CheiRank - Revision history

en>Shepelyansky at 11:58, 13 June 2014

2014-06-13T11:58:09Z

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	~~In [[mathematical analysis\|analysis]], a branch~~ of ~~mathematics~~, ~~'''Hilbert's inequality''' states that~~		With so many possible harm of ingesting unsafe water, communities these days are working out prerogatives for sophisticated approaches in filtration. Brisa window model evaporative coolers deliver much more CFM for each and every watt of power consumed than comparative competitive models. Cooler Master CM 690 II Advance with Transparent RC-692-KWN2 Mid Tower Chassis. Come across and evaluate costs on mastercool coolers from well-known brands and stores all in 1 location. Invest in coolers at BizRate, the greatest price tag comparison search engine on the net. Acquire a Danby or Haier wine cooler and get quickly and protected shipping at WineCoolers.<br><br><br><br>Place meals in zip-lock plastic bags or in plastic containers to keep it dry as the ice melts. For longer trips it's a fantastic concept to retain your beverages in a separate cooler that can be opened much more frequently. Place all of your meals in one more cooler and open it significantly less typically. Crushed ice cools products more quickly, but ice blocks last longer. Block ice is recommended for trips that are extra than 1 or two days. Air around the ice can be under 0degC.<br><br>Commonly the fans in inferior laptop coolers will be messed up within a year but not the super sturdy Zalman. Targus Lap Chill Mat AWE55US This laptop cooler is fairly big and can support a 17 inch laptop effortlessly devoid of any complications. If you treasured this article so you would like to receive more info concerning [http://www.bestcoolerreviewshq.com/best-cooler-for-the-money/ best cooler For the money - bestcoolerreviewshq.com] i implore you to visit our web-page. The rubber stops at the four edges is a good touch and prevents the laptop from slipping off the laptop cooler. I began this web-site to share my experiences with laptop coolers.<br><br>The very best way to prepare you cooler is to fill it with ice a day or two prior to you program to load it. When you are ready to load the cooler, dump the remaining ice and water and reload the cooler with food and fresh ice. The prior notion applies, if you load warm beer or soda into the cooler, it is going to burn via a lot of ice lowering the temperature of the cans. Load your ice and meals, then fill in any gaps with a lot more ice or even crumpled newspaper.<br><br>Some mid-range laptop coolers can perform as properly as the most pricey laptop cooler out there. Some laptop coolers can only last for a month or two just before their fan begins to get clogged up which slows down the cool air. Not all laptop coolers are durable so to get the most value for revenue, you must Ideal Camping Cooler For The Money choose one with a good create top quality. The laptop cooler fans really should be quiet and no far more than Very best Smaller Cooler For The Funds a pretty gentle murmur need to be even noticeable.

	~~: <math>~~
	~~\left\|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{r-s}\right\|\le\pi\displaystyle\sum_{r}\|u_{r}\|^2~~.
	~~</math>~~

	for ~~any sequence ''u''<sub>1</sub>,''u''<sub>2</sub>,~~... ~~of complex numbers. It was first demonstrated by [[David Hilbert]] with~~ the ~~constant 2{{pi}} instead of {{pi}};~~ the ~~sharp constant was found by [[Issai Schur]]~~. ~~It implies that the [[Hilbert transform#Discrete Hilbert transform\|discrete Hilbert transform]] is~~ a ~~bounded operator in ''ℓ''~~<~~sub~~>2<~~/sub~~>.

	~~==Formulation==~~

	~~Let (''u''~~<~~sub~~>~~''m''~~<~~/sub~~>~~) be a sequence of complex numbers~~. ~~If the sequence is infinite, assume that~~ it ~~is square-summable:~~

	~~:<math> \sum_m \|u_m\|^2 < \infty \, </math>~~

	~~Hilbert~~'s ~~inequality (see {{harvtxt\|Steele\|2004}}) asserts~~ that

	~~: <math>~~
	~~\left\|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{r-s}\right\|\le\pi\displaystyle\sum_{r}\|u_{r}\|^2~~.
	~~</math>~~

	~~==Extensions==~~

	~~In 1973, [[#MV\|Montgomery & Vaughan]] reported several generalizations~~ of ~~Hilbert's inequality, considering the bilinear forms~~

	~~: <math> \sum_{r\neq s}u_r\overline u_s\csc\pi(x_r-x_s) </math>~~

	and

	~~: <math>\sum_{r\neq s}\dfrac{u_r\overline u_s}{\lambda_r-\lambda_s}~~,~~</math>~~

	~~where ''x''<sub>~~1~~</sub>,''x''<sub>2</sub>,~~..~~.,''x''~~<~~sub~~>~~''m''~~<~~/sub~~> ~~are distinct real numbers modulo 1 (i~~.e. ~~they belong~~ to ~~distinct classes in the~~ [~~[quotient group]] '''R'''~~/~~'''Z''') and ''λ''<sub>1<~~/~~sub>,~~..~~.,''λ''<sub>''m''<~~/~~sub> are distinct real numbers~~. ~~[[#MV\|Montgomery & Vaughan~~]~~]'s generalizations of Hilbert's inequality are then given by~~

