Optical cavity - Revision history

en>Srleffler: too many or's

2014-11-04T01:36:17Z

too many or's

← Older revision		Revision as of 03:36, 4 November 2014
Line 1:		Line 1:
	~~__NOTOC__~~		Gabrielle Straub is what you can call me although it's not the quite a few feminine of names. Fish keeping is what I provide every week. Managing people is my month job now. My house is now with regard to South Carolina. Go to my website to find out more: http://circuspartypanama.com<br><br>Also visit my page; [http://circuspartypanama.com clash of clans hack download no survey no password]
	~~'''Szemerédi~~'s ~~theorem''' is a result In [[arithmetic combinatorics]], concerning [[arithmetic progression]]s in subset of~~ the ~~integers.~~
	In 1936, [[Paul Erdős\|Erdős]] and [[Paul Turán\|Turán]] conjectured<ref name="erdos turan">{{citation\|authorlink1=Paul Erdős\|first1=Paul\|last1=Erdős\|authorlink2=Paul Turán\|first2=Paul\|last2=Turán\|title=On some sequences of integers\|journal=[[Journal of the London Mathematical Society]]\|volume=11\|issue=4\|year=1936\|pages=261–264\|url=http://www.renyi.hu/~p_erdos/1936-05.pdf\|doi=10.1112/jlms/s1-11.4.261}}.</ref> that every set of integers ''A'' with positive [[natural density]] contains a ~~''k'' term [[arithmetic progression]] for every ''k''. This conjecture, which became Szemerédi's theorem, generalizes the statement~~ of ~~[[van der Waerden's theorem]]~~.

	~~==History==~~
	The cases ''k'' = 1 and ''k'' = 2 are trivial. The case ''k'' = 3 was established in 1953 by [[Klaus Roth]]<ref>{{citation\|authorlink=Klaus Friedrich Roth\|first=Klaus Friedrich\|last=Roth\|title=On certain sets of integers, I~~\|journal=[[Journal of the London Mathematical Society]]\|volume=28\|pages=104–109\|year=1953\|id=Zbl 0050~~.~~04002, {{MathSciNet\|id=0051853}}\|doi=10~~.1112/jlms/s1-28.1.104}}.</ref> via an adaptation of the [[Hardy–Littlewood circle method]]. The case ''k'' = 4 was established in 1969 by [[Endre Szemerédi]]<ref>{{citation\|authorlink=Endre Szemerédi\|first=Endre\|last=Szemerédi\|title=On sets of integers containing no four elements in arithmetic progression\|journal=Acta Math. ~~Acad. Sci. Hung.\|volume=20\|pages=89–104\|year=1969\|id=Zbl 0175.04301, {{MathSciNet\|id=0245555}}\|doi=10.1007/BF01894569}}.</ref> via a combinatorial method. Using an approach similar~~ to ~~the one he used for the case ''k'' = 3, Roth<ref>{{citation\|authorlink=Klaus Friedrich Roth\|first=Klaus Friedrich\|last=Roth\|title=Irregularities of sequences relative~~ to ~~arithmetic progressions, IV\|journal=Periodica Math. Hungar.\|volume=2\|pages=301–326\|year=1972\|id={{MathSciNet\|id=0369311}}\|doi=10.1007/BF02018670}}.</ref> gave a second proof for this in 1972.~~

	Finally, the case of general ''k'' was settled in 1975, also by Szemerédi,<ref>{{citation\|authorlink=Endre Szemerédi\|first=Endre\|last=Szemerédi\|title=On sets of integers containing no ''k'' elements in arithmetic progression\|journal=Acta Arithmetica\|volume=27\|pages=199–245\|year=1975\|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27132.pdf\|id=Zbl 0303.10056, {{MathSciNet\|id=0369312}}}}.</ref> via an ingenious and complicated extension of the previous combinatorial argument (called "a masterpiece of combinatorial reasoning" by [[Ronald Graham\|R. L. Graham]]). Several further proofs are now known, the most important amongst them being those by [[Hillel Furstenberg]]<ref>{{citation\|authorlink=Hillel Furstenberg\|first=Hillel\|last=Furstenberg\|title=Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions\|journal=J. D'Analyse Math.\|volume=31\|pages=204–256\|year=1977\|id={{MathSciNet\|id=0498471}}\|doi=10.1007/BF02813304}}.</ref> in 1977, using [[ergodic theory]], and by [[Timothy Gowers]]<ref name="gowers">{{citation\|authorlink=Timothy Gowers\|first=Timothy\|last=Gowers\|title=A new proof of Szemerédi's theorem\|journal=Geom. Funct. Anal.\|volume=11\|issue=3\|pages=465–588\|url=http://~~www.dpmms.cam.ac.uk/~wtg10/sz898.dvi\|year=2001\|id={{MathSciNet\|id=1844079}}\|doi=10.1007/s00039-001-0332-9}}~~.<~~/ref~~> ~~in 2001, using both [[Fourier analysis]] and [[combinatorics]].~~

