Periodic sequence - Revision history

en>David Eppstein: /* Examples */ cycle detection

2014-11-28T22:06:56Z

← Older revision		Revision as of 00:06, 29 November 2014
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	[~~[Image~~:~~Monkey Saddle Surface (Shaded)~~.~~png\|thumb\|~~The ~~monkey saddle\|300px]]~~		Formerly a [http://www.dailymail.co.uk/home/search.html?sel=site&searchPhrase=association+struggle association struggle] begins, you will see The most important particular War Map, a good map of this [http://thesaurus.com/browse/conflict conflict] area area association wars booty place. Welcoming territories will consistently often be on the left, with the adversary association when it comes to the right. Almost every boondocks anteroom on these war map represents some sort or other of war base.<br><br>
	In [~~[mathematics]~~], the ~~'''monkey saddle''' is~~ the ~~[[surface]] defined by the equation~~

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	It ~~belongs~~ to the ~~class~~ of ~~[[saddle surface]]s~~ and ~~its name derives from~~ the ~~observation~~ that ~~a [[saddle]]~~ for ~~a [[monkey]] requires three depressions: two for the legs~~, ~~and one for~~ the ~~tail~~. ~~The point (0,0,0)~~ on ~~the monkey saddle corresponds~~ to ~~a [[critical point (mathematics)\|degenerate critical point]] of~~ the ~~function ''z''(''x''~~,''~~y'')~~ at ~~(0, 0)~~. ~~The monkey saddle has an isolated~~ [~~[umbilical point]] with zero [[Gaussian curvature]~~] at the ~~origin, while the curvature is strictly negative at all other points~~.

	~~To show that~~ the ~~monkey saddle has three depressions, one~~ can ~~write the equation for ''z'' using [[complex number]]s as~~
	~~:<math> z(x,y) = \operatorname{Re} (x+iy)^3~~.<~~/math>~~
	~~It follows that ''z''(''tx'',''ty'') = ''t''<sup~~>3<~~/sup~~> ~~''z''(''x'',''y'')~~ for ~~''t'' ≥ 0~~, so the ~~surface is determined by ''z'' on~~ the ~~[[unit circle]]~~. ~~Parametrizing this by ''e''~~<~~sup~~>~~''i''φ~~<~~/sup~~>, ~~with φ ∈ [0, 2π), one can~~ see ~~that~~ on ~~the unit circle, ''z''(φ) = cos 3φ, so ''z'' has three depressions~~. ~~By replacing 3~~ with ~~any integer~~ '~~'k'' ≥ 1, one can create~~ a ~~saddle with ''k'' depressions~~.

	~~== Horse saddle ==~~

	~~The term ''horse saddle'' is used,~~ in ~~contrast~~ to ~~monkey saddle, to designate~~ a ~~[[saddle point]] that is a [[minimax]]~~, ~~that is to say a local minimum~~ or ~~maximum depending on the intersecting plane used~~. ~~The monkey saddle has just a [[point~~ of ~~inflection]]. To see this~~, ~~consider~~ a ~~line ''y'' = ''kx''. Along this direction~~ the ~~surface becomes simply~~ ''z'~~' = (1 − 3''k''<sup>2</sup>)''x''<sup>3</sup>, lacking any critical points.~~

	~~== See also==~~
	* [[Saddle point]]
	*[[Saddle surface]]

	~~==External links==~~
	* {{MathWorld \| urlname=MonkeySaddle \| title=Monkey Saddle}}

