The initial value (x<sub>0</sub>,y<sub>0</sub>) was (128873/350377,0.667751).]]
			The '''Kaplan–Yorke map''' is a [[discrete-time]] [[dynamical system]]. It is an example of a dynamical system that exhibits [[chaos theory\|chaotic behavior]]. The Kaplan–Yorke [[function (mathematics)\|map]] takes a point (''x<sub>n</sub>, y<sub>n</sub> '') in the [[plane (mathematics)\|plane]] and [[function (mathematics)\|maps]] it to a new point given by

			:<math>x_{n+1}=2x_n\ (\textrm{mod}~1)\,</math>
			:<math>y_{n+1}=\alpha y_n+\cos(4\pi x_n)\,</math>

			where ''mod'' is the [[modulo operation\|modulo operator]] with real arguments. The map depends on only the one [[Constant (mathematics)\|constant]] α.

			==Calculation method==
			Due to roundoff error, successive applications of the modulo operator will yield zero after some ten or twenty iterations when implemented as a floating point operation on a computer. It is better to implement the following equivalent algorithm:

			:<math>a_{n+1}=2a_n\ (\textrm{mod}~b)\,</math>
			:<math>x_{n+1}=a_n/b\,</math>
			:<math>y_{n+1}=\alpha y_n+\cos(4\pi x_n)\,</math>

			where the <math>a_n</math> and <math>b</math> are computational integers. It is also best to choose <math>b</math> to be a large [[prime number]] in order to get many different values of <math>x_n</math>.

			==References==
			* {{cite book \| author= J.L. Kaplan and J.A. Yorke \| title= Functional Differential Equations and Approximations of Fixed Points (Lecture notes in Mathematics 730) \|editor=H.O. Peitgen and H.O. Walther \| publisher=Springer-Verlag \| year=1979 \| isbn = 0-387-09518-7}}
			* {{cite journal \| author=P. Grassberger and I. Procaccia \| title=Measuring the strangeness of strange attractors \| journal=Physica \| year=1983 \| volume=9D \| issue=1-2\| pages=189–208 \| bibcode=1983PhyD....9..189G \| doi=10.1016/0167-2789(83)90298-1 }}

			{{Chaos theory}}

			{{DEFAULTSORT:Kaplan Yorke Map}}
			[[Category:Chaotic maps]]

			{{Mathapplied-stub}}
			{{physics-stub}}

193.52.24.28: /* Importance of P/poly */

2012-05-10T07:20:41Z

Importance of P/poly

New page

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