Church encoding - Revision history

en>Thepigdog: /* See also */

2014-11-08T12:19:11Z

en>Thepigdog at 05:06, 2 March 2014

2014-03-02T05:06:21Z

← Older revision		Revision as of 07:06, 2 March 2014
Line 1:		Line 1:
	~~{{Context\|date=September 2011}}~~		Alyson is what my spouse loves to contact me but I don't like when people use my full title. To play lacross is the thing I adore most of all. Since he was 18 he's been working as an information officer but he ideas on altering it. For years he's been residing in Alaska and he doesn't plan on changing it.<br><br>Here is my site :: clairvoyants ([http://www.chk.woobi.co.kr/xe/?document_srl=346069 www.chk.woobi.co.kr])
	~~In the [[calculus of variations]],~~ '~~''Γ-convergence''' ('''Gamma-convergence''')~~ is ~~a notion~~ of ~~convergence for [[Functional (mathematics)\|functionals]]~~. It was ~~introduced by [[Ennio de Giorgi]]~~.

	~~==Definition==~~
	~~Let <math>X</math> be a [[topological space]]~~ and ~~<math>F_n:X\to[0,+\infty)</math> a sequence of functionals~~ on ~~<math>X</math>~~. ~~Then~~ <~~math~~>~~F_n~~<~~/math~~> ~~are said to <math>\Gamma</math>-converge to the <math>\Gamma</math>-limit <math>F:X\to[0,+\infty)</math> if the following two conditions hold:~~
	* Lower bound inequality: ~~For every sequence <math>x_n\in X</math> such that <math>x_n\to x</math> as <math>n\to+\infty</math>,~~
	: ~~<math>F~~(~~x)\le\liminf_{n\to\infty} F_n(x_n).</math>~~
	* Upper bound inequality: For every <math>x\in X</math>, there is a sequence <math>x_n</math> converging to <math>x</math> such that
	: ~~<math>F(x)\ge\limsup_{n\to\infty} F_n(x_n)</math>~~

	~~The first condition means that <math>F<~~/~~math> provides an asymptotic common lower bound for the <math>F_n<~~/~~math>~~. ~~The second condition means that this lower bound is optimal~~.

	~~==Properties==~~
	* Minimizers converge to minimizers: If <math>F_n</math> <math>\Gamma</math>-converge to <math>F</math>, and <math>x_n</math> is a minimizer for <math>F_n</math>, then every cluster point of the sequence <math>x_n</math> is a minimizer of <math>F</math>.
	* <math>\Gamma</math>-limits are always [[Semi-continuity\|lower semicontinuous]].
	* <math>\Gamma</math>-convergence is stable under continuous perturbations: If <math>F_n</math> <math>\Gamma</math>-converges to <math>F</math> and <math>G:X\to[0,+\infty)</math> is continuous, then <math>F_n+G</math> will <math>\Gamma</~~math>-converge to <math>F+G<~~/~~math>.~~
	* A constant sequence of functionals <math>F_n=~~F</math> does not necessarily <math>\Gamma</math>-converge to <math>F</math>, but to the ''relaxation'' of <math>F</math>, the largest lower semicontinuous functional below <math>F</math>~~.

	~~==Applications==~~
	~~An important use for <math>\Gamma</math>-convergence is in [[homogenization (mathematics)\|homogenization theory]]~~. ~~It can also be used to rigorously justify the passage from discrete to continuum theories for materials, e~~.g. ~~in [[Elasticity (physics~~)~~\|elasticity]] theory.~~

	~~==See also==~~
	* [[Mosco convergence]]

	~~==References==~~
	* A. Braides: ''Γ-convergence for beginners''. Oxford University Press, 2002.
	* G. Dal Maso: ''An introduction to Γ-convergence''. Birkhäuser, Basel 1993.

	~~{{DEFAULTSORT:Gamma-Convergence}}~~
	~~[[Category:Calculus of variations]]~~
	~~[[Category:Variational analysis]]~~


	~~{{Mathanalysis-stub}}~~

en>Thepigdog at 13:53, 16 January 2014

2014-01-16T13:53:07Z

← Older revision		Revision as of 15:53, 16 January 2014
Line 1:		Line 1:
	~~Wilber Berryhill~~ is ~~what his spouse enjoys to call him and he completely enjoys this name~~. ~~I've~~ [~~http~~://~~jplusfn~~.~~Gaplus.kr~~/xe/~~qna~~/~~78647 psychic phone readings] always cherished living~~ in ~~Kentucky but now I'm contemplating other choices. I am really fond of~~ to go to ~~karaoke but I've been using on new things lately~~. ~~Credit authorising~~ is ~~how she tends~~ to ~~make a residing.~~<br><br>~~Take a look at my weblog - psychic readers [http~~://~~srncomm~~.~~com~~/~~blog~~/~~2014~~/08/25/~~relieve~~-~~that~~-~~stress~~-~~find~~-~~a-new-hobby~~/ ~~good psychic] -~~ [~~http:~~//~~www~~.~~prograd.uff.br~~/~~novo~~/~~facts~~-~~about~~-~~growing~~-~~greater~~-~~organic-garden www~~.~~prograd~~.~~uff.br~~],		{{Context\|date=September 2011}}
			In the [[calculus of variations]], '''Γ-convergence''' ('''Gamma-convergence''') is a notion of convergence for [[Functional (mathematics)\|functionals]]. It was introduced by [[Ennio de Giorgi]].

