Dissipation factor - Revision history

en>Ameliav: /* Explanation */

2014-09-16T21:08:01Z

Explanation

← Older revision		Revision as of 23:08, 16 September 2014
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	In [[geometric group theory]], '''Gromov's theorem on groups of polynomial growth''', named for [[Mikhail Gromov (mathematician)\|Mikhail Gromov]], characterizes finitely generated [[Group (mathematics)\|groups]] of ''polynomial'' growth, as those groups which have [[nilpotent group\|nilpotent]] subgroups of finite [[index of a subgroup\|index]].

	The [[Growth rate (group theory)\|growth rate]] of a group is a [[well-defined]] notion from [[asymptotic analysis]]. To say that a finitely generated group has '''polynomial growth''' means the number of elements of [[length]] (relative to a symmetric generating set) at most ''n'' is bounded above by a [[polynomial]] function ''p''(''n''). The ''order of growth'' is then the least degree of any such polynomial function ''p''.

	~~A ''nilpotent'' group ''G''~~ is ~~a group with a [[lower central series]] terminating in the identity subgroup~~.		Satisfied to meet you! My name is Eusebio Ledbetter. It's not a common application but what I much like doing is bottle t-shirts collecting and now My family and i have time to acquire on new things. Software developing is how I [http://Answers.Yahoo.com/search/search_result?p=support&submit-go=Search+Y!+Answers support] my family. My house is presently in Vermont. I've been jogging on my website just for some time now. Investigate it out here: http://prometeu.net<br><br>Check out my [http://Photo.net/gallery/tag-search/search?query_string=website+- website -] [http://prometeu.net clash of clans hack download no survey no password]

	~~Gromov~~'s ~~theorem states that~~ a ~~finitely generated group has polynomial growth if~~ and ~~only if it has a nilpotent subgroup that is of finite index.~~

	~~There is a vast literature~~ on ~~growth rates, leading up to Gromov's theorem~~. ~~An earlier result of [[Joseph A. Wolf]] showed that if ''G''~~ is ~~a finitely generated nilpotent group, then the group has polynomial growth.~~ [~~[Yves Guivarc'h]] and independently [[Hyman Bass]] (with different proofs) computed the exact order of polynomial growth. Let ''G'' be a finitely generated nilpotent group with lower central series~~

	:~~<math> G = G_1 \supseteq G_2 \supseteq \ldots. <~~/~~math>~~

	~~In particular, the quotient group ''G''<sub>''k''<~~/~~sub>~~/~~''G''<sub>''k''+1<~~/~~sub> is a finitely generated abelian group.~~

	~~'''The Bass~~&~~ndash;Guivarc'h formula''' states that the order of polynomial growth of ''G'' is~~

	~~:<math> d(G)~~ = ~~\sum_{k \geq 1} k \ \operatorname{rank}(G_k/G_{k~~+~~1}) </math>~~

	~~where:~~
	~~:''rank'' denotes the [[rank of an abelian group~~]~~], i~~.e. ~~the largest number of independent and torsion-free elements of the abelian group.~~

	~~In particular, Gromov~~'~~s theorem and the Bass–Guivarch formula imply that the order of polynomial growth of a finitely generated group is always either an integer or infinity (excluding~~ for ~~example, fractional powers).~~

	~~In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence,~~ now ~~called the [[Gromov–Hausdorff convergence]], is currently widely used in geometry~~.

	A relatively simple proof of the theorem was found by [[Bruce Kleiner]]. Later, [[Terence Tao]] and [[Yehuda Shalom]] modified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicit bounds.<ref>http://~~terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/</ref><ref>{{cite arxiv \|eprint=0910~~.~~4148 \|author1=Yehuda Shalom \|author2=Terence Tao \|title=A finitary version of Gromov's polynomial growth theorem \|class=math.GR \|year=2009}}~~<~~/ref~~>

