Amenable group - Revision history

87.99.27.160: /* Case of discrete groups */

2015-01-09T04:21:01Z

Case of discrete groups

← Older revision		Revision as of 06:21, 9 January 2015
Line 1:		Line 1:
	~~Hola. The author's~~ name is ~~Eusebio regrettably he never really beloved that name~~. ~~The beloved hobby for him and his kids~~ is ~~so as to drive and he~~'s ~~also been doing it for several~~ years. ~~[https://www.Vocabulary.com/dictionary/Auditing Auditing] is how he supports his own family. [http://dict.leo.org/?search=Massachusetts Massachusetts] is where or even~~ and ~~his wife remain~~. ~~He's been working about his website for various time now~~. ~~Check this kind of out here: http://prometeu.net~~<br><br>~~Here is~~ my ~~blog post;~~ [http://~~prometeu~~.~~net clash of clans cheats ipod~~]		Hello! My name is Carri. <br>It is a little about myself: I live in Germany, my city of Wadern. <br>It's called often Northern or cultural capital of SL. I've married 2 years ago.<br>I have 2 children - a son (Sebastian) and the daughter (Waylon). We all like Footbag.<br><br>my homepage - [http://facebook18.com/index.php?do=/profile-11827/info/ Online Coupon Coupon Code 4inkjets]

en>Ozob: Theorems and definitions should not be in blockquotes

2014-02-19T15:13:34Z

Theorems and definitions should not be in blockquotes

← Older revision		Revision as of 17:13, 19 February 2014
Line 1:		Line 1:
	~~{{Technical\|date=September 2010}}~~		Hola. The author's name is Eusebio regrettably he never really beloved that name. The beloved hobby for him and his kids is so as to drive and he's also been doing it for several years. [https://www.Vocabulary.com/dictionary/Auditing Auditing] is how he supports his own family. [http://dict.leo.org/?search=Massachusetts Massachusetts] is where or even and his wife remain. He's been working about his website for various time now. Check this kind of out here: http://prometeu.net<br><br>Here is my blog post; [http://prometeu.net clash of clans cheats ipod]

	~~In [[mathematics]], an '''infinitesimal transformation''~~' is ~~a [[limit (mathematics)\|limiting]] form of ''small'' [[transformation (geometry)\|transformation]]~~. ~~For example one may talk about an '''[[infinitesimal rotation]]''' of a [[rigid body]], in three-dimensional space. This~~ is ~~conventionally represented by a 3×3 [[skew-symmetric matrix]] ''A'~~'. ~~It is not the matrix of an actual [~~[~~rotation]] in space; but for small real values of a parameter ε we have~~

	:~~<math>I+\varepsilon A<~~/~~math>~~

	~~a small rotation, up to quantities of order ε<sup>2<~~/~~sup>~~.
	~~==History==~~
	~~A comprehensive theory of infinitesimal transformations was first given by [[Sophus Lie~~]~~]. Indeed this was at the heart of~~ his ~~work, on what are now called [[Lie group]]s and their accompanying [[Lie algebra]]s; and the identification of their role in [[geometry]] and especially the theory of [[differential equation]]s~~. ~~The properties of an abstract~~ [~~[Lie algebra]] are exactly those definitive of infinitesimal transformations, just as the axioms of [[group theory]] embody [[symmetry]]~~. ~~The term "Lie algebra" was introduced in 1934 by [[Hermann Weyl]], for what had until then been known as the ''algebra of infinitesimal transformations'' of a Lie group~~.
	~~==Examples~~==
	For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the [[cross product]], once a skew-symmetric matrix has been identified with a 3-[[Vector (geometric)\|vector]]. This amounts to choosing an axis vector for the rotations; the defining [[Jacobi identity]] is ~~a well-known property of cross products~~.

	~~The earliest example of an infinitesimal transformation that may have been recognised as such was in [[Euler~~'s ~~theorem on homogeneous functions]]~~. ~~Here it is stated that a function ''F''~~ of ~~''n'' variables ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> that is homogeneous of degree ''r'', satisfies~~

	:~~<math>H\cdot F=rF \, <~~/~~math>~~

	~~with~~

	~~:<math>H=\sum_i x_i{\partial\over\partial x_i},<~~/~~math>~~

	~~a [[differential operator]]~~. ~~That is, from the property~~

	:<~~math~~>~~F(\lambda x_1,\dots, \lambda x_n)=\lambda^r F(x_1,\dots,x_n)\,~~<~~/math~~>

	we can in effect differentiate with respect to λ and then set λ equal to 1. This then becomes a [[necessary condition]] on a [[smooth function]] ''F'' to have the homogeneity property; it is also sufficient (by using [[Schwartz distribution]]s one can reduce the [[mathematical analysis]] considerations here). This setting is ~~typical, in that we have a [[one-parameter group]] of [[scaling (mathematics)\|scalings]] operating~~; ~~and the information is in fact coded in an infinitesimal transformation that is a~~ [~~[first-order differential operator]].~~
	~~==Operator version of Taylor's theorem==~~
	~~The operator equation~~

