MDC-2 - Revision history

en>Cedar101: /* MDC-2DES hashes */

2014-08-04T06:20:18Z

MDC-2DES hashes

← Older revision		Revision as of 08:20, 4 August 2014
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	~~{{otheruses4\|rectifiable sets in measure theory\|rectifiable curves\|Arc length}}~~		I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Mississippi is exactly where his home is. Invoicing is what I do. What me and my family members adore is bungee jumping but I've been taking on new things recently.<br><br>Feel free to visit my web page spirit messages; [http://1.234.36.240/fxac/m001_2/7330 http://1.234.36.240/fxac/m001_2/7330],
	~~In [[mathematics]]~~, ~~a '''rectifiable set''~~' is ~~a set that~~ is ~~smooth in a certain [[measure theory\|measure-theoretic]] sense~~. It is an extension of the idea of a [[rectifiable curve]] to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth [[manifold]]s, including tangent spaces that are defined [[almost everywhere]]. ~~Rectifiable sets are the underlying object of study in [[geometric measure theory]].~~

	~~==Definition==~~
	~~A subset <math>E</math> of [[Euclidean space]] <math>\mathbb{R}^n</math>~~ is ~~said to be ''~~'~~<math>m</math>-rectifiable''' set if there exist a [[countable]] collection <math>\{f_i\}</math> of continuously differentiable maps~~

	~~:<math>f_i:\mathbb{R}^m \to \mathbb{R}^n</math>~~

	~~such that the <math>m</math>-[[Hausdorff measure]] <math>\mathcal{H}^m</math> of~~

	~~:<math>E\backslash \bigcup_{i=0}^\infty f_i\left(\mathbb{R}^m\right)</math>~~

	~~is zero~~. ~~The backslash here denotes the [[set difference]]. Equivalently, the~~ <~~math~~>~~f_i~~<~~/math~~> ~~may be taken~~ to be [~~[Lipschitz continuous]] without altering the definition.~~

	~~A set <math>E</math> is said to be '''purely <math>m</math>-unrectifiable''' if for ''every'' (continuous, differentiable) <math>f~~:~~\mathbb{R}^m \to \mathbb{R}^n<~~/~~math>, one has~~

	~~:<math>\mathcal{H}^m \left(E \cap f\left(\mathbb{R}^m\right)\right)=0.<~~/~~math>~~

	~~A standard example of a purely-~~1~~-unrectifiable set in two dimensions is the cross-product of the [[Smith-Volterra-Cantor set]] times itself~~.

	~~==References==~~
	* {{springer\|author=T.C.~~O'Neil\|id=G~~/~~g130040\|title=Geometric measure theory}}~~

	~~==External links==~~
	* [http://~~www~~.~~encyclopediaofmath~~.~~org/index~~.~~php/Rectifiable_set Rectifiable set] at [http:~~//~~www.encyclopediaofmath.org~~/ ~~Encyclopedia of Mathematics]~~

	~~[[Category:Measure theory]~~]

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3841252

2013-03-14T15:37:31Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q3841252

← Older revision		Revision as of 17:37, 14 March 2013
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	~~Hi there~~, ~~I am Alyson Boon even though it~~ is ~~not the name on my birth certificate~~. ~~The favorite pastime for him and his children~~ is to ~~perform lacross and he would by no means give~~ it up. ~~Ohio~~ is ~~where her house~~ is. ~~Office supervising~~ is ~~where her main income comes from.~~<br><br>~~Also visit my webpage~~ :: [http://~~Chorokdeul~~.Co.kr/index.php~~?document_srl=324263&mid=customer21 phone psychic readings~~]		{{otheruses4\|rectifiable sets in measure theory\|rectifiable curves\|Arc length}}
			In [[mathematics]], a '''rectifiable set''' is a set that is smooth in a certain [[measure theory\|measure-theoretic]] sense. It is an extension of the idea of a [[rectifiable curve]] to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth [[manifold]]s, including tangent spaces that are defined [[almost everywhere]]. Rectifiable sets are the underlying object of study in [[geometric measure theory]].

			==Definition==
			A subset <math>E</math> of [[Euclidean space]] <math>\mathbb{R}^n</math> is said to be '''<math>m</math>-rectifiable''' set if there exist a [[countable]] collection <math>\{f_i\}</math> of continuously differentiable maps

			:<math>f_i:\mathbb{R}^m \to \mathbb{R}^n</math>

			such that the <math>m</math>-[[Hausdorff measure]] <math>\mathcal{H}^m</math> of

			:<math>E\backslash \bigcup_{i=0}^\infty f_i\left(\mathbb{R}^m\right)</math>

			is zero. The backslash here denotes the [[set difference]]. Equivalently, the <math>f_i</math> may be taken to be [[Lipschitz continuous]] without altering the definition.

			A set <math>E</math> is said to be '''purely <math>m</math>-unrectifiable''' if for ''every'' (continuous, differentiable) <math>f:\mathbb{R}^m \to \mathbb{R}^n</math>, one has

			:<math>\mathcal{H}^m \left(E \cap f\left(\mathbb{R}^m\right)\right)=0.</math>

			A standard example of a purely-1-unrectifiable set in two dimensions is the cross-product of the [[Smith-Volterra-Cantor set]] times itself.

			==References==
			* {{springer\|author=T.C.O'Neil\|id=G/g130040\|title=Geometric measure theory}}

			==External links==
			* [http://www.encyclopediaofmath.org/index.php/Rectifiable_set Rectifiable set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

			[[Category:Measure theory]]

en>Yobot: WP:CHECKWIKI error fixes + general fixes using AWB (8024)

2012-03-19T16:12:14Z

WP:CHECKWIKI error fixes + general fixes using AWB (8024)

New page

Hi there, I am Alyson Boon even though it is not the name on my birth certificate. The favorite pastime for him and his children is to perform lacross and he would by no means give it up. Ohio is where her house is. Office supervising is where her main income comes from.<br><br>Also visit my webpage :: [http://Chorokdeul.Co.kr/index.php?document_srl=324263&mid=customer21 phone psychic readings]