Ramanujan theta function - Revision history

en>Vhsatheeshkumar at 02:35, 1 March 2014

2014-03-01T02:35:16Z

← Older revision		Revision as of 04:35, 1 March 2014
Line 1:		Line 1:
	~~In [[mathematics]]~~, ~~specifically in [[topology]], a '''pseudo-Anosov map''' is a type of a [[diffeomorphism]] or [[homeomorphism]] of a [[surface]]~~. It is a generalization of a linear [[Anosov diffeomorphism]] of the [[torus]]. Its definition relies on the notion of a '''measured foliation''' invented by [[William Thurston]], who also coined the term "pseudo-Anosov diffeomorphism" when he proved his ~~[[Nielsen–Thurston classification\|classification of diffeomorphisms of a surface]]~~.		Hi there, I am Alyson Pomerleau and I think it sounds quite great when you say it. Mississippi is exactly where his house is. Office supervising is where her main earnings comes from. To perform lacross is the thing I adore most of all.<br><br>my web-site :: live psychic reading ([http://koreanyelp.com/index.php?document_srl=1798&mid=SchoolNews visit the next website])

	~~== Definition of a measured foliation ==~~
	~~A '''measured foliation''' ''F'' on a closed surface ''S''~~ is ~~a geometric structure on ''S'' which consists of a singular [[foliation]] and a measure in the transverse direction~~. ~~In some neighborhood of a regular point of ''F'', there~~ is ~~a "flow box" ''φ'': ''U'' → '''R'''<sup>2</sup> which sends~~ the ~~leaves~~ of ~~''F'' to the horizontal lines in '''R'''<sup>2</sup>~~. ~~If two such neighborhoods ''U''~~<~~sub~~>~~''i''~~<~~/sub~~> ~~and ''U''<sub>''j''</sub> overlap then there is a '''transition function''' ''φ''<sub>''ij''</sub> defined on ''φ''<sub>''j''</sub>(''U''<sub>''j''</sub>), with the standard property~~

	: ~~<math> \phi_{ij}\circ\phi_j=\phi_i,</math>~~

	~~which must have the form~~

	: ~~<math> \phi~~(~~x,y)=(f(x,y),c\pm y)</math>~~

	for some constant ''c''. This assures that along a simple curve, the variation in ''y''-coordinate, measured locally in every chart, is a geometric quantity (i.e. independent of the chart) and permits the definition of a total variation along a simple closed curve on ''S''. A finite number of singularities of ''F'' of the type of "''p''-pronged saddle", ''p''≥3, are allowed. At such a singular point, the differentiable structure of the surface is modified to make the point into a conical point with the total angle ''πp''. The notion of a diffeomorphism of ''S'' is redefined with respect to this modified differentiable structure. With some technical modifications, these definitions extend to the case of a surface with boundary.

	~~== Definition of a pseudo-Anosov map ==~~
	~~A homeomorphism~~

	:~~<math>f: S \to S <~~/~~math>~~

	~~of a closed surface ''S'' is called '''pseudo-Anosov''' if there exists a transverse pair of measured foliations on ''S'', ''F''<sup>''s''<~~/~~sup> (stable) and ''F''<sup>''u''</sup> (unstable), and a real number ''λ'' > 1 such that the foliations are preserved by ''f'' and their transverse measures are multiplied by 1~~/~~''λ'' and ''λ''~~. ~~The number ''~~&~~lambda;'' is called the '''stretch factor''' or '''dilatation''' of ''f''.~~

	== ~~Significance ==~~
	Thurston constructed a compactification of the [[Teichmüller space]] ''T''(''S'') of a surface ''S'' such that the action induced on ''T''(''S'') by any diffeomorphism ''f'' of ''S'' extends to a homeomorphism of the Thurston compactification. The dynamics of this homeomorphism is the simplest when ''f'' is a pseudo-Anosov map: in this case, there are two fixed points on the Thurston boundary, one attracting and one repelling, and the homeomorphism behaves similarly to a hyperbolic automorphism of the [[Poincaré half-plane]]. A "generic" diffeomorphism of a surface of genus at least two is isotopic to a pseudo-Anosov diffeomorphism.