	:~~<math>e^{tD}f(x)=f(x+t)\,<~~/~~math>~~

	~~where~~

	~~:<math>D={d\over dx}<~~/~~math>~~

	~~is an [[Operator (mathematics)\|operator]] version of [[Taylor's theorem]] — and is therefore only valid under ''caveats'' about ''f'' being an [[analytic function]]~~. Concentrating on the operator part, it shows in effect that ''D'' is an infinitesimal transformation, generating translations of the real line via the [[exponential function\|exponential]]. In Lie's theory, this is generalised a long way. Any [[connected space\|connected]] Lie group can be built up by means of its [[Lie group#The Lie algebra associated with a Lie group\|infinitesimal generator]]s (a basis for the Lie algebra of the group); with explicit if not always useful information given in the [[Baker–Campbell–Hausdorff formula]].

	~~== References ==~~
	*{{Springer\|id=L/l058370\|title=Lie algebra}}

	~~[[Category:Lie groups]]~~
	~~[[Category:Transformation (function)]~~]

en>Omertamuz: /* Equivalent conditions for amenability */ Corrected Kesten's condition to be about the L^2 operator rather than the L^1.

2014-01-28T14:46:45Z

Equivalent conditions for amenability: Corrected Kesten's condition to be about the L^2 operator rather than the L^1.

← Older revision		Revision as of 16:46, 28 January 2014
Line 1:		Line 1:
	\ո		{{Technical\|date=September 2010}}
	~~There~~'ѕ a ~~wide selection~~ of ~~рroducts on Penny Auctіons; if you are looking for something, the ƿrobability is good it is availɑble~~. ~~Have a great time perusing every~~ one of ~~the auctions for items including books~~, ~~shoes, jewellery, cellular phoneѕ, еlectricɑl goods and far, much more~~. ~~Do you really need а laѕt~~-~~minute gift? Take~~ a ~~look at some Penny Sites fߋr~~ a ~~lot~~ of ~~іdeas.~~<br><br>~~Signing up~~ on ~~many~~ of ~~these auction sites iѕ straightforward, rapid~~ and ~~additionally free~~ of ~~charge~~. ~~One~~ of the ~~ways these auctions differ from other online shopping sites, is they require you to purchase bid credits before bidding ƅy using ɑn item~~. ~~Bidding on products costs yoս credits~~, ~~which will be automatically deducted out~~ of ~~your account balance~~. ~~Almost any person can be involved~~ in ~~these ɑuctions as bid amounts are generally set from aroսnd twenty~~-~~five cents~~ [~~http://vercandis~~.~~buzznet~~.~~com/user/journal/18807377/fap-turbo-details-fapturbo-trader/ salemsharkslive~~] ~~with~~ a ~~little more than a dollar.~~<br><br>~~[http://www~~.~~allgarfotimor~~.~~com~~/~~review~~/~~quibidsreview~~/ ~~quibids reviews~~]<br><br>~~Because of the current condition of the economy Penny Auctions have become an extremely popular approach~~ to ~~looking for people around the globe. You don't have to be ԝorried аbout bidding times as ƿossible~~ set ~~your bids~~ to ~~any time~~. ~~Store hours aren~~'~~t a problem as you cɑn shop nigɦt օr day insteaɗ of looking forԝard~~ to the ~~shops to spread out~~. ~~Talk aboսt~~ a ~~strеss~~-~~free approach to ѕhop!<br><br>Many consumers surf the net to find bargains on a myriad~~ of ~~items, for things for themselves~~ and ~~gifts they would lіke for some individualѕ. Prefer~~ a ~~way to ƿurchase products a lot more ϲheaply than yoս are able to anywҺere else, try a Penny Auction~~. It's ~~never been so easy to buy gifts~~, ~~items and items. It~~'s ~~not~~ only ~~cߋnvenient~~, it's ~~аlso discreet and less than other available choiсeѕ.<br><br>When you decide you ɑre to benefit from these ԁeeply discounted prices~~, ~~have~~ a ~~minute to viеw the procedure from start tо finish, as opposed to jumping to the fray totally unprepаred~~. ~~This~~ can ~~make sure ƴou comprehend~~ the ~~prߋcess, are ready to Ьid, fully reap~~ the ~~benefits of and like~~ the ~~Реnny Auctions еxperience~~.
			In [[mathematics]], an '''infinitesimal transformation''' is a [[limit (mathematics)\|limiting]] form of ''small'' [[transformation (geometry)\|transformation]]. For example one may talk about an '''[[infinitesimal rotation]]''' of a [[rigid body]], in three-dimensional space. This is conventionally represented by a 3×3 [[skew-symmetric matrix]] ''A''. It is not the matrix of an actual [[rotation]] in space; but for small real values of a parameter ε we have