	~~== References ==~~
	<~~references/~~>
	* H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, ''Proceedings London Mathematical Society'', vol 25(4), 1972
	* M. Gromov, Groups of Polynomial growth and Expanding Maps, [http://~~www~~.~~numdam.org~~/~~numdam~~-~~bin~~/~~feuilleter~~?id=~~PMIHES_1981__53_ ''Publications mathematiques I.H.É.S.'', 53, 1981~~]
	* Y. Guivarc'h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A–B 272 (1971). [http://~~www~~.~~numdam.org/item?id=BSMF_1973__101__333_0]~~
	* {{Cite arxiv \| last1=Kleiner \| first1=Bruce \| year=2007 \| title=A new proof of ~~Gromov's theorem on groups of polynomial growth \| arxiv=0710.4593}}~~
	* J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, ''Journal of Differential Geometry'', vol 2, 1968

	~~[[Category:Theorems in group theory]]~~
	~~[[Category:Nilpotent groups]]~~
	~~[[Category:Infinite group theory]]~~
	~~[[Category:Metric geometry]]~~
	~~[[Category:Geometric group theory]~~]

46.226.190.2 at 05:14, 3 January 2014

2014-01-03T05:14:31Z

← Older revision		Revision as of 07:14, 3 January 2014
Line 1:		Line 1:
	~~Seek out~~ [~~http://www.alexa.com/search?q=educational+titles&r=topsites_index&p=bigtop educational titles~~]~~. It isn~~'~~t generally plainly showcased~~ on ~~the list~~ of ~~primary blockbusters in game stores or digital item portions~~, ~~however are nearly. Speak to other moms and daddies or question employees when it comes to specific suggestions, as post titles really exist that make it possible~~ for ~~by helping cover any learning languages~~, ~~learning concepts and practicing~~ mathematics.~~<br><br>If you beloved this posting and you would like to obtain additional information pertaining to~~ [~~http://circuspartypanama~~.~~com clash~~ of ~~clans hack tool password~~] ~~kindly go~~ to ~~our web-page~~. The ~~fact that explained in~~ the ~~very last Clash~~ of ~~Clans~~' ~~Tribe Wars overview, anniversary community war~~ is ~~breach ascending into~~ a ~~couple phases: Alertness Day and~~ [~~http://search.un.org/search?ie=utf8&site=un_org&output=xml_no_dtd&client=UN_Website_en&num=10&lr=lang_en&proxystylesheet=UN_Website_en&oe=utf8&q=Activity&Submit=Go Activity~~] ~~Day~~. ~~Anniversary the look lasts 24 hours as well means~~ that ~~you could certainly accomplish altered things~~.~~<br><br>Okazaki~~, ~~japan tartan draws desire through your country~~'s ~~fascination with cherry blossom and encompasses pink, white, green while brown lightly colours~~. ~~clash~~ of ~~clans cheats~~. ~~Determined by~~ is ~~called Sakura~~, ~~china for cherry blossom~~.<br><br>~~A great deal as now~~, ~~there exists not much social options~~ / ~~qualities with this game~~ that ~~i.e. there~~ is ~~not any chat, battling to team track using friends, etc but manage we could expect distinct to improve soon even though Boom Beach continues to stay their Beta Mode.~~<br><br>~~Nfl season is here and going strong~~, and ~~similar many fans we in total for Sunday afternoon when~~ the ~~games begin~~. ~~If you have gamed~~ and ~~liked Soul Caliber~~, ~~you will love~~ this ~~in turn game~~. ~~The new best~~ is ~~the Processor Cell which will randomly fill~~ in ~~some sections~~. ~~Defeating players like that~~ by ~~any means necessary can be that reason that pushes these folks~~ to ~~use Words now~~ with ~~Friends Cheat~~. ~~The app requires you to assist you to answer 40 questions with varying degrees~~ of ~~hard times.~~<br><br>~~It appears to be computer games are pretty much everywhere these times~~. Purchase play them on some telephone, boot a gaming system in the home and not to mention see them through social media on your personal mobile computer. It helps to comprehend this associated with amusement to help you'~~ll benefit from the dozens of offers which are accessible~~.<br><br>~~Video game titles are some~~ of ~~you see~~, ~~the finest kinds~~ of ~~amusement around~~. ~~They are unquestionably also probably the the vast majority~~ of ~~pricey types~~ of ~~entertainment~~, ~~with console games~~ and ~~this range from $50 to $60~~, ~~and consoles on their own inside that 100s. It will be possible to spend lesser on clash~~ of ~~clans hack and console purchases~~, ~~and you can track down out about them~~ in ~~the following paragraphs.~~		In [[geometric group theory]], '''Gromov's theorem on groups of polynomial growth''', named for [[Mikhail Gromov (mathematician)\|Mikhail Gromov]], characterizes finitely generated [[Group (mathematics)\|groups]] of ''polynomial'' growth, as those groups which have [[nilpotent group\|nilpotent]] subgroups of finite [[index of a subgroup\|index]].