	~~==Finitary version==~~
	~~Let ''k'' be a positive integer and let 0~~ < ~~δ ≤ 1/2. A [[finitary]] version of the theorem states that there exists a positive integer~~

	~~:<math~~>~~N = N(k,\delta)\,</math>~~

	~~such that every subset of {1, 2, ..., ''N''} of size at least δ''N'' contains an arithmetic progression of length ''k''.~~
	~~The best known bounds for ''N''(''k'', δ) are~~

	~~:<math>C^{\log(1/\delta)^{k-1}} \leq N(k,\delta) \leq 2^{2^{\delta^{-2^{2^{k+9}}}}}</math>~~

	with ''C'' > 1. The lower bound is due to [[Felix A. Behrend\|Behrend]]<ref>{{citation\|authorlink=Felix A. Behrend\|first=Felix A.\|last=Behrend\|title=On the sets of integers which contain no three in arithmetic progression\|journal=[[Proceedings of the National Academy of Sciences]]\|volume=23\|issue=12\|pages=331–332\|year=1946\|id=Zbl 0060.10302\|doi=10.1073/pnas.32.12.331}}.</ref> (for ''k'' = 3) and [[Robert Alexander Rankin\|Rankin]],<ref>{{citation\|authorlink=Robert A. Rankin\|first=Robert A.\|last=Rankin\|title=Sets of integers containing not more than a given number of terms in arithmetical progression\|journal=Proc. Roy. Soc. Edinburgh Sect. A\|volume=65\|pages=332–344\|year=1962\|id=Zbl 0104.03705, {{MathSciNet\|id=0142526}}}}.</ref> and the upper bound is due to Gowers.<ref name="gowers"/>
	~~When ''k'' = 3 better upper bounds are known~~; [[Jean Bourgain\|Bourgain]]<ref>{{citation\|authorlink=Jean Bourgain\|first=Jean\|last=Bourgain\|title=On triples in arithmetic progression\|journal=Geom. Func. Anal.\|volume=9\|issue=5\|year=1999\|pages=968–984\|id={{MathSciNet\|id=1726234}}\|doi=10.1007/s000390050105}}</ref> has proved that

	~~:<math>N(3,\delta) \leq C^{\delta^{-2}\log(1/\delta)},</math>~~

	This was improved by [[Tom Sanders (mathematician)\|Sanders]]<ref>{{citation\|authorlink=Tom Sanders (mathematician)\|first=Tom\|last=Sanders\|title=On Roth's theorem on progressions\|journal=Annals of Mathematics\|volume=174\|issue=1\|year=2011\|pages=619–636\|id={{MathSciNet\|id=2811612}}\|doi=10.4007/annals.2011.174.1.20}}</ref> to

	~~:<math>N(3,\delta) \leq C^{\delta^{-1}(\log(1/\delta))^5}</math>~~

	~~==See also==~~
	* [[Problems involving arithmetic progressions]]
	* [[Ergodic Ramsey theory]]
	* [[Arithmetic combinatorics]]
	* [[Szemerédi regularity lemma]]
	* [[Green–Tao theorem]]
	* [[Erdős conjecture on arithmetic progressions]]

