Bitcrusher - Revision history

76.76.228.219 at 06:56, 4 July 2014

2014-07-04T06:56:57Z

← Older revision		Revision as of 08:56, 4 July 2014
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	~~In [[Riemannian geometry]]~~, ~~[[Mikhail Gromov (mathematician)\|Gromov]]~~'~~s optimal stable 2-[[systolic geometry\|systolic]] inequality is the inequality~~		I would like to introduce myself to you, I am Andrew and my spouse doesn't like it at all. For many years she's been living in Kentucky but her spouse wants them to move. Doing ballet is some thing she would by no means give up. Distributing production has been his occupation for some time.<br><br>My web site :: clairvoyants ([http://Galab-Work.Cs.Pusan.Ac.kr/Sol09B/?document_srl=1489804 Suggested Web site])

	~~: <math>\mathrm{stsys}_2{}^n \leq n!~~
	~~\;\mathrm{vol}_{2n}(\mathbb{CP}^n)</math>,~~

	~~valid for an arbitrary Riemannian metric on the [[complex projective space]], where the optimal bound is attained~~
	by the symmetric [[Fubini-Study metric]], providing a natural geometrisation of [[quantum mechanics]]. Here <math>\operatorname{stsys_2}</math> is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line <math>\mathbb{CP}^1 \subset \mathbb{CP}^n</math> in 2-dimensional homology.

	~~The inequality first appeared in Gromov~~'s ~~1981 book entitled ''Structures métriques pour les variétés riemanniennes'' (Theorem 4.36).~~

	~~The proof of Gromov's inequality relies on the [[Wirtinger inequality (2-forms)\|Wirtinger inequality for exterior 2-forms]]~~.

	~~==Projective planes over division algebras <math> \mathbb{R,C,H}</math>==~~

	In the special case n=2, Gromov's inequality becomes <math>\mathrm{stsys}_2{}^2 \leq 2 \mathrm{vol}_4(\mathbb{CP}^2)</math>. This inequality can be thought of as an analog of [[Pu's inequality\|Pu's inequality for the real projective plane]] <math>\mathbb{RP}^2</math>. In both cases, the boundary case of equality is ~~attained~~ by ~~the symmetric metric of the projective plane~~. ~~Meanwhile, in the quaternionic case, the symmetric metric on <math>\mathbb{HP}^2</math> is not its systolically optimal metric~~. ~~In other words, the manifold~~ <~~math~~>~~\mathbb{HP}^2~~<~~/math~~> ~~admits Riemannian metrics with higher systolic ratio <math>\mathrm{stsys}_4{}^2/\mathrm{vol}_8</math> than for its symmetric metric, see Bangert et al. (2009).~~

	~~==See also==~~
	*[[Loewner's torus inequality]]
	*[[Pu's inequality]]
	*[[Gromov's inequality]]
	*[[Gromov's systolic inequality for essential manifolds]]
	*[[Systolic geometry]]

	~~==References==~~
	*[[Victor Bangert\|Bangert, V]]; [[Mikhail Katz\|Katz, M.]]; [[Steve Shnider\|Shnider, S.]]; [[Shmuel Weinberger\|Weinberger, S.]]: ~~[[E7~~ (~~mathematics)\|E_7]], [~~[~~Wirtinger inequality\|Wirtinger inequalities]], Cayley 4-form, and homotopy. [[Duke Mathematical Journal]] 146 (2009), no. 1, 35-70. See arXiv~~:~~math.DG~~/~~0608006~~
	*Gromov, M.: Structures métriques pour les variétés riemanniennes. Edited by J. Lafontaine and P. Pansu. Textes Mathématiques, 1. CEDIC, Paris, 1981 (first edition of [[Metric Structures for Riemannian and Non-~~Riemannian Spaces]])~~.
	*{{Citation \| last1=[[Mikhail Katz\|Katz]] \| first1=Mikhail G. ~~\| title=Systolic geometry and topology\|pages=19 \| publisher=[[American Mathematical Society]] \| location=Providence, R~~.I. ~~\| series~~=~~Mathematical Surveys and Monographs \| isbn=978-0-8218-4177-8 \| year=2007 \| volume=137}}~~

	~~{{Systolic geometry navbox}}~~

	~~[[Category:Geometric inequalities]]~~
	~~[[Category:Differential geometry]]~~
	~~[[Category:Riemannian geometry]]~~
	~~[[Category:Systolic geometry]~~]