			==Definition==
			Let <math>X</math> be a [[topological space]] and <math>F_n:X\to[0,+\infty)</math> a sequence of functionals on <math>X</math>. Then <math>F_n</math> are said to <math>\Gamma</math>-converge to the <math>\Gamma</math>-limit <math>F:X\to[0,+\infty)</math> if the following two conditions hold:
			* Lower bound inequality: For every sequence <math>x_n\in X</math> such that <math>x_n\to x</math> as <math>n\to+\infty</math>,
			: <math>F(x)\le\liminf_{n\to\infty} F_n(x_n).</math>
			* Upper bound inequality: For every <math>x\in X</math>, there is a sequence <math>x_n</math> converging to <math>x</math> such that
			: <math>F(x)\ge\limsup_{n\to\infty} F_n(x_n)</math>

			The first condition means that <math>F</math> provides an asymptotic common lower bound for the <math>F_n</math>. The second condition means that this lower bound is optimal.

			==Properties==
			* Minimizers converge to minimizers: If <math>F_n</math> <math>\Gamma</math>-converge to <math>F</math>, and <math>x_n</math> is a minimizer for <math>F_n</math>, then every cluster point of the sequence <math>x_n</math> is a minimizer of <math>F</math>.
			* <math>\Gamma</math>-limits are always [[Semi-continuity\|lower semicontinuous]].
			* <math>\Gamma</math>-convergence is stable under continuous perturbations: If <math>F_n</math> <math>\Gamma</math>-converges to <math>F</math> and <math>G:X\to[0,+\infty)</math> is continuous, then <math>F_n+G</math> will <math>\Gamma</math>-converge to <math>F+G</math>.
			* A constant sequence of functionals <math>F_n=F</math> does not necessarily <math>\Gamma</math>-converge to <math>F</math>, but to the ''relaxation'' of <math>F</math>, the largest lower semicontinuous functional below <math>F</math>.

			==Applications==
			An important use for <math>\Gamma</math>-convergence is in [[homogenization (mathematics)\|homogenization theory]]. It can also be used to rigorously justify the passage from discrete to continuum theories for materials, e.g. in [[Elasticity (physics)\|elasticity]] theory.

			==See also==
			* [[Mosco convergence]]

			==References==
			* A. Braides: ''Γ-convergence for beginners''. Oxford University Press, 2002.
			* G. Dal Maso: ''An introduction to Γ-convergence''. Birkhäuser, Basel 1993.

			{{DEFAULTSORT:Gamma-Convergence}}
			[[Category:Calculus of variations]]
			[[Category:Variational analysis]]


			{{Mathanalysis-stub}}

en>Paul1andrews: /* Definition */

2012-08-19T22:57:05Z

Definition

New page

Wilber Berryhill is what his spouse enjoys to call him and he completely enjoys this name. I've [http://jplusfn.Gaplus.kr/xe/qna/78647 psychic phone readings] always cherished living in Kentucky but now I'm contemplating other choices. I am really fond of to go to karaoke but I've been using on new things lately. Credit authorising is how she tends to make a residing.<br><br>Take a look at my weblog - psychic readers [http://srncomm.com/blog/2014/08/25/relieve-that-stress-find-a-new-hobby/ good psychic] - [http://www.prograd.uff.br/novo/facts-about-growing-greater-organic-garden www.prograd.uff.br],

← Older revision		Revision as of 14:19, 8 November 2014
Line 1:		Line 1:
	~~Alyson is what my spouse loves to contact me but I don~~'~~t like when people use my full~~ title. To play lacross ~~is the thing I adore most of all. Since he was 18~~ he'~~s been working as an information officer but he ideas on altering~~ it. For years he's been ~~residing~~ in Alaska and he doesn't ~~plan~~ on ~~changing~~ it.<br><br>~~Here is~~ my ~~site~~ :: ~~clairvoyants (~~[http://~~www~~.~~chk~~.~~woobi.co.kr~~/xe/~~?document_srl=346069 www.chk.woobi.co~~.kr])		The writer's title is Andera and she believes it seems fairly great. The favorite hobby for him and his children is to play lacross and he'll be beginning some thing else alongside with it. Invoicing is my occupation. For many years he's been living in Alaska and he doesn't strategy on altering it.<br><br>Feel free to surf to my web blog :: psychic readings; [http://breenq.com/index.php?do=/profile-1144/info/ breenq.com],