	~~== Generalization ==~~
	Using the theory of '''train tracks''', the notion of a pseudo-Anosov map has been extended to self-maps of graphs (on the topological side) and outer automorphisms of [[free group]]s (on the algebraic side). This leads to an analogue of Thurston classification for the ~~case of automorphisms of free groups, developed by [[Mladen Bestvina\|Bestvina~~]~~] and Handel.~~

	~~==References==~~
	*A. Casson, S. Bleiler, "Automorphisms of Surfaces after Nielsen and Thurston", (London Mathematical Society Student Texts 9)~~, (1988).~~
	*A. Fathi, F. Laudenbach, and [[Valentin Poénaru\|V. Poénaru]], "Travaux de Thurston sur les surfaces," Asterisque, Vols. 66 and 67 (1979).
	*R.C. Penner. "A construction of pseudo-Anosov homeomorphisms", Trans. Amer. Math. Soc., 310 (1988) No 1, 179–197
	*{{Citation \| last1=Thurston \| first1=William P. \| author1-link=William Thurston \| title=On the geometry and dynamics of diffeomorphisms of surfaces \| doi=10.1090/S0273-0979-1988-15685-6 \| mr=956596 \| year=1988 \| journal=American Mathematical Society. Bulletin. New Series \| issn=0002-9904 \| volume=19 \| issue=2 \| pages=417–431}}

	~~{{DEFAULTSORT:Pseudo-Anosov Map}}~~
	~~[[Category:Dynamical systems]]~~
	~~[[Category:Geometric topology]]~~
	~~[[Category:Homeomorphisms]]~~

141.161.225.182: /* Definition */ one fix for f(q,q^3) formula. see Mathworld.

2013-03-25T00:17:47Z

Definition: one fix for f(q,q^3) formula. see Mathworld.

← Older revision		Revision as of 02:17, 25 March 2013
Line 1:		Line 1:
	~~Getting paid money to work online isn't the easiest thing to do~~ in ~~the world~~, ~~but it~~ is ~~possible~~. ~~If this~~ is ~~something you wish to work with, then~~ the ~~tips presented above should have helped you~~. ~~Take some time, do things~~ the ~~right way and then you can succeed. Here~~'~~s more information about~~ [~~http://ganhedinheironainternet.comoganhardinheiro101.com/ como ganhar dinheiro~~] ~~stop by ganhedinheironainternet.comoganhardinheiro101.com/ Start your online~~ [~~http://www~~.~~comoganhardinheiro101.com/?p~~=~~66 como ganhar dinheiro na internet~~] ~~earning today by following~~ the ~~great advice discussed in this article~~. ~~Earning money~~ is ~~not as hard as it may seem, you just need to know how to get started. By choosing to put your right foot forward, you are heading off to~~ a ~~great start earning money~~ to ~~make ends meet~~. <br><br><br>~~Make money online by selling your talents. Good music~~ is ~~always in demand and with today~~'~~s technological advances~~, ~~anyone~~ with ~~[http~~://~~comoganhardinheironainternet~~.~~comoganhardinheiro101~~.~~com/ musical] talent can make music~~ and ~~offer it for sale [http://www~~.~~comoganhardinheiro101~~.~~com/ganhar-dinheiro-pela-internet-com-opcoes-binarias/ como conseguir dinheiro]~~ to a ~~broad audience~~. ~~By [http://comoficarrico~~.~~comoganhardinheiro101.com ganhar dinheiro] setting up your own website and using social media for promotion~~, ~~you can share your music with others and sell downloads~~ with a ~~free PayPal account.~~ <br><br><br>~~fee to watch your webinar at their convenience. Once it is in place, [http:~~/~~/www.comoganhardinheiro101.com como ganhar dinheiro] promotion~~ and ~~possibly answering questions will be your only tasks.~~<br><br>It'~~s easy to make money online~~. ~~There~~ is ~~truth to~~ the ~~fact~~ that ~~you can start making money~~ on the ~~Internet as soon as you~~'~~re done with~~ this ~~article. After all~~, ~~so many others~~ are ~~making money online~~, ~~why not you? Keep your mind open~~ and ~~you can make~~ a ~~lot~~ of ~~money. As you~~ [~~http://www~~.~~comoganhardinheiro101~~.~~com/ganhar~~-~~dinheiro~~-~~pela-internet-com-opcoes-binarias/ ganhar dinheiro pela internet~~] ~~can see, there are many ways~~ to ~~approach~~ the ~~world~~ of ~~online income~~. ~~With various streams~~ of ~~income available~~, ~~you are sure to find one~~, ~~or two~~, ~~that can help you with your income needs~~. ~~Take this information to heart~~, ~~put it to use~~ and ~~build your own online success story~~.		In [[mathematics]], specifically in [[topology]], a '''pseudo-Anosov map''' is a type of a [[diffeomorphism]] or [[homeomorphism]] of a [[surface]]. It is a generalization of a linear [[Anosov diffeomorphism]] of the [[torus]]. Its definition relies on the notion of a '''measured foliation''' invented by [[William Thurston]], who also coined the term "pseudo-Anosov diffeomorphism" when he proved his [[Nielsen–Thurston classification\|classification of diffeomorphisms of a surface]].

