SameGame - Revision history

en>Pioneersaigon at 21:16, 26 October 2014

2014-10-26T21:16:16Z

← Older revision		Revision as of 23:16, 26 October 2014
Line 1:		Line 1:
	~~Major may~~ be the ~~name he loves in order to called with and he totally digs that determine. For years she's been~~ [http://~~en.~~search.~~wordpress~~.com/?q=living~~+citrus~~ living ~~citrus]~~. ~~My day job is~~ a data processing [http://~~data~~.~~gov~~.uk/~~data~~/~~search?q=officer officer] as well as the salary recently been really potentially bountiful. To play hockey is what i do obtain. If you want to learn more about check out his website:~~ http://Www.~~evolutisweb~~.com/~~index.php~~/~~User:FionaVallery~~<br><br>~~My web~~ page :: [http://~~Www~~.~~evolutisweb~~.com/~~index.php~~/~~User:FionaVallery snoop dogg~~ miley cyrus]		Deedra will be the [http://search.Huffingtonpost.com/search?q=she%27s+called&s_it=header_form_v1 she's called] although it is far from her birth name. Alabama has for ages been my living place so love regularly living these. It's not a common thing but what I recommend doing will be always to play hockey but Cannot make it my profession really. For years I've been working like a data processing officer even so plan on changing the following. His wife and she maintain web page. You might to be able to check it out: http://www.Ourabilitywiki.com/dining-and-night-life/[http://Www.Tumblr.com/tagged/diamond-tape-seemed-at-a-glance diamond-tape-seemed-at-a-glance]<br><br>Take a look at my page - [http://www.Ourabilitywiki.com/dining-and-night-life/diamond-tape-seemed-at-a-glance miley cyrus wrecking ball]

197.76.137.151: /* Versions */

2014-03-02T18:21:39Z

Versions

← Older revision		Revision as of 20:21, 2 March 2014
Line 1:		Line 1:
	~~__NOTOC__~~		Major may be the name he loves in order to called with and he totally digs that determine. For years she's been [http://en.search.wordpress.com/?q=living+citrus living citrus]. My day job is a data processing [http://data.gov.uk/data/search?q=officer officer] as well as the salary recently been really potentially bountiful. To play hockey is what i do obtain. If you want to learn more about check out his website: http://Www.evolutisweb.com/index.php/User:FionaVallery<br><br>My web page :: [http://Www.evolutisweb.com/index.php/User:FionaVallery snoop dogg miley cyrus]
	~~[[Image:Kochsim.gif\|thumb\|right\|250px\|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.]]~~
	~~[[File:Standard self-similarity.png\|thumb\|300px\|Standard (trivial) self-similarity.<ref>Mandelbrot, Benoit B. (1982)~~. '~~'The Fractal Geometry of Nature'', p.44. ISBN 978-0716711865.</ref>]]~~

	In [[mathematics]], a '''self-similar''' object is exactly or approximately [[similarity (geometry)\|similar]] to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as [[coastline]]s, are statistically self-similar: parts of them show the same statistical properties at many scales.<ref>{{cite web\|
	~~url=~~http://~~www~~.~~sciencemag~~.~~org/content/156/3775~~/~~636\|~~
	~~title=How Long Is the Coast of Britain~~? ~~Statistical Self-Similarity and Fractional Dimension\|~~
	~~author~~=~~Benoit Mandelbrot\|~~
	~~publisher=Science Magazine\|~~
	~~date=May 1967\|~~
	~~authorlink=Benoit Mandelbrot}}</ref> Self-similarity is a typical property of [[fractal~~]]s.

	~~[[Scale invariance]] is an exact form of self-similarity where at any magnification there~~ is a ~~smaller piece of the object that is~~ [~~[Similarity (geometry)\|similar]] to the whole. For instance, a side of the [[Koch snowflake]] is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape~~.