	~~==Notes==~~
	~~{{reflist\|colwidth=30em}}~~

	~~==Further reading==~~
	* {{Citation \| first=Terence \| last=Tao \| authorlink=Terence Tao \| chapter=The ergodic and combinatorial approaches to Szemerédi's theorem \| pages=145–193 \| editor1-first=Andrew \| editor1-last=Granville \| editor2-first=Melvyn B. \| editor2-last=Nathanson\| editor3-first=József \| editor3-last=Solymosi \| title=Additive Combinatorics \| series= CRM Proceedings & Lecture Notes \| volume=43 \| publisher=[[American Mathematical Society]] \| year=2007 \| isbn=978-0-8218-4351-2 \| zbl=1159.11005 }}

	~~==External links==~~
	* [http://planetmath.org/encyclopedia/SzemeredisTheorem.html PlanetMath source for initial version of this page]
	* [http://~~www.math.ucla.edu/~tao/whatsnew.html Announcement by Ben Green and Terence Tao] – the preprint is available at [http://front.math.ucdavis.edu/math.NT/0404188 math.NT/0404188]~~
	* [http://in-theory.blogspot.com~~/2006/06/szemeredis-theorem.html Discussion~~ of ~~Szemerédi's theorem (part 1 of 5)]~~
	* Ben Green and Terence Tao: [http://www.scholarpedia.org/article/Szemeredi%27s_Theorem Szemerédi's theorem] on [[Scholarpedia]]
	* {{MathWorld\|SzemeredisTheorem\|SzemeredisTheorem}}
	* {{cite web\|last=Grime\|first=James\|title=6,000,000: Endre Szemerédi wins the Abel Prize\|url=http://www.numberphile.com/videos/abel_prize.html\|work=Numberphile\|year=2012\|publisher=[[Brady Haran]]\|coauthors=Hodge, David}}

	~~{{DEFAULTSORT:Szemeredi's theorem}}~~
	~~[[Category:Ramsey theory]]~~
	~~[[Category:Theorems in combinatorics]]~~
	~~[[Category:Theorems in number theory]~~]

en>IanWilliamson: /* See also */

2013-09-22T20:12:12Z

See also

← Older revision		Revision as of 22:12, 22 September 2013
Line 1:		Line 1:
	~~Eusebio~~ is the name ~~persons use to call me and my~~ [http://~~Www~~.~~Bing~~.~~com~~/~~search?q~~=~~friends~~&~~form~~=~~MSNNWS~~&~~mkt~~=~~en-us~~&pq=~~friends friends~~] ~~and~~ I ~~think getting this done sounds quite good when you say it~~. I used to ~~be unemployed but now I am~~ a ~~major cashier. My house is~~ this in ~~South Carolina~~ in ~~addition I don~~'~~t plan directly~~ on ~~changing it~~. It's ~~not a common place but what I desire doing is bottle top [~~http://~~Www~~.~~dict~~.cc/?s=~~collecting collecting~~] and ~~now My hubby~~ and ~~i have time~~ to ~~consume on new things~~. I'm not ~~high-quality at webdesign but your preferred retail stores want~~ to ~~check the actual website: http~~://~~prometeu~~.~~net~~<br><br>~~Feel free~~ to ~~surf to my weblog~~ - ~~how to hack clash~~ of ~~clans~~ ([http://~~prometeu~~.~~net read page~~])		__NOTOC__
			'''Szemerédi's theorem''' is a result In [[arithmetic combinatorics]], concerning [[arithmetic progression]]s in subset of the integers.
			In 1936, [[Paul Erdős\|Erdős]] and [[Paul Turán\|Turán]] conjectured<ref name="erdos turan">{{citation\|authorlink1=Paul Erdős\|first1=Paul\|last1=Erdős\|authorlink2=Paul Turán\|first2=Paul\|last2=Turán\|title=On some sequences of integers\|journal=[[Journal of the London Mathematical Society]]\|volume=11\|issue=4\|year=1936\|pages=261–264\|url=http://www.renyi.hu/~p_erdos/1936-05.pdf\|doi=10.1112/jlms/s1-11.4.261}}.</ref> that every set of integers ''A'' with positive [[natural density]] contains a ''k'' term [[arithmetic progression]] for every ''k''. This conjecture, which became Szemerédi's theorem, generalizes the statement of [[van der Waerden's theorem]].