			== Definition of a measured foliation ==
			A '''measured foliation''' ''F'' on a closed surface ''S'' is a geometric structure on ''S'' which consists of a singular [[foliation]] and a measure in the transverse direction. In some neighborhood of a regular point of ''F'', there is a "flow box" ''φ'': ''U'' → '''R'''<sup>2</sup> which sends the leaves of ''F'' to the horizontal lines in '''R'''<sup>2</sup>. If two such neighborhoods ''U''<sub>''i''</sub> and ''U''<sub>''j''</sub> overlap then there is a '''transition function''' ''φ''<sub>''ij''</sub> defined on ''φ''<sub>''j''</sub>(''U''<sub>''j''</sub>), with the standard property

			: <math> \phi_{ij}\circ\phi_j=\phi_i,</math>

			which must have the form

			: <math> \phi(x,y)=(f(x,y),c\pm y)</math>

			for some constant ''c''. This assures that along a simple curve, the variation in ''y''-coordinate, measured locally in every chart, is a geometric quantity (i.e. independent of the chart) and permits the definition of a total variation along a simple closed curve on ''S''. A finite number of singularities of ''F'' of the type of "''p''-pronged saddle", ''p''≥3, are allowed. At such a singular point, the differentiable structure of the surface is modified to make the point into a conical point with the total angle ''πp''. The notion of a diffeomorphism of ''S'' is redefined with respect to this modified differentiable structure. With some technical modifications, these definitions extend to the case of a surface with boundary.

			== Definition of a pseudo-Anosov map ==
			A homeomorphism

			:<math>f: S \to S </math>

			of a closed surface ''S'' is called '''pseudo-Anosov''' if there exists a transverse pair of measured foliations on ''S'', ''F''<sup>''s''</sup> (stable) and ''F''<sup>''u''</sup> (unstable), and a real number ''λ'' > 1 such that the foliations are preserved by ''f'' and their transverse measures are multiplied by 1/''λ'' and ''λ''. The number ''λ'' is called the '''stretch factor''' or '''dilatation''' of ''f''.