	~~The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales~~. ~~As a [[counterexample]], whereas any portion of a [[straight line]] may resemble the whole, further detail is not revealed.~~

	~~==Definition==~~
	~~A [[Compact space\|compact]] [[topological space]] ''X'' is self-similar if there exists a [[finite set]] ''S'' indexing a set of non-[[surjective]] [[homeomorphism]]s <math>\{ f_s \}_{s\in S}<~~/~~math> for which~~

	~~:<math>X=\cup_{s\in S} f_s(X)<~~/~~math>~~

	~~If <math>X\subset Y</math>, we call ''X'' self-similar if it is the only [[Non-empty set\|non-empty]] [[subset]] of ''Y'' such that the equation above holds for <math>\{ f_s \}_{s\in S}</math>. We call~~

	~~:<math>\mathfrak{L}~~=~~(X,S,\{ f_s \}_{s\in S})</math>~~

	~~a ''self-similar structure''. The homeomorphisms may be [[iterated function\|iterated~~]~~], resulting in an [[iterated function system]]. The composition of functions creates the algebraic structure of a [[monoid]]. When the set ''S'' has only two elements, the monoid is known~~ as ~~the [[dyadic monoid]]. The dyadic monoid can be visualized~~ as ~~an infinite [[binary tree]]; more generally, if the set ''S'' has ''p'' elements, then~~ the ~~monoid may be represented as a [[p-adic number\|p-adic]] tree~~.

	~~The [[automorphism]]s of the dyadic monoid~~ is ~~the [[modular group]]; the automorphisms can be pictured as [[Hyperbolic coordinates\|hyperbolic rotation]]s of the binary tree~~.

	~~==Examples==~~
	~~[[Image:Feigenbaumzoom.gif\|left\|thumb\|201px\|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)]]~~
	~~[[Image~~:~~Fractal fern explained.png\|thumb\|right\|200px\|An image of a fern which exhibits [[affine transformation\|affine]] self-similarity]]~~

	~~The [[Mandelbrot set]] is also self-similar around [[Misiurewicz point]]s.~~

	Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in [[teletraffic engineering]], [[packet switched]] data traffic patterns seem to be statistically self-similar.<ref>Leland ''et al.'' "On the self-similar nature of Ethernet traffic", ''IEEE/ACM Transactions on Networking'', Volume '''2''', Issue 1 (February 1994)</ref> This property means that simple models using a [[Poisson distribution]] are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways.

	Similarly, [[stock market]] movements are described as displaying [[self-affinity]], i.e. they appear self-similar when transformed via an appropriate [[affine transformation]] for the level of detail being shown.<ref>{{cite web\|
	~~url=~~http://~~www~~.~~sciam~~.com/~~article~~.~~cfm?id=multifractals-explain-wall-street\|~~
	~~title=How Fractals Can Explain What's Wrong with Wall Street\|~~
	~~author=Benoit Mandelbrot\|~~
	~~publisher=Scientific American\|~~
	~~date=February 1999\|~~
	~~authorlink=Benoit Mandelbrot}}<~~/~~ref>~~

	~~[[Finite subdivision rules]] are a powerful technique for building self-similar sets, including the [[Cantor set]] and the [[Sierpinski triangle]].~~

	~~[[File~~:~~RepeatedBarycentricSubdivision.png\|thumb\|A triangle subdivided repeatedly using [[barycentric subdivision]]. The [[complement]] of the large circles is becoming a [[Sierpinski carpet]]]]~~

	~~{{clear\|left}} [[Andrew Lo]] describes Stock Market log return self-similarity in [[Econometrics]].~~<~~ref~~>~~Campbell, Lo and MacKinlay (1991) "[[Econometrics]] of Financial Markets ", Princeton University Press! iSBN 978-0691043012~~<~~/ref~~>

	~~=== In nature ===~~

	~~[[File~~:~~Flickr - cyclonebill - Romanesco.jpg\|thumb\|right\|200px\|Close-up of a [[Romanesco broccoli]].]]~~

	Self-similarity can be found in nature, as well. To the right is a mathematically-generated, perfectly self-similar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as [[Romanesco broccoli]], exhibit strong self-similarity.

	~~=== In music ===~~

	*A [[Shepard tone]] is self-similar in the frequency or wavelength domains.
	* The [[Denmark\|Danish]] [[composer]] [[Per Nørgård]] has made use of a self-similar [[integer sequence]] named the 'infinity series' in much of his music.

