Laser rangefinder - Revision history

en>Wesleynlevy: /* Sports */ added difference between western & eastern US hunting

2015-01-13T03:49:10Z

Sports: added difference between western & eastern US hunting

← Older revision		Revision as of 05:49, 13 January 2015
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	: '~~'For other inequalities named after Wirtinger, see [[Wirtinger's inequality]]~~.''		Irwin Butts is what my wife enjoys to contact me although I don't really like becoming known as like that. What I adore performing is doing ceramics but I haven't produced a dime with it. Years ago we moved to [http://chastity.com/chastity-qa/stds/infections/whats-most-common-std/whats-most North Dakota] and I love every working day residing right here. Hiring has been my occupation for some home std test time but I've already applied for an additional one.<br><br>Feel free at home std test to visit my homepage ... at home std testing ([http://tomport.ru/node/621726 click to read])

	~~In [[mathematics]], historically '''Wirtinger's inequality''' for real functions was an [[inequality (mathematics)\|inequality]] used in [[Fourier analysis]]~~. ~~It was named after [[Wilhelm Wirtinger]]. It was used in 1904~~ to ~~prove the [~~[~~isoperimetric inequality]]. A variety of closely related results are today known as Wirtinger's inequality.~~

	~~==Theorem==~~
	~~===First version===~~
	~~Let <math>f~~ : ~~\mathbb{R} \to \mathbb{R}<~~/~~math> be a [[periodic function]] of period 2π, which is continuous and has a continuous derivative throughout '''R''', and such that~~

	~~:<math>\int_0^{2\pi}f(x) \, dx = 0~~.</~~math>~~

	~~Then~~

	~~:<math>\int_0^{2\pi}f'^2(x) \, dx \ge \int_0^{2\pi}f^2(x) \, dx<~~/~~math>~~

	~~with equality [[if~~ and ~~only if]] ''f''(''x'') = ''a'' sin(''x'') + ''b'' cos(''x'')~~ for some '~~'a'' and ''b'' (or equivalently ''f''(''x'') = ''c'' sin (''x'' + ''d'')~~ for ~~some ''c'' and ''d'').~~

	~~This version of the Wirtinger inequality is the~~ one~~-dimensional [[Poincaré inequality]], with optimal constant~~.

	~~===Second version===~~
	~~The following related inequality is also called Wirtinger's inequality {{harv\|Dym\|McKean\|1985}}:~~

	:<~~math~~>~~\pi^{2}\int_0^a \|f\|^2 \le a^2 \int_0^a\|f'\|^2~~<~~/math~~>

	~~whenever ''f'' is a C<sup>1</sup> function such that ''f''(0) = ''f''(''a'') = 0.~~ ~~In this form, Wirtinger's inequality is seen as the one-dimensional version of [[Friedrichs' inequality]]~~.

	~~===Proof===~~
	~~The proof of the two versions are similar~~. ~~Here is a proof of the first version of the inequality~~. ~~Since~~ [~~[Dirichlet's conditions]] are met, we can write~~

	:~~<math>f(x)=\frac{1}{2}a_0+\sum_{n\ge 1}\left(a_n\frac{\sin nx}{\sqrt{\pi}}+b_n\frac{\cos nx}{\sqrt{\pi}}\right),<~~/~~math>~~

	~~and moreover ''a''<sub>0<~~/~~sub> = 0 since the integral of ''f'' vanishes~~. ~~By [[Parseval's identity]],~~

	~~:<math>\int_0^{2\pi}f^2(x)dx=\sum_{n=1}^\infty(a_n^2+b_n^2)<~~/~~math>~~

	~~and~~

	~~:<math>\int_0^{2\pi}f'^2(x) \, dx = \sum_{n=1}^\infty n^2(a_n^2+b_n^2)<~~/~~math>~~

	~~and since the summands are all ≥ 0, we get the desired inequality, with equality if and only if ''a<sub>n</sub>'' = ''b<sub>n</sub>'' = 0 for all ''n'' ≥ 2.~~

	~~==References==~~
	*{{citation\|first1=H\|last1=Dym\|authorlink1=Harry Dym\|first2=H\|last2=McKean\|title=Fourier series and integrals\|publisher=Academic press\|year=1985\|isbn=978-0-12-226451-1}}
	*[[Vadim Komkov\|Komkov, Vadim]~~] (1983~~) ~~Euler's buckling formula and Wirtinger's inequality. Internat. J. Math. Ed. Sci. Tech. 14, no. 6, 661—668.~~