	~~[[Category:Multivariable calculus]]~~
	~~[[Category:Surfaces]]~~

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3503302

2013-03-13T10:29:21Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3503302

← Older revision		Revision as of 12:29, 13 March 2013
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	~~Until today enjoying a brand upcoming clash of clans identify tool~~, ~~see~~ the ~~cheat book. Most video game possess a book you can buy individually. You need to definitely think about doing a and studying it before you play, or even when you~~'~~re playing~~. ~~In this manner~~, ~~you can complete out of your game play.~~<~~br><br~~>~~People may possibly play game titles~~ to ~~rest following~~ a ~~very long working day in your workplace. Some like socializing by tinkering sufficient reason~~ for ~~friends~~ and ~~family~~. ~~If you have whichever inquiries about where and in what ways~~ to ~~use Clash~~ of ~~a Clans Cheat~~, ~~you can build contact with us~~ at ~~just our web site~~. ~~Other individuals perform associated~~ with ~~when they're jobless with require something for removing their brains away those scenario. No item reasons why you enjoy~~, ~~this information will assist you to engage in usual way which~~ is ~~more relaxed~~.<br><br>~~A real clash of clans hack equipment and tricks could be skillfully handy in the function~~ that ~~you are a keen player of~~ the ~~diversion. The finest hack apparatuses tackle just about any doing work shape - Android and Apple iOS. You also can exploit~~ the ~~obtain that can be used on your apple i phone or Android telephones~~.<br><br>~~Seen the evaluations and~~ see the ~~trailers before buying a relevant video game~~. ~~If you have~~ any ~~inquiries~~ with ~~regards~~ to ~~where by and how~~ to ~~use~~ [~~http://prometeu.net clash of clans hack no download No survey~~], ~~you can call us at~~ the ~~webpage~~. ~~Help it become one thing you are looking at before you get it~~. ~~These video games aren't low-cost~~, ~~and also you likely to get nearly as abundant cash whenever you markets inside~~ a ~~employed video game which you have few utilized several times~~.<br><br>Its important to agenda your apple is consistently confident from association war dilemmas because association wars end up being fought inside a improved breadth absolutely -- the following war zone. On the war region, your adapt and advance rivalry bases instead of given villages; therefore, your villages resources, trophies, and absorber are never in risk.<br><br>~~Were you aware that some [http://Search.Un~~.~~org/search?ie~~=~~utf8&site~~=~~un_org&output~~=~~xml_no_dtd&client~~=~~UN_Website_en&num~~=~~10&lr~~=~~lang_en&proxystylesheet~~=~~UN_Website_en&oe~~=~~utf8&q~~=~~personal+games&Submit~~=~~Go personal games~~] ~~are~~ [~~http~~:~~//Mondediplo.com/spip.php?page=recherche&recherche=educational educational~~] tools? If you know a children that likes to have fun with video games, educational opportunities are a fantastic tactics to combine learning with entertaining. The World wide web can connect you while using thousands of parents with similar values and are unquestionably more than willing to help share their reviews combined with notions with you.<br><br>An individual are are playing a flaunting activity, and you too don't possess knowledge of it, establish the ailment stage to rookie. This should help owners pick-up in the awesome options that come while using the game and discover nearer round the field. Should you set things more than that, in all probability you'll get frustrated and has not possess fun.		[[Image:Monkey Saddle Surface (Shaded).png\|thumb\|The monkey saddle\|300px]]
			In [[mathematics]], the '''monkey saddle''' is the [[surface]] defined by the equation

			:<math> z = x^3 - 3xy^2. \, </math>

			It belongs to the class of [[saddle surface]]s and its name derives from the observation that a [[saddle]] for a [[monkey]] requires three depressions: two for the legs, and one for the tail. The point (0,0,0) on the monkey saddle corresponds to a [[critical point (mathematics)\|degenerate critical point]] of the function ''z''(''x'',''y'') at (0, 0). The monkey saddle has an isolated [[umbilical point]] with zero [[Gaussian curvature]] at the origin, while the curvature is strictly negative at all other points.

			To show that the monkey saddle has three depressions, one can write the equation for ''z'' using [[complex number]]s as
			:<math> z(x,y) = \operatorname{Re} (x+iy)^3.</math>
			It follows that ''z''(''tx'',''ty'') = ''t''<sup>3</sup> ''z''(''x'',''y'') for ''t'' ≥ 0, so the surface is determined by ''z'' on the [[unit circle]]. Parametrizing this by ''e''<sup>''i''φ</sup>, with φ ∈ [0, 2π), one can see that on the unit circle, ''z''(φ) = cos 3φ, so ''z'' has three depressions. By replacing 3 with any integer ''k'' ≥ 1, one can create a saddle with ''k'' depressions.

			== Horse saddle ==

			The term ''horse saddle'' is used, in contrast to monkey saddle, to designate a [[saddle point]] that is a [[minimax]], that is to say a local minimum or maximum depending on the intersecting plane used. The monkey saddle has just a [[point of inflection]]. To see this, consider a line ''y'' = ''kx''. Along this direction the surface becomes simply ''z'' = (1 − 3''k''<sup>2</sup>)''x''<sup>3</sup>, lacking any critical points.

			== See also==
			* [[Saddle point]]
			*[[Saddle surface]]

			==External links==
			* {{MathWorld \| urlname=MonkeySaddle \| title=Monkey Saddle}}

			[[Category:Multivariable calculus]]
			[[Category:Surfaces]]

en>Luckas-bot: robot Adding: fr:Suite périodique

2010-04-02T08:16:41Z

robot Adding: fr:Suite périodique

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