	~~: <math>~~
	~~\left\|\sum_{r\neq s} u_r \overline{u_s}\csc\pi(x_r-x_s)\right\|\le\delta^{~~-~~1}\sum_r \|u_r\|^2~~.
	~~</math>~~

	and

	~~: <math>~~
	~~\left\|\sum_{r\neq s}\dfrac{u_r\overline{u_s}}{\lambda_r~~-~~\lambda_s}\right\|\le\pi\tau^{-1} \sum_r \|u_r\|^2~~.
	<~~/math~~>

	~~where~~

	:<~~math~~>~~\delta={\min_{r,s}}{}_{+}\\|x_{r}-x_{s}\\|, \quad \tau=\min_{r,s}{}_{+}\\|\lambda_r-\lambda_s\\|, \, </math>~~

	~~:<math>\\|s\\|= \min_{m\in\mathbb{Z}}\|s-m\|</math>~~

	is ~~the distance from ''s''~~ to the ~~nearest integer~~, and ~~min<sub>+</sub> denotes~~ the ~~smallest positive value~~. ~~Moreover~~, if

	~~:<math>0<\delta_r \le {\min_s}{}_{+}\\|x_r-x_s\\| \quad \text{~~and~~} \quad 0<\tau_{r}\le {\min_{s}}{}_{+}\\|\lambda_r-\lambda_s\\|,\~~,~~</math>~~

	then ~~the following inequalities hold:~~

	~~: <math>~~
	~~\left\|\sum_{r\neq s} u_r\overline{u_s}\csc\pi(x_r-x_s)\right\|\le\dfrac{3}{2} \sum_r \|u_r\|^2 \delta_r^{-1}~~.
	<~~/math~~>

	~~and~~

	: <~~math~~>~~\left\|\sum_{r\neq s}\dfrac{u_r \overline{u_s}}{\lambda_r~~-~~\lambda_s}\right\|\le \dfrac{3}{2} \pi \sum_r \|u_r\|^2\tau_r^{-1}~~.
	~~</math>~~

	~~==References==~~

	* Online book chapter [http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_HilbertandCompensatingDifficulties.pdf Hilbert’s Inequality and Compensating Difficulties] extracted from {{cite book \|title=The Cauchy-Schwarz master class: an introduction to the ~~art of mathematical inequalities \|first=J~~. ~~Michael\|last= Steele \|pages=155–165 \|chapter=Chapter 10: Hilbert’s Inequality~~ and ~~Compensating Difficulties \|url=http://books.google.com/?id=bvgBdZKEYAEC&pg=PA155~~ ~~\|year=2004 \|isbn=0-521-54677-X \|publisher=Cambridge University Press\|ref=harv}}.~~

	* {{cite journal
	~~\| last1 = Montgomery~~
	~~\| first1 = H~~. L.
	~~\| author-link = Hugh Montgomery (mathematician)~~
	~~\| last2 = Vaughan~~
	~~\| first2 = R. C.~~
	~~\| title = Hilbert's inequality~~
	~~\| journal = J. London Math. Soc. (2)~~
	~~\| volume = 8~~
	~~\| year = 1974~~
	~~\| pages = 73–82~~
	~~\| issn = 0024-6107~~
	~~\| ref = MV~~
	}}

	~~==External links==~~
	* {{SpringerEOM\|title=Hilbert inequality\|id=Hilbert_inequality\| last=Godunova\|first=E.K.}}

	~~[[Category:Inequalities]]~~
	~~[[Category:Complex analysis]]~~
	~~[[Category:Number theory]]~~

en>Meatsgains: Added {{no footnotes}} tag to article (TW)

2013-03-24T02:21:19Z

Added {{no footnotes}} tag to article (TW)

← Older revision		Revision as of 04:21, 24 March 2013
Line 1:		Line 1:
	~~Greetings~~. ~~The author~~'~~s name is Nadine Chou~~. ~~He currently lives in New Hampshire and he loves each and every day living in~~ that ~~location. He~~ is ~~currently an accounting officer~~. ~~One~~ of the ~~perfect things in this field for him~~ is ~~cooking but he can~~'~~t render it his line~~ of ~~business. Check out~~ the ~~latest news on his website~~: ~~http~~://~~www~~.~~flar~~.us/~~ray~~-~~ban~~-~~7019~~-on-~~sale~~.~~html~~<br><br>~~Have a look at my web site~~ :: [http://www.~~flar~~.us/~~ray~~-~~ban~~-~~7019~~-on-~~sale~~.~~html Ray Ban 7019~~]		In [[mathematical analysis\|analysis]], a branch of mathematics, '''Hilbert's inequality''' states that

			: <math>
			\left\|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{r-s}\right\|\le\pi\displaystyle\sum_{r}\|u_{r}\|^2.
			</math>

			for any sequence ''u''<sub>1</sub>,''u''<sub>2</sub>,... of complex numbers. It was first demonstrated by [[David Hilbert]] with the constant 2{{pi}} instead of {{pi}}; the sharp constant was found by [[Issai Schur]]. It implies that the [[Hilbert transform#Discrete Hilbert transform\|discrete Hilbert transform]] is a bounded operator in ''ℓ''<sub>2</sub>.