	~~==See also==~~
	* [[Droste effect]]
	* [[Long-range dependency]]
	* [[Non-well-founded set theory]]
	* [[Recursion]]
	* [[Self-dissimilarity]]
	* [[Self-reference]]
	* [[Tweedie distributions]]
	* [[Zipf's law]]

	~~==References==~~
	~~{{reflist}}~~

	~~==External links==~~
	*[http://~~www~~.~~ericbigas~~.com/~~fractals/cc "Copperplate Chevrons"] — a self-similar fractal zoom movie~~
	*[http://pi.~~314159.ru~~/~~longlist.htm "Self-Similarity"] — New articles about the Self-Similarity. Waltz Algorithm~~

	~~{{Fractals}}~~

	~~{{DEFAULTSORT:Self-Similarity}}~~
	~~[[Category:Fractals]]~~
	~~[[Category:Scaling symmetries]]~~
	~~[[Category~~:~~Homeomorphisms]~~]

en>Chris the speller: /* Rules variations */per WP:HYPHEN, sub-subsection 3, points 3,4,5, replaced: randomly- → randomly using AWB

2013-11-15T03:15:11Z

Rules variations: per WP:HYPHEN, sub-subsection 3, points 3,4,5, replaced: randomly- → randomly using AWB

← Older revision		Revision as of 05:15, 15 November 2013
Line 1:		Line 1:
	~~Greetings! I am Marvella and I feel comfortable~~ when ~~people use~~ the ~~complete~~ title. ~~The favorite hobby for my kids and me~~ is to ~~perform baseball~~ and I'~~m attempting to make~~ it a ~~occupation~~. ~~North Dakota~~ is ~~her beginning place but she will have~~ to ~~transfer~~ 1 ~~working day or~~ an ~~additional~~. ~~She~~ is a ~~librarian but she's always wanted her own company~~.<br><br>~~Also visit my website~~ ... [http://~~Agaolgu~~.com/~~weightlossfooddelivery24650 Agaolgu~~.~~com~~]		__NOTOC__
			[[Image:Kochsim.gif\|thumb\|right\|250px\|A [[Koch curve]] has an infinitely repeating self-similarity when it is magnified.]]
			[[File:Standard self-similarity.png\|thumb\|300px\|Standard (trivial) self-similarity.<ref>Mandelbrot, Benoit B. (1982). ''The Fractal Geometry of Nature'', p.44. ISBN 978-0716711865.</ref>]]

			In [[mathematics]], a '''self-similar''' object is exactly or approximately [[similarity (geometry)\|similar]] to a part of itself (i.e. the whole has the same shape as one or more of the parts). Many objects in the real world, such as [[coastline]]s, are statistically self-similar: parts of them show the same statistical properties at many scales.<ref>{{cite web\|
			url=http://www.sciencemag.org/content/156/3775/636\|
			title=How Long Is the Coast of Britain? Statistical Self-Similarity and Fractional Dimension\|
			author=Benoit Mandelbrot\|
			publisher=Science Magazine\|
			date=May 1967\|
			authorlink=Benoit Mandelbrot}}</ref> Self-similarity is a typical property of [[fractal]]s.

			[[Scale invariance]] is an exact form of self-similarity where at any magnification there is a smaller piece of the object that is [[Similarity (geometry)\|similar]] to the whole. For instance, a side of the [[Koch snowflake]] is both symmetrical and scale-invariant; it can be continually magnified 3x without changing shape.

			The non-trivial similarity evident in fractals is distinguished by their fine structure, or detail on arbitrarily small scales. As a [[counterexample]], whereas any portion of a [[straight line]] may resemble the whole, further detail is not revealed.

			==Definition==
			A [[Compact space\|compact]] [[topological space]] ''X'' is self-similar if there exists a [[finite set]] ''S'' indexing a set of non-[[surjective]] [[homeomorphism]]s <math>\{ f_s \}_{s\in S}</math> for which

			:<math>X=\cup_{s\in S} f_s(X)</math>

			If <math>X\subset Y</math>, we call ''X'' self-similar if it is the only [[Non-empty set\|non-empty]] [[subset]] of ''Y'' such that the equation above holds for <math>\{ f_s \}_{s\in S}</math>. We call

			:<math>\mathfrak{L}=(X,S,\{ f_s \}_{s\in S})</math>

			a ''self-similar structure''. The homeomorphisms may be [[iterated function\|iterated]], resulting in an [[iterated function system]]. The composition of functions creates the algebraic structure of a [[monoid]]. When the set ''S'' has only two elements, the monoid is known as the [[dyadic monoid]]. The dyadic monoid can be visualized as an infinite [[binary tree]]; more generally, if the set ''S'' has ''p'' elements, then the monoid may be represented as a [[p-adic number\|p-adic]] tree.