			== Significance ==
			Thurston constructed a compactification of the [[Teichmüller space]] ''T''(''S'') of a surface ''S'' such that the action induced on ''T''(''S'') by any diffeomorphism ''f'' of ''S'' extends to a homeomorphism of the Thurston compactification. The dynamics of this homeomorphism is the simplest when ''f'' is a pseudo-Anosov map: in this case, there are two fixed points on the Thurston boundary, one attracting and one repelling, and the homeomorphism behaves similarly to a hyperbolic automorphism of the [[Poincaré half-plane]]. A "generic" diffeomorphism of a surface of genus at least two is isotopic to a pseudo-Anosov diffeomorphism.

			== Generalization ==
			Using the theory of '''train tracks''', the notion of a pseudo-Anosov map has been extended to self-maps of graphs (on the topological side) and outer automorphisms of [[free group]]s (on the algebraic side). This leads to an analogue of Thurston classification for the case of automorphisms of free groups, developed by [[Mladen Bestvina\|Bestvina]] and Handel.

			==References==
			*A. Casson, S. Bleiler, "Automorphisms of Surfaces after Nielsen and Thurston", (London Mathematical Society Student Texts 9), (1988).
			*A. Fathi, F. Laudenbach, and [[Valentin Poénaru\|V. Poénaru]], "Travaux de Thurston sur les surfaces," Asterisque, Vols. 66 and 67 (1979).
			*R.C. Penner. "A construction of pseudo-Anosov homeomorphisms", Trans. Amer. Math. Soc., 310 (1988) No 1, 179–197
			*{{Citation \| last1=Thurston \| first1=William P. \| author1-link=William Thurston \| title=On the geometry and dynamics of diffeomorphisms of surfaces \| doi=10.1090/S0273-0979-1988-15685-6 \| mr=956596 \| year=1988 \| journal=American Mathematical Society. Bulletin. New Series \| issn=0002-9904 \| volume=19 \| issue=2 \| pages=417–431}}

			{{DEFAULTSORT:Pseudo-Anosov Map}}
			[[Category:Dynamical systems]]
			[[Category:Geometric topology]]
			[[Category:Homeomorphisms]]

en>RibotBOT: Robot: Modifying pt:Função teta de Ramanujan

2012-08-19T14:53:33Z

Robot: Modifying pt:Função teta de Ramanujan

New page

Getting paid money to work online isn't the easiest thing to do in the world, but it is possible. If this is something you wish to work with, then the tips presented above should have helped you. Take some time, do things the right way and then you can succeed. Here's more information about [http://ganhedinheironainternet.comoganhardinheiro101.com/ como ganhar dinheiro] stop by ganhedinheironainternet.comoganhardinheiro101.com/ Start your online [http://www.comoganhardinheiro101.com/?p=66 como ganhar dinheiro na internet] earning today by following the great advice discussed in this article. Earning money is not as hard as it may seem, you just need to know how to get started. By choosing to put your right foot forward, you are heading off to a great start earning money to make ends meet. Make money online by selling your talents. Good music is always in demand and with today's technological advances, anyone with [http://comoganhardinheironainternet.comoganhardinheiro101.com/ musical] talent can make music and offer it for sale [http://www.comoganhardinheiro101.com/ganhar-dinheiro-pela-internet-com-opcoes-binarias/ como conseguir dinheiro] to a broad audience. By [http://comoficarrico.comoganhardinheiro101.com ganhar dinheiro] setting up your own website and using social media for promotion, you can share your music with others and sell downloads with a free PayPal account. fee to watch your webinar at their convenience. Once it is in place, [http://www.comoganhardinheiro101.com como ganhar dinheiro] promotion and possibly answering questions will be your only tasks. It's easy to make money online. There is truth to the fact that you can start making money on the Internet as soon as you're done with this article. After all, so many others are making money online, why not you? Keep your mind open and you can make a lot of money. As you [http://www.comoganhardinheiro101.com/ganhar-dinheiro-pela-internet-com-opcoes-binarias/ ganhar dinheiro pela internet] can see, there are many ways to approach the world of online income. With various streams of income available, you are sure to find one, or two, that can help you with your income needs. Take this information to heart, put it to use and build your own online success story.