			==Formulation==

			Let (''u''<sub>''m''</sub>) be a sequence of complex numbers. If the sequence is infinite, assume that it is square-summable:

			:<math> \sum_m \|u_m\|^2 < \infty \, </math>

			Hilbert's inequality (see {{harvtxt\|Steele\|2004}}) asserts that

			: <math>
			\left\|\sum_{r\neq s}\dfrac{u_{r}\overline{u_{s}}}{r-s}\right\|\le\pi\displaystyle\sum_{r}\|u_{r}\|^2.
			</math>

			==Extensions==

			In 1973, [[#MV\|Montgomery & Vaughan]] reported several generalizations of Hilbert's inequality, considering the bilinear forms

			: <math> \sum_{r\neq s}u_r\overline u_s\csc\pi(x_r-x_s) </math>

			and

			: <math>\sum_{r\neq s}\dfrac{u_r\overline u_s}{\lambda_r-\lambda_s},</math>

			where ''x''<sub>1</sub>,''x''<sub>2</sub>,...,''x''<sub>''m''</sub> are distinct real numbers modulo 1 (i.e. they belong to distinct classes in the [[quotient group]] '''R'''/'''Z''') and ''λ''<sub>1</sub>,...,''λ''<sub>''m''</sub> are distinct real numbers. [[#MV\|Montgomery & Vaughan]]'s generalizations of Hilbert's inequality are then given by

			: <math>
			\left\|\sum_{r\neq s} u_r \overline{u_s}\csc\pi(x_r-x_s)\right\|\le\delta^{-1}\sum_r \|u_r\|^2.
			</math>

			and

			: <math>
			\left\|\sum_{r\neq s}\dfrac{u_r\overline{u_s}}{\lambda_r-\lambda_s}\right\|\le\pi\tau^{-1} \sum_r \|u_r\|^2.
			</math>

			where

			:<math>\delta={\min_{r,s}}{}_{+}\\|x_{r}-x_{s}\\|, \quad \tau=\min_{r,s}{}_{+}\\|\lambda_r-\lambda_s\\|, \, </math>

			:<math>\\|s\\|= \min_{m\in\mathbb{Z}}\|s-m\|</math>

			is the distance from ''s'' to the nearest integer, and min<sub>+</sub> denotes the smallest positive value. Moreover, if

			:<math>0<\delta_r \le {\min_s}{}_{+}\\|x_r-x_s\\| \quad \text{and} \quad 0<\tau_{r}\le {\min_{s}}{}_{+}\\|\lambda_r-\lambda_s\\|,\,</math>

			then the following inequalities hold:

			: <math>
			\left\|\sum_{r\neq s} u_r\overline{u_s}\csc\pi(x_r-x_s)\right\|\le\dfrac{3}{2} \sum_r \|u_r\|^2 \delta_r^{-1}.
			</math>

			and

			: <math>\left\|\sum_{r\neq s}\dfrac{u_r \overline{u_s}}{\lambda_r-\lambda_s}\right\|\le \dfrac{3}{2} \pi \sum_r \|u_r\|^2\tau_r^{-1}.
			</math>

			==References==

			* Online book chapter [http://www-stat.wharton.upenn.edu/~steele/Publications/Books/CSMC/CSMC_HilbertandCompensatingDifficulties.pdf Hilbert’s Inequality and Compensating Difficulties] extracted from {{cite book \|title=The Cauchy-Schwarz master class: an introduction to the art of mathematical inequalities \|first=J. Michael\|last= Steele \|pages=155–165 \|chapter=Chapter 10: Hilbert’s Inequality and Compensating Difficulties \|url=http://books.google.com/?id=bvgBdZKEYAEC&pg=PA155 \|year=2004 \|isbn=0-521-54677-X \|publisher=Cambridge University Press\|ref=harv}}.

			* {{cite journal
			\| last1 = Montgomery
			\| first1 = H. L.
			\| author-link = Hugh Montgomery (mathematician)
			\| last2 = Vaughan
			\| first2 = R. C.
			\| title = Hilbert's inequality
			\| journal = J. London Math. Soc. (2)
			\| volume = 8
			\| year = 1974
			\| pages = 73–82
			\| issn = 0024-6107
			\| ref = MV
			}}

			==External links==
			* {{SpringerEOM\|title=Hilbert inequality\|id=Hilbert_inequality\| last=Godunova\|first=E.K.}}

			[[Category:Inequalities]]
			[[Category:Complex analysis]]
			[[Category:Number theory]]

en>Rjwilmsi: /* References */Format plain DOIs using AWB (8068)

2012-05-30T06:32:48Z

References: Format plain DOIs using AWB (8068)

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Greetings. The author's name is Nadine Chou. He currently lives in New Hampshire and he loves each and every day living in that location. He is currently an accounting officer. One of the perfect things in this field for him is cooking but he can't render it his line of business. Check out the latest news on his website: http://www.flar.us/ray-ban-7019-on-sale.html<br><br>Have a look at my web site :: [http://www.flar.us/ray-ban-7019-on-sale.html Ray Ban 7019]