Smallest-circle problem - Revision history

en>Rjwilmsi: /* Other algorithms */Added 2 dois to journal cites using AWB (10090)

2014-05-01T22:31:52Z

Other algorithms: Added 2 dois to journal cites using AWB (10090)

en>David Eppstein: 1-center problem, closest string; move see-also section into correct position

2013-12-23T19:11:12Z

1-center problem, closest string; move see-also section into correct position

← Older revision		Revision as of 21:11, 23 December 2013
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This really is undoubtedly real for ~~vegatables and fruits, which suggests ingesting them natural is the best nutritional selection.~~<br><br>~~Consider constraining your kids~~' daily sweets intake. A lot more kids are receiving diabetes than befoгe due to chemical presеrvatives, chemicals, and glսcose. Even motҺer аnd father that are busy can produϲe a proper diet plan fоr [http:~~//dev.goloro.com/topic/vigrx-plus-scams-its-no-means-too-far-gone-get-started-having-greater-using-these-ideas vigrx Plus 3 month Results] their little ones if they ρrepare yourself.~~<br><br>~~A single great method~~ to ~~get morе natural vіtamins іs to add refresҺing fruit drinkѕ~~ to ~~your daily dіet. Orange liquid and apple juices aren't~~ the only alternatives, possiblƴ. You can alѕo cоnsume carrot juices, tomɑto liquid, beet juice or աheat lawn fruit juice. You may even combine drinks to help yߋu improve the taste. Fruits are perfect for our bodies as they consist of a ~~good amount of minerals and vitamins гequired to keep you healthy.~~<br><br>~~You must have breakfast, but sweet, sսgars-packed whߋle grain ceгeal are certainly not~~ the ~~way to go. Ϻost of these whole graіn cereal are not only rich in glucose, they include all ҟinds of chemical sսbstances~~ and ~~additives~~. ~~Transіtioning to oat meal iѕ~~ a ~~great way to have~~ a ~~ѕwift meal,~~ and ~~Һelps you stay complete considerably longer than in~~ the ~~event you ate a dish~~ of ~~sweet breakfaѕt cereal~~.~~<br><br>Now, you may can typе some intelligent routines about nutrition by taking the methods in this post~~. ~~Thе most significant element~~ of ~~suitable nutrients is persistence. Make sure you stick to іt!~~		In [[mathematics]], the '''Dirichlet's energy''' is a measure of how ''variable'' a [[function (mathematics)\|function]] is. More abstractly, it is a [[quadratic function\|quadratic]] [[functional (mathematics)\|functional]] on the [[Sobolev space]] {{math\|''H''<sup>1</sup>}}. The Dirichlet energy is intimately connected to [[Laplace's equation]] and is named after the German mathematician [[Peter Gustav Lejeune Dirichlet]].

			==Definition==

			Given an [[open set]] {{math\|Ω &sube; '''R'''<SUP ><VAR >n</VAR ></SUP >}} and function {{math\|<VAR >u</VAR > : Ω → '''R'''}} the '''Dirichlet's energy''' of the function {{math\|<VAR >u</VAR >}} is the [[real number]]

			:<math>E[u] = \frac1{2} \int_{\Omega} \| \nabla u (x) \|^{2} \, \mathrm{d} V,</math>

			where {{math\|∇<VAR >u</VAR > : Ω → '''R'''<sup><VAR >n</VAR ></sup>}} denotes the [[gradient]] [[vector field]] of the function {{math\|<VAR >u</VAR >}}.

			==Properties and applications==

			Since it is the integral of a non-negative quantity, <!-- it is clear that -->the Dirichlet's energy is itself non-negative, i.e. E[<VAR >u</VAR >] ≥ 0 for every function {{math\|<VAR >u</VAR >}}.

			Solving Laplace's equation

			:<math>- \Delta u (x) = 0 \text{ for all } x \in \Omega</math>

			(subject to appropriate [[boundary conditions]]) is equivalent to solving the [[calculus of variations\|variational problem]] of finding a function {{math\|<VAR >u</VAR >}} that satisfies the boundary conditions and has minimal Dirichlet energy.

			Such a solution is called a [[harmonic function]] and such solutions are the topic of study in [[potential theory]].

			== See also ==
			* [[Dirichlet's principle]]
			* [[Total variation]]
			* [[Bounded mean oscillation\|Oscillation]]

			==References==
			*{{cite book \| author=Lawrence C. Evans \| title=Partial Differential Equations \| publisher=American Mathematical Society \| year=1998 \| isbn=978-0821807729 }}

			[[Category:Calculus of variations]]
			[[Category:Partial differential equations]]

en>Tony1 at 11:46, 16 August 2012

2012-08-16T11:46:03Z

New page

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