Two envelopes problem - Revision history

en>FrescoBot: Bot: link syntax and minor changes

2014-12-27T23:37:50Z

Bot: link syntax and minor changes

← Older revision		Revision as of 01:37, 28 December 2014
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	In [~~[mathematics]], the '''equidistribution theorem''' is the statement that the sequence~~		Andera is what [http://checkmates.co.za/index.php?do=/profile-56347/info/ free psychic reading] you can contact her but she never really favored that name. North Carolina is the place he loves most but now he is contemplating other options. Credit authorising is how she tends to make a living. The favorite pastime for him [http://www.zavodpm.ru/blogs/glennmusserrvji/14565-great-hobby-advice-assist-allow-you-get-going live psychic reading] and his kids is to play lacross and he would by no means give it up.<br><br>Also visit my page :: tarot readings ([http://www.herandkingscounty.com/content/information-and-facts-you-must-know-about-hobbies www.herandkingscounty.com])

	:~~''a'', 2''a'', 3''a'', .~~.. ~~mod 1~~

	~~is [[Equidistributed sequence\|uniformly distributed]] on the [[circle]] <math>\mathbb{R}/\mathbb{Z}<~~/~~math>, when ''a'' is an [[irrational number]]~~. ~~It is a special case of the [[ergodic theorem]] where one takes the normalized angle measure <math>\mu~~=~~\frac{d\theta}{2\pi}<~~/~~math>.~~

	~~==History==~~
	~~While this theorem was proved in 1909 and 1910 separately by [[Hermann Weyl]], [[Wacław Sierpiński]] and [[Piers Bohl]], variants of this theorem continue to be studied to this day.~~

	~~In 1916, Weyl proved that the sequence ''a'', 2<sup>2<~~/~~sup>''a'', 3<sup>2<~~/~~sup>''a'', ... mod 1 is uniformly distributed on the unit interval. In 1935, [[Ivan Vinogradov~~]~~] proved~~ that ~~the sequence ''p''<sub>''n''</sub> ''a'' mod 1 is uniformly distributed, where ''p''<sub>''n''</sub>~~ is the ~~''n''th [[prime number\|prime]]. Vinogradov's proof was a byproduct of the [[odd Goldbach conjecture]], that every sufficiently large odd number~~ is ~~the sum of three primes~~.

	~~[[George Birkhoff]], in 1931, and [[Aleksandr Khinchin]], in 1933, proved that the generalization ''x'' + ''na'', for [[almost all]] ''x'',~~ is ~~equidistributed on any [[Lebesgue measurable]] subset of the unit interval~~. The ~~corresponding generalizations~~ for ~~the Weyl and Vinogradov results were proven by~~ [~~[Jean Bourgain]] in 1988.~~

	~~Specifically, Khinchin showed that the identity~~

	:~~<math>\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n~~
	~~f( (x+ka) \mod 1 ) = \int_0^1 f(y)\,dy <~~/~~math>~~

	~~holds for almost all ''x'' and any Lebesgue integrable function ƒ~~. ~~In modern formulations, it is asked under what conditions the identity~~

	~~:<math>\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n~~
	~~f( (x+b_ka) \mod 1 ) = \int_0^1 f(y)\,dy </math>~~

	~~might hold, given some general [[sequence]] ''b''<sub>''k''</sub>~~.

	~~One noteworthy result is that the sequence 2<sup>''k''<~~/~~sup>''a'' mod 1 is uniformly distributed for almost all, but not all, irrational ''a''. Similarly, for the sequence ''b''<sub>''k''<~~/~~sub> = 2<sup> ''k''<~~/~~sup>, for every irrational ''a'',~~ and ~~almost all ''x'', there exists a function ƒ for which the sum diverges. In this sense, this sequence~~ is ~~considered~~ to ~~be a '''universally bad averaging sequence''', as opposed to ''b''~~<~~sub~~>~~''k''~~<~~/sub~~>~~ = ''k'', which is termed a '''universally good averaging sequence''', because it does not have the latter shortcoming.~~

	A powerful general result is [[Weyl's criterion]], which shows that equidistribution is equivalent to having a non-trivial estimate for the [[exponential sum]]s formed with the sequence as exponents. For the case of multiples of ''a'', Weyl's criterion reduces the problem to summing finite [[geometric series]].