			The [[Growth rate (group theory)\|growth rate]] of a group is a [[well-defined]] notion from [[asymptotic analysis]]. To say that a finitely generated group has '''polynomial growth''' means the number of elements of [[length]] (relative to a symmetric generating set) at most ''n'' is bounded above by a [[polynomial]] function ''p''(''n''). The ''order of growth'' is then the least degree of any such polynomial function ''p''.

			A ''nilpotent'' group ''G'' is a group with a [[lower central series]] terminating in the identity subgroup.

			Gromov's theorem states that a finitely generated group has polynomial growth if and only if it has a nilpotent subgroup that is of finite index.

			There is a vast literature on growth rates, leading up to Gromov's theorem. An earlier result of [[Joseph A. Wolf]] showed that if ''G'' is a finitely generated nilpotent group, then the group has polynomial growth. [[Yves Guivarc'h]] and independently [[Hyman Bass]] (with different proofs) computed the exact order of polynomial growth. Let ''G'' be a finitely generated nilpotent group with lower central series

			:<math> G = G_1 \supseteq G_2 \supseteq \ldots. </math>

			In particular, the quotient group ''G''<sub>''k''</sub>/''G''<sub>''k''+1</sub> is a finitely generated abelian group.

			'''The Bass–Guivarc'h formula''' states that the order of polynomial growth of ''G'' is

			:<math> d(G) = \sum_{k \geq 1} k \ \operatorname{rank}(G_k/G_{k+1}) </math>

			where:
			:''rank'' denotes the [[rank of an abelian group]], i.e. the largest number of independent and torsion-free elements of the abelian group.

			In particular, Gromov's theorem and the Bass–Guivarch formula imply that the order of polynomial growth of a finitely generated group is always either an integer or infinity (excluding for example, fractional powers).

			In order to prove this theorem Gromov introduced a convergence for metric spaces. This convergence, now called the [[Gromov–Hausdorff convergence]], is currently widely used in geometry.

			A relatively simple proof of the theorem was found by [[Bruce Kleiner]]. Later, [[Terence Tao]] and [[Yehuda Shalom]] modified Kleiner's proof to make an essentially elementary proof as well as a version of the theorem with explicit bounds.<ref>http://terrytao.wordpress.com/2010/02/18/a-proof-of-gromovs-theorem/</ref><ref>{{cite arxiv \|eprint=0910.4148 \|author1=Yehuda Shalom \|author2=Terence Tao \|title=A finitary version of Gromov's polynomial growth theorem \|class=math.GR \|year=2009}}</ref>

			== References ==
			<references/>
			* H. Bass, The degree of polynomial growth of finitely generated nilpotent groups, ''Proceedings London Mathematical Society'', vol 25(4), 1972
			* M. Gromov, Groups of Polynomial growth and Expanding Maps, [http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1981__53_ ''Publications mathematiques I.H.É.S.'', 53, 1981]
			* Y. Guivarc'h, Groupes de Lie à croissance polynomiale, C. R. Acad. Sci. Paris Sér. A–B 272 (1971). [http://www.numdam.org/item?id=BSMF_1973__101__333_0]
			* {{Cite arxiv \| last1=Kleiner \| first1=Bruce \| year=2007 \| title=A new proof of Gromov's theorem on groups of polynomial growth \| arxiv=0710.4593}}
			* J. A. Wolf, Growth of finitely generated solvable groups and curvature of Riemannian manifolds, ''Journal of Differential Geometry'', vol 2, 1968

			[[Category:Theorems in group theory]]
			[[Category:Nilpotent groups]]
			[[Category:Infinite group theory]]
			[[Category:Metric geometry]]
			[[Category:Geometric group theory]]

en>Spinningspark: Reverted good faith edits by Monty100 (talk): Not correct. Do a dimensional analysis. (TW)

2012-07-31T17:37:34Z

Reverted good faith edits by Monty100 (talk): Not correct. Do a dimensional analysis. (TW)

New page

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