	~~{{PlanetMath attribution\|id=5393\|title=Wirtinger's inequality}}~~

	~~[[Category:Fourier analysis]]~~
	~~[[Category:Inequalities]]~~
	~~[[Category:Theorems in analysis]]~~

en>Kbog: /* Industry production processes */

2013-11-15T03:28:52Z

Industry production processes

← Older revision		Revision as of 05:28, 15 November 2013
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	~~Hello~~. ~~Allow me introduce~~ the ~~author~~. ~~Her title is Emilia Shroyer but it~~'s ~~not the most female title out there~~. ~~Managing individuals~~ has ~~been his working day job~~ for a ~~while~~. ~~His wife doesn't like it~~ the ~~way he does but what he truly likes performing~~ is ~~to do aerobics and he~~'s ~~been performing it for fairly~~ a ~~whilst~~. ~~For~~ a ~~whilst she~~'s ~~been in South Dakota.~~<br><br>~~my page ::~~ [~~https~~://~~Tumblr.com~~/~~blog~~/~~mmattchuu diet meal delivery~~]		: ''For other inequalities named after Wirtinger, see [[Wirtinger's inequality]].''

			In [[mathematics]], historically '''Wirtinger's inequality''' for real functions was an [[inequality (mathematics)\|inequality]] used in [[Fourier analysis]]. It was named after [[Wilhelm Wirtinger]]. It was used in 1904 to prove the [[isoperimetric inequality]]. A variety of closely related results are today known as Wirtinger's inequality.

			==Theorem==
			===First version===
			Let <math>f : \mathbb{R} \to \mathbb{R}</math> be a [[periodic function]] of period 2π, which is continuous and has a continuous derivative throughout '''R''', and such that

			:<math>\int_0^{2\pi}f(x) \, dx = 0.</math>

			Then

			:<math>\int_0^{2\pi}f'^2(x) \, dx \ge \int_0^{2\pi}f^2(x) \, dx</math>

			with equality [[if and only if]] ''f''(''x'') = ''a'' sin(''x'') + ''b'' cos(''x'') for some ''a'' and ''b'' (or equivalently ''f''(''x'') = ''c'' sin (''x'' + ''d'') for some ''c'' and ''d'').

			This version of the Wirtinger inequality is the one-dimensional [[Poincaré inequality]], with optimal constant.

			===Second version===
			The following related inequality is also called Wirtinger's inequality {{harv\|Dym\|McKean\|1985}}:

			:<math>\pi^{2}\int_0^a \|f\|^2 \le a^2 \int_0^a\|f'\|^2</math>

			whenever ''f'' is a C<sup>1</sup> function such that ''f''(0) = ''f''(''a'') = 0. In this form, Wirtinger's inequality is seen as the one-dimensional version of [[Friedrichs' inequality]].

			===Proof===
			The proof of the two versions are similar. Here is a proof of the first version of the inequality. Since [[Dirichlet's conditions]] are met, we can write

			:<math>f(x)=\frac{1}{2}a_0+\sum_{n\ge 1}\left(a_n\frac{\sin nx}{\sqrt{\pi}}+b_n\frac{\cos nx}{\sqrt{\pi}}\right),</math>

			and moreover ''a''<sub>0</sub> = 0 since the integral of ''f'' vanishes. By [[Parseval's identity]],

			:<math>\int_0^{2\pi}f^2(x)dx=\sum_{n=1}^\infty(a_n^2+b_n^2)</math>

			and

			:<math>\int_0^{2\pi}f'^2(x) \, dx = \sum_{n=1}^\infty n^2(a_n^2+b_n^2)</math>

			and since the summands are all ≥ 0, we get the desired inequality, with equality if and only if ''a<sub>n</sub>'' = ''b<sub>n</sub>'' = 0 for all ''n'' ≥ 2.

			==References==
			*{{citation\|first1=H\|last1=Dym\|authorlink1=Harry Dym\|first2=H\|last2=McKean\|title=Fourier series and integrals\|publisher=Academic press\|year=1985\|isbn=978-0-12-226451-1}}
			*[[Vadim Komkov\|Komkov, Vadim]] (1983) Euler's buckling formula and Wirtinger's inequality. Internat. J. Math. Ed. Sci. Tech. 14, no. 6, 661—668.

			{{PlanetMath attribution\|id=5393\|title=Wirtinger's inequality}}

			[[Category:Fourier analysis]]
			[[Category:Inequalities]]
			[[Category:Theorems in analysis]]

en>NOrbeck: Removed link to commercial site in "see also" section

2012-07-27T22:05:10Z

Removed link to commercial site in "see also" section

New page

Hello. Allow me introduce the author. Her title is Emilia Shroyer but it's not the most female title out there. Managing individuals has been his working day job for a while. His wife doesn't like it the way he does but what he truly likes performing is to do aerobics and he's been performing it for fairly a whilst. For a whilst she's been in South Dakota.<br><br>my page :: [https://Tumblr.com/blog/mmattchuu diet meal delivery]