			==History==
			The cases ''k'' = 1 and ''k'' = 2 are trivial. The case ''k'' = 3 was established in 1953 by [[Klaus Roth]]<ref>{{citation\|authorlink=Klaus Friedrich Roth\|first=Klaus Friedrich\|last=Roth\|title=On certain sets of integers, I\|journal=[[Journal of the London Mathematical Society]]\|volume=28\|pages=104–109\|year=1953\|id=Zbl 0050.04002, {{MathSciNet\|id=0051853}}\|doi=10.1112/jlms/s1-28.1.104}}.</ref> via an adaptation of the [[Hardy–Littlewood circle method]]. The case ''k'' = 4 was established in 1969 by [[Endre Szemerédi]]<ref>{{citation\|authorlink=Endre Szemerédi\|first=Endre\|last=Szemerédi\|title=On sets of integers containing no four elements in arithmetic progression\|journal=Acta Math. Acad. Sci. Hung.\|volume=20\|pages=89–104\|year=1969\|id=Zbl 0175.04301, {{MathSciNet\|id=0245555}}\|doi=10.1007/BF01894569}}.</ref> via a combinatorial method. Using an approach similar to the one he used for the case ''k'' = 3, Roth<ref>{{citation\|authorlink=Klaus Friedrich Roth\|first=Klaus Friedrich\|last=Roth\|title=Irregularities of sequences relative to arithmetic progressions, IV\|journal=Periodica Math. Hungar.\|volume=2\|pages=301–326\|year=1972\|id={{MathSciNet\|id=0369311}}\|doi=10.1007/BF02018670}}.</ref> gave a second proof for this in 1972.

			Finally, the case of general ''k'' was settled in 1975, also by Szemerédi,<ref>{{citation\|authorlink=Endre Szemerédi\|first=Endre\|last=Szemerédi\|title=On sets of integers containing no ''k'' elements in arithmetic progression\|journal=Acta Arithmetica\|volume=27\|pages=199–245\|year=1975\|url=http://matwbn.icm.edu.pl/ksiazki/aa/aa27/aa27132.pdf\|id=Zbl 0303.10056, {{MathSciNet\|id=0369312}}}}.</ref> via an ingenious and complicated extension of the previous combinatorial argument (called "a masterpiece of combinatorial reasoning" by [[Ronald Graham\|R. L. Graham]]). Several further proofs are now known, the most important amongst them being those by [[Hillel Furstenberg]]<ref>{{citation\|authorlink=Hillel Furstenberg\|first=Hillel\|last=Furstenberg\|title=Ergodic behavior of diagonal measures and a theorem of Szemerédi on arithmetic progressions\|journal=J. D'Analyse Math.\|volume=31\|pages=204–256\|year=1977\|id={{MathSciNet\|id=0498471}}\|doi=10.1007/BF02813304}}.</ref> in 1977, using [[ergodic theory]], and by [[Timothy Gowers]]<ref name="gowers">{{citation\|authorlink=Timothy Gowers\|first=Timothy\|last=Gowers\|title=A new proof of Szemerédi's theorem\|journal=Geom. Funct. Anal.\|volume=11\|issue=3\|pages=465–588\|url=http://www.dpmms.cam.ac.uk/~wtg10/sz898.dvi\|year=2001\|id={{MathSciNet\|id=1844079}}\|doi=10.1007/s00039-001-0332-9}}.</ref> in 2001, using both [[Fourier analysis]] and [[combinatorics]].

			==Finitary version==
			Let ''k'' be a positive integer and let 0 < δ ≤ 1/2. A [[finitary]] version of the theorem states that there exists a positive integer

			:<math>N = N(k,\delta)\,</math>

			such that every subset of {1, 2, ..., ''N''} of size at least δ''N'' contains an arithmetic progression of length ''k''.
			The best known bounds for ''N''(''k'', δ) are