	~~==See also==~~
	* [[Diophantine approximation]]
	* [[Low-discrepancy sequence]]

	~~==References==~~
	~~===Historical references===~~
	* P. Bohl, (~~1909) ''Über ein in der Theorie der säkutaren Störungen vorkommendes Problem'', ''J. reine angew. Math.'' '''135''', pp, 189–283.~~
	* {{cite journal \| last1 = Weyl \| first1 = H. \| year = 1910 \| title = Über die Gibbs'sche Erscheinung und verwandte Konvergenzphänomene \| url = \| journal = [[~~Rendiconti del Circolo Matematico di Palermo]] \| volume = 330 \| issue = \| pages = 377–407 }}~~
	* W. Sierpinski, (1910) ''Sur la valeur asymptotique d'une certaine somme'', ''Bull Intl. Acad. Polonmaise des Sci. et des Lettres'' (Cracovie) '''series A''', pp. 9–11.
	* {{cite journal\|last=Weyl\|first=H.\|title=Ueber die Gleichverteilung von Zahlen mod. Eins,\|journal=Math. Ann.\|year=1916\|volume=77\|pages=313–352 \| doi = 10.1007/BF01475864 \| issue = 3 \|postscript = .}}
	* {{cite journal \| first1= G. D. \|last1=Birkhoff \|url=http://www.~~pnas~~.~~org~~/~~cgi~~/reprint/17/12/656 \|title = Proof of the ergodic theorem \|year=1931 \|volume = 17 \|issue = 12 \|pages= 656–660 \| pmc = 1076138 \|postscript = . \|pmid=16577406 \|journal=Proc. Natl. Acad. Sci. U.S.A. \|doi=10.1073/pnas.17.12.656}}
	* {{cite journal \| first1= A. \|last1 = Ya. Khinchin \|title = Zur Birkhoff's Lösung des Ergodensproblems \|year= 1933 \| journal = Math. Ann. \| volume = 107\| pages = 485–488 \|postscript = . \| doi = 10.1007/BF01448905}}

	~~===Modern references===~~
	* Joseph M. Rosenblatt and Máté Weirdl, ''Pointwise ergodic theorems via harmonic analysis'', (1993) appearing in ''Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference'', (1995) Karl E. Petersen and Ibrahim A. Salama, ''eds.'', Cambridge University Press, Cambridge, ISBN 0-~~521~~-~~45999~~-0. ~~''(An extensive survey of the ergodic properties of generalizations of the equidistribution theorem of [[shift map~~]~~]s on the [[unit interval]]. Focuses on methods developed by Bourgain.~~)''
	* Elias M. Stein and Rami Shakarchi, ''Fourier Analysis. An Introduction'', (2003) Princeton University Press, pp 105–113 ''(Proof of the Weyl's theorem based on Fourier Analysis)''

	~~[[Category:Ergodic theory]]~~
	~~[[Category:Diophantine approximation]]~~
	~~[[Category:Theorems in number theory]]~~

en>Monkbot: /* Introduction to resolutions based on Bayesian probability theory */Fix CS1 deprecated date parameter errors

2014-01-31T03:52:40Z

Introduction to resolutions based on Bayesian probability theory: Fix CS1 deprecated date parameter errors

← Older revision		Revision as of 05:52, 31 January 2014
Line 1:		Line 1:
	~~Andrew Berryhill~~ is ~~what his spouse enjoys~~ to ~~contact him and he completely digs~~ that ~~title~~. ~~Credit authorising~~ is ~~how she makes~~ a ~~residing~~. ~~To perform lacross~~ is ~~something he would never give up~~. ~~Alaska~~ is ~~where I~~'~~ve usually been living~~.<br><br>~~My site real psychic readings~~ [[http://~~jplusfn~~.~~gaplus~~.kr/xe/~~qna~~/~~78647 jplusfn~~.~~gaplus~~.kr]]		In [[mathematics]], the '''equidistribution theorem''' is the statement that the sequence

			:''a'', 2''a'', 3''a'', ... mod 1

			is [[Equidistributed sequence\|uniformly distributed]] on the [[circle]] <math>\mathbb{R}/\mathbb{Z}</math>, when ''a'' is an [[irrational number]]. It is a special case of the [[ergodic theorem]] where one takes the normalized angle measure <math>\mu=\frac{d\theta}{2\pi}</math>.

			==History==
			While this theorem was proved in 1909 and 1910 separately by [[Hermann Weyl]], [[Wacław Sierpiński]] and [[Piers Bohl]], variants of this theorem continue to be studied to this day.

			In 1916, Weyl proved that the sequence ''a'', 2<sup>2</sup>''a'', 3<sup>2</sup>''a'', ... mod 1 is uniformly distributed on the unit interval. In 1935, [[Ivan Vinogradov]] proved that the sequence ''p''<sub>''n''</sub> ''a'' mod 1 is uniformly distributed, where ''p''<sub>''n''</sub> is the ''n''th [[prime number\|prime]]. Vinogradov's proof was a byproduct of the [[odd Goldbach conjecture]], that every sufficiently large odd number is the sum of three primes.