58.8.101.134: /* Examples */ that's likely downsampling not bitcrush

2013-12-26T07:31:10Z

Examples: that's likely downsampling not bitcrush

← Older revision		Revision as of 09:31, 26 December 2013
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	~~Hi there~~, ~~I am Alyson Boon although it~~ is ~~not~~ the ~~title~~ on ~~my beginning certificate~~. It's ~~not a typical thing but what I like performing is to climb but I don~~'~~t have~~ the ~~time lately~~. ~~Distributing production~~ is ~~how he tends to make a living~~. ~~Kentucky~~ is ~~exactly where I've always been living~~.<br><br>~~my homepage :~~: [~~http~~:/~~/www~~.~~sirudang~~.~~com/siroo_Notice/2110 phone psychic~~]		In [[Riemannian geometry]], [[Mikhail Gromov (mathematician)\|Gromov]]'s optimal stable 2-[[systolic geometry\|systolic]] inequality is the inequality

			: <math>\mathrm{stsys}_2{}^n \leq n!
			\;\mathrm{vol}_{2n}(\mathbb{CP}^n)</math>,

			valid for an arbitrary Riemannian metric on the [[complex projective space]], where the optimal bound is attained
			by the symmetric [[Fubini-Study metric]], providing a natural geometrisation of [[quantum mechanics]]. Here <math>\operatorname{stsys_2}</math> is the stable 2-systole, which in this case can be defined as the infimum of the areas of rational 2-cycles representing the class of the complex projective line <math>\mathbb{CP}^1 \subset \mathbb{CP}^n</math> in 2-dimensional homology.

			The inequality first appeared in Gromov's 1981 book entitled ''Structures métriques pour les variétés riemanniennes'' (Theorem 4.36).

			The proof of Gromov's inequality relies on the [[Wirtinger inequality (2-forms)\|Wirtinger inequality for exterior 2-forms]].

			==Projective planes over division algebras <math> \mathbb{R,C,H}</math>==

			In the special case n=2, Gromov's inequality becomes <math>\mathrm{stsys}_2{}^2 \leq 2 \mathrm{vol}_4(\mathbb{CP}^2)</math>. This inequality can be thought of as an analog of [[Pu's inequality\|Pu's inequality for the real projective plane]] <math>\mathbb{RP}^2</math>. In both cases, the boundary case of equality is attained by the symmetric metric of the projective plane. Meanwhile, in the quaternionic case, the symmetric metric on <math>\mathbb{HP}^2</math> is not its systolically optimal metric. In other words, the manifold <math>\mathbb{HP}^2</math> admits Riemannian metrics with higher systolic ratio <math>\mathrm{stsys}_4{}^2/\mathrm{vol}_8</math> than for its symmetric metric, see Bangert et al. (2009).

			==See also==
			*[[Loewner's torus inequality]]
			*[[Pu's inequality]]
			*[[Gromov's inequality]]
			*[[Gromov's systolic inequality for essential manifolds]]
			*[[Systolic geometry]]

			==References==
			*[[Victor Bangert\|Bangert, V]]; [[Mikhail Katz\|Katz, M.]]; [[Steve Shnider\|Shnider, S.]]; [[Shmuel Weinberger\|Weinberger, S.]]: [[E7 (mathematics)\|E_7]], [[Wirtinger inequality\|Wirtinger inequalities]], Cayley 4-form, and homotopy. [[Duke Mathematical Journal]] 146 (2009), no. 1, 35-70. See arXiv:math.DG/0608006
			*Gromov, M.: Structures métriques pour les variétés riemanniennes. Edited by J. Lafontaine and P. Pansu. Textes Mathématiques, 1. CEDIC, Paris, 1981 (first edition of [[Metric Structures for Riemannian and Non-Riemannian Spaces]]).
			*{{Citation \| last1=[[Mikhail Katz\|Katz]] \| first1=Mikhail G. \| title=Systolic geometry and topology\|pages=19 \| publisher=[[American Mathematical Society]] \| location=Providence, R.I. \| series=Mathematical Surveys and Monographs \| isbn=978-0-8218-4177-8 \| year=2007 \| volume=137}}

			{{Systolic geometry navbox}}

			[[Category:Geometric inequalities]]
			[[Category:Differential geometry]]
			[[Category:Riemannian geometry]]
			[[Category:Systolic geometry]]

24.154.94.117 at 03:08, 5 March 2012

2012-03-05T03:08:24Z

New page

Hi there, I am Alyson Boon although it is not the title on my beginning certificate. It's not a typical thing but what I like performing is to climb but I don't have the time lately. Distributing production is how he tends to make a living. Kentucky is exactly where I've always been living.<br><br>my homepage :: [http://www.sirudang.com/siroo_Notice/2110 phone psychic]