Graph enumeration - Revision history

en>Rjwilmsi: Added 1 doi to a journal cite using AWB (10213)

2014-05-24T16:00:08Z

Added 1 doi to a journal cite using AWB (10213)

← Older revision		Revision as of 18:00, 24 May 2014
Line 1:		Line 1:
	~~Andera is what you can contact her but she by no means truly liked that name. My working day job is a travel agent. What me~~ and ~~my family love is bungee leaping but~~ I~~'ve been using on new issues lately~~. For many years ~~she~~'s been ~~residing in Kentucky but~~ her ~~husband wants them to transfer.<br><br>My site - accurate psychic predictions (~~[http://~~Brazil~~.~~Amor~~-~~amore~~.~~com~~/~~irboothe amor-amore~~.~~com])~~		Greetings! I am Marvella and I really feel comfortable when people use the full name. For many years I've been operating as a payroll clerk. California is exactly where her [http://www.sex-porn-tube.ch/user/JJeannere home std test] is but she requirements to move simply because of her family members. He is truly fond of performing ceramics but he is struggling to find time for it.

Salix alba: fix tags, replaced: ^{''n'' − 2 → ^{''n'' − 2} using AWB}

2014-03-01T08:49:07Z

fix tags, replaced: ''n'' − 2 → ''n'' − 2 using AWB

← Older revision		Revision as of 10:49, 1 March 2014
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	~~In [[mathematics]], the '''Khintchine inequality''', named after [[Aleksandr Khinchin]] and spelled in multiple ways in the Roman alphabet,~~ is ~~a theorem from [[probability]], and is also frequently used in [[mathematical analysis\|analysis]]. Heuristically, it says~~ that if we pick <math> N </math> [[complex numbers]] <math> x_1,\dots,x_N \in\mathbb{C}</math>, and add them together each multiplied by a random sign <math>\pm 1 </math>, then the [[expected value]] of its [[absolute value\|modulus]], or the modulus it will be closest to on average, will be not too far off from <math> \sqrt{\|x_1\|^{2}+\cdots + \|x_N\|^{2}}</math>.		Andera is what you can contact her but she by no means truly liked that name. My working day job is a travel agent. What me and my family love is bungee leaping but I've been using on new issues lately. For many years she's been residing in Kentucky but her husband wants them to transfer.<br><br>My site - accurate psychic predictions ([http://Brazil.Amor-amore.com/irboothe amor-amore.com])

	~~==Statement of theorem==~~

	~~Let <math> \{\epsilon_{n}\}_{n=1}^{N} </math> be [[i.i.d.]] [[random variables]]~~
	~~with <math>P(\epsilon_n=\pm1)=\frac12</math> for every <math>n=1\ldots N</math>,~~
	~~i.e.,~~ a ~~sequence with [[Rademacher distribution]]~~.
	~~Let <math> 0<p<\infty</math>~~ and ~~let <math> x_1,...,x_N\in \mathbb{C}</math>.~~
	~~Then~~

	:<math> A_p \left( \sum_{n=1}^{N}\|x_{n}\|^{2} \right)^{\frac{1}{2}} \leq \left(\mathbb{E}\Big\|\sum_{n=1}^{N}\epsilon_{n}x_{n}\Big\|^{p} \right)^{1/p} \leq B_p \left(\sum_{n=1}^{N}\|x_{n}\|^{2}\right)^{\frac{1}{2}} </math>

	~~for some constants <math> A_p,B_p>0 </math> depending only~~ on ~~<math>p</math> (see [[Expected value]] for notation)~~. ~~The sharp values of the constants <math>A_p,B_p</math> were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof).~~

	~~==Uses~~ in ~~analysis==~~

	~~The uses of this inequality are not limited~~ to ~~applications in [[probability theory]]~~. ~~One example of its use in [[Mathematical Analysis\|analysis]] is the following: if we let~~ <~~math~~>T<~~/math~~> ~~be a [[linear operator]] between two [[Lp space\|L<sup>''p''</sup> spaces]] <math> L^p(X,\mu)</math> and <math> L^p~~(~~Y,\nu) </math>, <math>1\leq p<\infty</math>, with bounded~~ [~~[operator norm\|norm]] <math> \\|T\\|<\infty </math>, then one can use Khintchine's inequality to show that~~

	:~~<math> \left\\|\left(\sum_{n=1}^{N}\|Tf_n\|^{2} \right)^{\frac{1}{2}}\right\\|_{L^p(Y,\nu)}\leq C_p\left\\|\left(\sum_{n=1}^{N}\|f_{n}\|^{2}\right)^{\frac{1}{2}}\right\\|_{L^p(X,\mu)} <~~/~~math>~~

	~~for some constant <math>C_p>0<~~/~~math> depending only on <math>p</math> and <math>\\|T\\|</math>~~.

	~~== See also ==~~
	* [[Marcinkiewicz–Zygmund inequality]]

	~~==References==~~
	~~#[[Thomas Wolff\|Thomas H~~. ~~Wolff]], "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0~~-~~8218-3449-5~~
	~~#Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math~~. ~~70 (1981~~)~~, no. 3, 231–283 (1982).~~
	#[[Fedor Nazarov]] and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247–267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.