			[[George Birkhoff]], in 1931, and [[Aleksandr Khinchin]], in 1933, proved that the generalization ''x'' + ''na'', for [[almost all]] ''x'', is equidistributed on any [[Lebesgue measurable]] subset of the unit interval. The corresponding generalizations for the Weyl and Vinogradov results were proven by [[Jean Bourgain]] in 1988.

			Specifically, Khinchin showed that the identity

			:<math>\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n
			f( (x+ka) \mod 1 ) = \int_0^1 f(y)\,dy </math>

			holds for almost all ''x'' and any Lebesgue integrable function ƒ. In modern formulations, it is asked under what conditions the identity

			:<math>\lim_{n\to\infty} \frac{1}{n} \sum_{k=1}^n
			f( (x+b_ka) \mod 1 ) = \int_0^1 f(y)\,dy </math>

			might hold, given some general [[sequence]] ''b''<sub>''k''</sub>.

			One noteworthy result is that the sequence 2<sup>''k''</sup>''a'' mod 1 is uniformly distributed for almost all, but not all, irrational ''a''. Similarly, for the sequence ''b''<sub>''k''</sub> = 2<sup> ''k''</sup>, for every irrational ''a'', and almost all ''x'', there exists a function ƒ for which the sum diverges. In this sense, this sequence is considered to be a '''universally bad averaging sequence''', as opposed to ''b''<sub>''k''</sub> = ''k'', which is termed a '''universally good averaging sequence''', because it does not have the latter shortcoming.

			A powerful general result is [[Weyl's criterion]], which shows that equidistribution is equivalent to having a non-trivial estimate for the [[exponential sum]]s formed with the sequence as exponents. For the case of multiples of ''a'', Weyl's criterion reduces the problem to summing finite [[geometric series]].

			==See also==
			* [[Diophantine approximation]]
			* [[Low-discrepancy sequence]]

			==References==
			===Historical references===
			* P. Bohl, (1909) ''Über ein in der Theorie der säkutaren Störungen vorkommendes Problem'', ''J. reine angew. Math.'' '''135''', pp, 189–283.
			* {{cite journal \| last1 = Weyl \| first1 = H. \| year = 1910 \| title = Über die Gibbs'sche Erscheinung und verwandte Konvergenzphänomene \| url = \| journal = [[Rendiconti del Circolo Matematico di Palermo]] \| volume = 330 \| issue = \| pages = 377–407 }}
			* W. Sierpinski, (1910) ''Sur la valeur asymptotique d'une certaine somme'', ''Bull Intl. Acad. Polonmaise des Sci. et des Lettres'' (Cracovie) '''series A''', pp. 9–11.
			* {{cite journal\|last=Weyl\|first=H.\|title=Ueber die Gleichverteilung von Zahlen mod. Eins,\|journal=Math. Ann.\|year=1916\|volume=77\|pages=313–352 \| doi = 10.1007/BF01475864 \| issue = 3 \|postscript = .}}
			* {{cite journal \| first1= G. D. \|last1=Birkhoff \|url=http://www.pnas.org/cgi/reprint/17/12/656 \|title = Proof of the ergodic theorem \|year=1931 \|volume = 17 \|issue = 12 \|pages= 656–660 \| pmc = 1076138 \|postscript = . \|pmid=16577406 \|journal=Proc. Natl. Acad. Sci. U.S.A. \|doi=10.1073/pnas.17.12.656}}
			* {{cite journal \| first1= A. \|last1 = Ya. Khinchin \|title = Zur Birkhoff's Lösung des Ergodensproblems \|year= 1933 \| journal = Math. Ann. \| volume = 107\| pages = 485–488 \|postscript = . \| doi = 10.1007/BF01448905}}

			===Modern references===
			* Joseph M. Rosenblatt and Máté Weirdl, ''Pointwise ergodic theorems via harmonic analysis'', (1993) appearing in ''Ergodic Theory and its Connections with Harmonic Analysis, Proceedings of the 1993 Alexandria Conference'', (1995) Karl E. Petersen and Ibrahim A. Salama, ''eds.'', Cambridge University Press, Cambridge, ISBN 0-521-45999-0. ''(An extensive survey of the ergodic properties of generalizations of the equidistribution theorem of [[shift map]]s on the [[unit interval]]. Focuses on methods developed by Bourgain.)''
			* Elias M. Stein and Rami Shakarchi, ''Fourier Analysis. An Introduction'', (2003) Princeton University Press, pp 105–113 ''(Proof of the Weyl's theorem based on Fourier Analysis)''

			[[Category:Ergodic theory]]
			[[Category:Diophantine approximation]]
			[[Category:Theorems in number theory]]

en>Byelf2007: grammar

2012-08-06T05:59:46Z

grammar

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