			:<math>C^{\log(1/\delta)^{k-1}} \leq N(k,\delta) \leq 2^{2^{\delta^{-2^{2^{k+9}}}}}</math>

			with ''C'' > 1. The lower bound is due to [[Felix A. Behrend\|Behrend]]<ref>{{citation\|authorlink=Felix A. Behrend\|first=Felix A.\|last=Behrend\|title=On the sets of integers which contain no three in arithmetic progression\|journal=[[Proceedings of the National Academy of Sciences]]\|volume=23\|issue=12\|pages=331–332\|year=1946\|id=Zbl 0060.10302\|doi=10.1073/pnas.32.12.331}}.</ref> (for ''k'' = 3) and [[Robert Alexander Rankin\|Rankin]],<ref>{{citation\|authorlink=Robert A. Rankin\|first=Robert A.\|last=Rankin\|title=Sets of integers containing not more than a given number of terms in arithmetical progression\|journal=Proc. Roy. Soc. Edinburgh Sect. A\|volume=65\|pages=332–344\|year=1962\|id=Zbl 0104.03705, {{MathSciNet\|id=0142526}}}}.</ref> and the upper bound is due to Gowers.<ref name="gowers"/>
			When ''k'' = 3 better upper bounds are known; [[Jean Bourgain\|Bourgain]]<ref>{{citation\|authorlink=Jean Bourgain\|first=Jean\|last=Bourgain\|title=On triples in arithmetic progression\|journal=Geom. Func. Anal.\|volume=9\|issue=5\|year=1999\|pages=968–984\|id={{MathSciNet\|id=1726234}}\|doi=10.1007/s000390050105}}</ref> has proved that

			:<math>N(3,\delta) \leq C^{\delta^{-2}\log(1/\delta)},</math>

			This was improved by [[Tom Sanders (mathematician)\|Sanders]]<ref>{{citation\|authorlink=Tom Sanders (mathematician)\|first=Tom\|last=Sanders\|title=On Roth's theorem on progressions\|journal=Annals of Mathematics\|volume=174\|issue=1\|year=2011\|pages=619–636\|id={{MathSciNet\|id=2811612}}\|doi=10.4007/annals.2011.174.1.20}}</ref> to

			:<math>N(3,\delta) \leq C^{\delta^{-1}(\log(1/\delta))^5}</math>

			==See also==
			* [[Problems involving arithmetic progressions]]
			* [[Ergodic Ramsey theory]]
			* [[Arithmetic combinatorics]]
			* [[Szemerédi regularity lemma]]
			* [[Green–Tao theorem]]
			* [[Erdős conjecture on arithmetic progressions]]

			==Notes==
			{{reflist\|colwidth=30em}}

			==Further reading==
			* {{Citation \| first=Terence \| last=Tao \| authorlink=Terence Tao \| chapter=The ergodic and combinatorial approaches to Szemerédi's theorem \| pages=145–193 \| editor1-first=Andrew \| editor1-last=Granville \| editor2-first=Melvyn B. \| editor2-last=Nathanson\| editor3-first=József \| editor3-last=Solymosi \| title=Additive Combinatorics \| series= CRM Proceedings & Lecture Notes \| volume=43 \| publisher=[[American Mathematical Society]] \| year=2007 \| isbn=978-0-8218-4351-2 \| zbl=1159.11005 }}

			==External links==
			* [http://planetmath.org/encyclopedia/SzemeredisTheorem.html PlanetMath source for initial version of this page]
			* [http://www.math.ucla.edu/~tao/whatsnew.html Announcement by Ben Green and Terence Tao] – the preprint is available at [http://front.math.ucdavis.edu/math.NT/0404188 math.NT/0404188]
			* [http://in-theory.blogspot.com/2006/06/szemeredis-theorem.html Discussion of Szemerédi's theorem (part 1 of 5)]
			* Ben Green and Terence Tao: [http://www.scholarpedia.org/article/Szemeredi%27s_Theorem Szemerédi's theorem] on [[Scholarpedia]]
			* {{MathWorld\|SzemeredisTheorem\|SzemeredisTheorem}}
			* {{cite web\|last=Grime\|first=James\|title=6,000,000: Endre Szemerédi wins the Abel Prize\|url=http://www.numberphile.com/videos/abel_prize.html\|work=Numberphile\|year=2012\|publisher=[[Brady Haran]]\|coauthors=Hodge, David}}

			{{DEFAULTSORT:Szemeredi's theorem}}
			[[Category:Ramsey theory]]
			[[Category:Theorems in combinatorics]]
			[[Category:Theorems in number theory]]

en>Headbomb: Various citation cleanup + AWB fixes for Book:Resonance using AWB (8062)

2012-07-31T03:36:17Z

Various citation cleanup + AWB fixes for Book:Resonance using AWB (8062)

New page

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