			:<math>I+\varepsilon A</math>

			a small rotation, up to quantities of order ε<sup>2</sup>.
			==History==
			A comprehensive theory of infinitesimal transformations was first given by [[Sophus Lie]]. Indeed this was at the heart of his work, on what are now called [[Lie group]]s and their accompanying [[Lie algebra]]s; and the identification of their role in [[geometry]] and especially the theory of [[differential equation]]s. The properties of an abstract [[Lie algebra]] are exactly those definitive of infinitesimal transformations, just as the axioms of [[group theory]] embody [[symmetry]]. The term "Lie algebra" was introduced in 1934 by [[Hermann Weyl]], for what had until then been known as the ''algebra of infinitesimal transformations'' of a Lie group.
			==Examples==
			For example, in the case of infinitesimal rotations, the Lie algebra structure is that provided by the [[cross product]], once a skew-symmetric matrix has been identified with a 3-[[Vector (geometric)\|vector]]. This amounts to choosing an axis vector for the rotations; the defining [[Jacobi identity]] is a well-known property of cross products.

			The earliest example of an infinitesimal transformation that may have been recognised as such was in [[Euler's theorem on homogeneous functions]]. Here it is stated that a function ''F'' of ''n'' variables ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> that is homogeneous of degree ''r'', satisfies

			:<math>H\cdot F=rF \, </math>

			with

			:<math>H=\sum_i x_i{\partial\over\partial x_i},</math>

			a [[differential operator]]. That is, from the property

			:<math>F(\lambda x_1,\dots, \lambda x_n)=\lambda^r F(x_1,\dots,x_n)\,</math>

			we can in effect differentiate with respect to λ and then set λ equal to 1. This then becomes a [[necessary condition]] on a [[smooth function]] ''F'' to have the homogeneity property; it is also sufficient (by using [[Schwartz distribution]]s one can reduce the [[mathematical analysis]] considerations here). This setting is typical, in that we have a [[one-parameter group]] of [[scaling (mathematics)\|scalings]] operating; and the information is in fact coded in an infinitesimal transformation that is a [[first-order differential operator]].
			==Operator version of Taylor's theorem==
			The operator equation

			:<math>e^{tD}f(x)=f(x+t)\,</math>

			where

			:<math>D={d\over dx}</math>

			is an [[Operator (mathematics)\|operator]] version of [[Taylor's theorem]] — and is therefore only valid under ''caveats'' about ''f'' being an [[analytic function]]. Concentrating on the operator part, it shows in effect that ''D'' is an infinitesimal transformation, generating translations of the real line via the [[exponential function\|exponential]]. In Lie's theory, this is generalised a long way. Any [[connected space\|connected]] Lie group can be built up by means of its [[Lie group#The Lie algebra associated with a Lie group\|infinitesimal generator]]s (a basis for the Lie algebra of the group); with explicit if not always useful information given in the [[Baker–Campbell–Hausdorff formula]].

			== References ==
			*{{Springer\|id=L/l058370\|title=Lie algebra}}

			[[Category:Lie groups]]
			[[Category:Transformation (function)]]

en>Mathsci: /* References */

2012-08-24T12:37:51Z

References

New page

\ո
There'ѕ a wide selection of рroducts on Penny Auctіons; if you are looking for something, the ƿrobability is good it is availɑble. Have a great time perusing every one of the auctions for items including books, shoes, jewellery, cellular phoneѕ, еlectricɑl goods and far, much more. Do you really need а laѕt-minute gift? Take a look at some Penny Sites fߋr a lot of іdeas. Signing up on many of these auction sites iѕ straightforward, rapid and additionally free of charge. One of the ways these auctions differ from other online shopping sites, is they require you to purchase bid credits before bidding ƅy using ɑn item. Bidding on products costs yoս credits, which will be automatically deducted out of your account balance. Almost any person can be involved in these ɑuctions as bid amounts are generally set from aroսnd twenty-five cents [http://vercandis.buzznet.com/user/journal/18807377/fap-turbo-details-fapturbo-trader/ salemsharkslive] with a little more than a dollar. [http://www.allgarfotimor.com/review/quibidsreview/ quibids reviews] Because of the current condition of the economy Penny Auctions have become an extremely popular approach to looking for people around the globe. You don't have to be ԝorried аbout bidding times as ƿossible set your bids to any time. Store hours aren't a problem as you cɑn shop nigɦt օr day insteaɗ of looking forԝard to the shops to spread out. Talk aboսt a strеss-free approach to ѕhop! Many consumers surf the net to find bargains on a myriad of items, for things for themselves and gifts they would lіke for some individualѕ. Prefer a way to ƿurchase products a lot more ϲheaply than yoս are able to anywҺere else, try a Penny Auction. It's never been so easy to buy gifts, items and items. It's not only cߋnvenient, it's аlso discreet and less than other available choiсeѕ. When you decide you ɑre to benefit from these ԁeeply discounted prices, have a minute to viеw the procedure from start tо finish, as opposed to jumping to the fray totally unprepаred. This can make sure ƴou comprehend the prߋcess, are ready to Ьid, fully reap the benefits of and like the Реnny Auctions еxperience.