	~~[[Category:Mathematical analysis]]~~
	~~[[Category:Probabilistic inequalities]]~~

en>Trappist the monk: rename venn to acad; remove deprecated parameters; using AWB

2013-12-10T15:21:41Z

rename venn to acad; remove deprecated parameters; using AWB

← Older revision		Revision as of 17:21, 10 December 2013
Line 1:		Line 1:
	~~Nice to meet you~~, ~~my name~~ is ~~Figures Held although I don't really like becoming known as like that. Bookkeeping~~ is ~~what I do~~. ~~The factor she adores most is body developing~~ and ~~now she is trying~~ to ~~make cash~~ with it. ~~Years~~ in the ~~past he moved~~ to ~~North Dakota~~ and ~~his family loves it.~~<br><br>~~Also visit my web page:~~ [~~http~~://~~www~~.~~uknewsx~~.~~com/what~~-~~you~~-~~want~~-~~to-do-when-confronted-with-candida-albicans/ at home std testing~~]		In [[mathematics]], the '''Khintchine inequality''', named after [[Aleksandr Khinchin]] and spelled in multiple ways in the Roman alphabet, is a theorem from [[probability]], and is also frequently used in [[mathematical analysis\|analysis]]. Heuristically, it says that if we pick <math> N </math> [[complex numbers]] <math> x_1,\dots,x_N \in\mathbb{C}</math>, and add them together each multiplied by a random sign <math>\pm 1 </math>, then the [[expected value]] of its [[absolute value\|modulus]], or the modulus it will be closest to on average, will be not too far off from <math> \sqrt{\|x_1\|^{2}+\cdots + \|x_N\|^{2}}</math>.

			==Statement of theorem==

			Let <math> \{\epsilon_{n}\}_{n=1}^{N} </math> be [[i.i.d.]] [[random variables]]
			with <math>P(\epsilon_n=\pm1)=\frac12</math> for every <math>n=1\ldots N</math>,
			i.e., a sequence with [[Rademacher distribution]].
			Let <math> 0<p<\infty</math> and let <math> x_1,...,x_N\in \mathbb{C}</math>.
			Then

			:<math> A_p \left( \sum_{n=1}^{N}\|x_{n}\|^{2} \right)^{\frac{1}{2}} \leq \left(\mathbb{E}\Big\|\sum_{n=1}^{N}\epsilon_{n}x_{n}\Big\|^{p} \right)^{1/p} \leq B_p \left(\sum_{n=1}^{N}\|x_{n}\|^{2}\right)^{\frac{1}{2}} </math>

			for some constants <math> A_p,B_p>0 </math> depending only on <math>p</math> (see [[Expected value]] for notation). The sharp values of the constants <math>A_p,B_p</math> were found by Haagerup (Ref. 2; see Ref. 3 for a simpler proof).

			==Uses in analysis==

			The uses of this inequality are not limited to applications in [[probability theory]]. One example of its use in [[Mathematical Analysis\|analysis]] is the following: if we let <math>T</math> be a [[linear operator]] between two [[Lp space\|L<sup>''p''</sup> spaces]] <math> L^p(X,\mu)</math> and <math> L^p(Y,\nu) </math>, <math>1\leq p<\infty</math>, with bounded [[operator norm\|norm]] <math> \\|T\\|<\infty </math>, then one can use Khintchine's inequality to show that

			:<math> \left\\|\left(\sum_{n=1}^{N}\|Tf_n\|^{2} \right)^{\frac{1}{2}}\right\\|_{L^p(Y,\nu)}\leq C_p\left\\|\left(\sum_{n=1}^{N}\|f_{n}\|^{2}\right)^{\frac{1}{2}}\right\\|_{L^p(X,\mu)} </math>

			for some constant <math>C_p>0</math> depending only on <math>p</math> and <math>\\|T\\|</math>.

			== See also ==
			* [[Marcinkiewicz–Zygmund inequality]]

			==References==
			#[[Thomas Wolff\|Thomas H. Wolff]], "Lectures on Harmonic Analysis". American Mathematical Society, University Lecture Series vol. 29, 2003. ISBN 0-8218-3449-5
			#Uffe Haagerup, "The best constants in the Khintchine inequality", Studia Math. 70 (1981), no. 3, 231–283 (1982).
			#[[Fedor Nazarov]] and Anatoliy Podkorytov, "Ball, Haagerup, and distribution functions", Complex analysis, operators, and related topics, 247–267, Oper. Theory Adv. Appl., 113, Birkhäuser, Basel, 2000.

			[[Category:Mathematical analysis]]
			[[Category:Probabilistic inequalities]]

en>Myasuda: added missing diacritic

2012-05-16T13:13:12Z

added missing diacritic

New page

Nice to meet you, my name is Figures Held although I don't really like becoming known as like that. Bookkeeping is what I do. The factor she adores most is body developing and now she is trying to make cash with it. Years in the past he moved to North Dakota and his family loves it. Also visit my web page: [http://www.uknewsx.com/what-you-want-to-do-when-confronted-with-candida-albicans/ at home std testing]

Graph enumeration - Revision history

en>Rjwilmsi: Added 1 doi to a journal cite using AWB (10213)

Salix alba: fix tags, replaced: ''n'' − 2 → ''n'' − 2 using AWB

en>Trappist the monk: rename venn to acad; remove deprecated parameters; using AWB

en>Myasuda: added missing diacritic

Salix alba: fix tags, replaced: ^{''n'' − 2 → ^{''n'' − 2} using AWB}