			The [[automorphism]]s of the dyadic monoid is the [[modular group]]; the automorphisms can be pictured as [[Hyperbolic coordinates\|hyperbolic rotation]]s of the binary tree.

			==Examples==
			[[Image:Feigenbaumzoom.gif\|left\|thumb\|201px\|Self-similarity in the [[Mandelbrot set]] shown by zooming in on the Feigenbaum point at (−1.401155189..., 0)]]
			[[Image:Fractal fern explained.png\|thumb\|right\|200px\|An image of a fern which exhibits [[affine transformation\|affine]] self-similarity]]

			The [[Mandelbrot set]] is also self-similar around [[Misiurewicz point]]s.

			Self-similarity has important consequences for the design of computer networks, as typical network traffic has self-similar properties. For example, in [[teletraffic engineering]], [[packet switched]] data traffic patterns seem to be statistically self-similar.<ref>Leland ''et al.'' "On the self-similar nature of Ethernet traffic", ''IEEE/ACM Transactions on Networking'', Volume '''2''', Issue 1 (February 1994)</ref> This property means that simple models using a [[Poisson distribution]] are inaccurate, and networks designed without taking self-similarity into account are likely to function in unexpected ways.

			Similarly, [[stock market]] movements are described as displaying [[self-affinity]], i.e. they appear self-similar when transformed via an appropriate [[affine transformation]] for the level of detail being shown.<ref>{{cite web\|
			url=http://www.sciam.com/article.cfm?id=multifractals-explain-wall-street\|
			title=How Fractals Can Explain What's Wrong with Wall Street\|
			author=Benoit Mandelbrot\|
			publisher=Scientific American\|
			date=February 1999\|
			authorlink=Benoit Mandelbrot}}</ref>

			[[Finite subdivision rules]] are a powerful technique for building self-similar sets, including the [[Cantor set]] and the [[Sierpinski triangle]].

			[[File:RepeatedBarycentricSubdivision.png\|thumb\|A triangle subdivided repeatedly using [[barycentric subdivision]]. The [[complement]] of the large circles is becoming a [[Sierpinski carpet]]]]

			{{clear\|left}} [[Andrew Lo]] describes Stock Market log return self-similarity in [[Econometrics]].<ref>Campbell, Lo and MacKinlay (1991) "[[Econometrics]] of Financial Markets ", Princeton University Press! iSBN 978-0691043012</ref>

			=== In nature ===

			[[File:Flickr - cyclonebill - Romanesco.jpg\|thumb\|right\|200px\|Close-up of a [[Romanesco broccoli]].]]

			Self-similarity can be found in nature, as well. To the right is a mathematically-generated, perfectly self-similar image of a fern, which bears a marked resemblance to natural ferns. Other plants, such as [[Romanesco broccoli]], exhibit strong self-similarity.

			=== In music ===

			*A [[Shepard tone]] is self-similar in the frequency or wavelength domains.
			* The [[Denmark\|Danish]] [[composer]] [[Per Nørgård]] has made use of a self-similar [[integer sequence]] named the 'infinity series' in much of his music.

			==See also==
			* [[Droste effect]]
			* [[Long-range dependency]]
			* [[Non-well-founded set theory]]
			* [[Recursion]]
			* [[Self-dissimilarity]]
			* [[Self-reference]]
			* [[Tweedie distributions]]
			* [[Zipf's law]]

			==References==
			{{reflist}}

			==External links==
			*[http://www.ericbigas.com/fractals/cc "Copperplate Chevrons"] — a self-similar fractal zoom movie
			*[http://pi.314159.ru/longlist.htm "Self-Similarity"] — New articles about the Self-Similarity. Waltz Algorithm

			{{Fractals}}

			{{DEFAULTSORT:Self-Similarity}}
			[[Category:Fractals]]
			[[Category:Scaling symmetries]]
			[[Category:Homeomorphisms]]

en>Jagged 85: /* History */ re-word

2012-02-19T20:19:53Z

History: re-word

New page

Greetings! I am Marvella and I feel comfortable when people use the complete title. The favorite hobby for my kids and me is to perform baseball and I'm attempting to make it a occupation. North Dakota is her beginning place but she will have to transfer 1 working day or an additional. She is a librarian but she's always wanted her own company.<br><br>Also visit my website ... [http://Agaolgu.com/weightlossfooddelivery24650 Agaolgu.com]