Bayesian average - Revision history

en>Rpyle731: stub sort

2014-11-03T02:07:44Z

stub sort

← Older revision		Revision as of 04:07, 3 November 2014
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	~~In [[complex geometry]], a '''Hopf manifold''' {{harv\|Hopf\|1948}} is obtained~~		They call me Emilia. Puerto Rico is where he and his wife live. Hiring is my occupation. To gather cash is what his family and him enjoy.<br><br>my website; at home std test ([http://www.videoworld.com/blog/211112 visit web site])
	~~as a quotient of the complex [[vector space]]~~
	~~(with zero deleted) <math>({\Bbb C}^n\backslash 0)</math>~~
	~~by a [[Group action\|free action]] of the [[Group (mathematics)\|group]] <math>\Gamma \cong {\Bbb Z}</math> of~~
	~~[[integer]]s, with the generator <math>\gamma</math>~~
	~~of <math>\Gamma</math> acting by holomorphic [[Contraction mapping\|contractions]]~~. ~~Here, a ''holomorphic contraction''~~
	is ~~a map <math>\gamma:\; {\Bbb C}^n \mapsto {\Bbb C}^n</math>~~
	~~such that a sufficiently big iteration <math>\;\gamma^N</math>~~
	~~puts any given [[compact subset]] <math>{\Bbb C}^n</math>~~
	~~onto an arbitrarily small [[Neighbourhood (mathematics)\|neighbourhood]] of 0.~~

	~~Two dimensional Hopf manifolds are called [[Hopf surface]]s~~.

	~~== Examples ==~~
	~~In a typical situation, <math>\Gamma</math>~~ is ~~generated~~
	~~by a linear contraction, usually a [[diagonal matrix]]~~
	~~<math>q\cdot Id</math>, with <math>q\in {\Bbb C}</math>~~
	~~a complex number, <math>0<\|q\|<1</math>~~. ~~Such manifold~~
	is ~~called ''a classical Hopf manifold''~~.

	~~== Properties ==~~
	~~A Hopf manifold~~ <~~math~~>~~H:=({\Bbb C}^n\backslash 0)/{\Bbb Z}~~<~~/math~~>
	is [~~[diffeomorphic]] to <math>S^{2n-1}\times S^1<~~/~~math>.~~
	~~For <math>n\geq 2<~~/~~math>, it is non-[[Kähler manifold\|Kähler]]~~. ~~Indeed,~~
	~~the first [[cohomology group]] of ''H''~~
	~~is odd-dimensional~~. ~~By [[Hodge decomposition]],~~
	~~odd cohomology of a compact [[Kähler manifold]]~~
	~~are always even-dimensional.~~

	~~== Hypercomplex structure ==~~
	~~Even-dimensional Hopf manifolds admit~~
	~~[[Hypercomplex manifold\|hypercomplex structure]].~~
	~~The Hopf surface is the only compact [[hypercomplex manifold]] of quaternionic dimension 1 which is not [[hyperkähler manifold\|hyperkähler]].~~

	~~== References ==~~
	*{{Citation \| last1=Hopf \| first1=Heinz \| author1-link=Heinz Hopf \| title=Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 \| publisher=Interscience Publishers, Inc., New York \| id={{MathSciNet \| id = 0023054}} \| year=1948 \| chapter=Zur Topologie der komplexen Mannigfaltigkeiten \| pages=167–185}}
	*{{eom\|id=H/~~h110270\|first=L. \|last=Ornea}}~~

	~~[[Category:Complex manifolds]~~]

en>Chawette: Added symbol in example for better understanding of formula

2014-01-24T12:42:29Z

Added symbol in example for better understanding of formula

← Older revision		Revision as of 14:42, 24 January 2014
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	~~Nice to satisfy you~~, ~~my title~~ is ~~Refugia~~. ~~To collect badges~~ is ~~what her family and her appreciate~~. ~~Bookkeeping~~ is ~~her day occupation now~~. ~~California~~ is ~~our beginning place~~.<br><br>~~Feel free~~ to ~~surf to my site :: std testing at home ([http:~~//~~www~~.~~neweracinema~~.~~com~~/~~tube/user/D69E simply click the following webpage~~])		In [[complex geometry]], a '''Hopf manifold''' {{harv\|Hopf\|1948}} is obtained
			as a quotient of the complex [[vector space]]
			(with zero deleted) <math>({\Bbb C}^n\backslash 0)</math>
			by a [[Group action\|free action]] of the [[Group (mathematics)\|group]] <math>\Gamma \cong {\Bbb Z}</math> of
			[[integer]]s, with the generator <math>\gamma</math>
			of <math>\Gamma</math> acting by holomorphic [[Contraction mapping\|contractions]]. Here, a ''holomorphic contraction''
			is a map <math>\gamma:\; {\Bbb C}^n \mapsto {\Bbb C}^n</math>
			such that a sufficiently big iteration <math>\;\gamma^N</math>
			puts any given [[compact subset]] <math>{\Bbb C}^n</math>
			onto an arbitrarily small [[Neighbourhood (mathematics)\|neighbourhood]] of 0.

			Two dimensional Hopf manifolds are called [[Hopf surface]]s.

			== Examples ==
			In a typical situation, <math>\Gamma</math> is generated
			by a linear contraction, usually a [[diagonal matrix]]
			<math>q\cdot Id</math>, with <math>q\in {\Bbb C}</math>
			a complex number, <math>0<\|q\|<1</math>. Such manifold
			is called ''a classical Hopf manifold''.

			== Properties ==
			A Hopf manifold <math>H:=({\Bbb C}^n\backslash 0)/{\Bbb Z}</math>
			is [[diffeomorphic]] to <math>S^{2n-1}\times S^1</math>.
			For <math>n\geq 2</math>, it is non-[[Kähler manifold\|Kähler]]. Indeed,
			the first [[cohomology group]] of ''H''
			is odd-dimensional. By [[Hodge decomposition]],
			odd cohomology of a compact [[Kähler manifold]]
			are always even-dimensional.

			== Hypercomplex structure ==
			Even-dimensional Hopf manifolds admit
			[[Hypercomplex manifold\|hypercomplex structure]].
			The Hopf surface is the only compact [[hypercomplex manifold]] of quaternionic dimension 1 which is not [[hyperkähler manifold\|hyperkähler]].

			== References ==
			*{{Citation \| last1=Hopf \| first1=Heinz \| author1-link=Heinz Hopf \| title=Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, 1948 \| publisher=Interscience Publishers, Inc., New York \| id={{MathSciNet \| id = 0023054}} \| year=1948 \| chapter=Zur Topologie der komplexen Mannigfaltigkeiten \| pages=167–185}}
			*{{eom\|id=H/h110270\|first=L. \|last=Ornea}}

			[[Category:Complex manifolds]]

en>Andrewaskew: Fixed order of categorisation.

2012-07-03T01:10:22Z

Fixed order of categorisation.

New page

Nice to satisfy you, my title is Refugia. To collect badges is what her family and her appreciate. Bookkeeping is her day occupation now. California is our beginning place.<br><br>Feel free to surf to my site :: std testing at home ([http://www.neweracinema.com/tube/user/D69E simply click the following webpage])