Gyroradius - Revision history

en>Tom.Reding: Gen fixes (page/s, endash, , et al., unicodify, concising wikilinks, etc.), ref cleanup using AWB

2014-12-29T06:02:00Z

Gen fixes (page/s, endash, &nbsp, et al., unicodify, concising wikilinks, etc.), ref cleanup using AWB

← Older revision		Revision as of 08:02, 29 December 2014
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	~~Hi there, I am~~ Alyson ~~Pomerleau~~ and I believe it ~~sounds~~ fairly ~~great~~ when you say it. ~~To perform lacross is the thing I love most of all. He is an information officer~~. Ohio is where her home is.<br><br>my web ~~page good psychic,~~ [http://~~mybrandcp~~.~~com~~/~~xe/board_XmDx25/107997 mybrandcp.com~~],		Alyson is the title individuals use to contact me and I believe it seems fairly good when you say it. Distributing manufacturing has been his occupation for some time. Ohio is where her home is. What me and my family love is bungee leaping but I've been taking on new things recently.<br><br>my web blog [http://kard.dk/?p=24252 psychic readers]

en>Balljust: Added gyrokinetics see also

2014-02-24T17:34:25Z

Added gyrokinetics see also

← Older revision		Revision as of 19:34, 24 February 2014
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	~~In [[mathematics]]~~, specifically in [[symplectic geometry]], the '''symplectic cut''' is a geometric modification on [[symplectic manifold]]s. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the [[symplectic sum]], that glues two manifolds together into one. The symplectic cut can also be viewed as a generalization of symplectic [[Blowing up\|blow up]]. The cut was introduced in 1995 by Eugene Lerman, who used it ~~to study the [[moment map\|symplectic quotient]] and other operations on manifolds.~~		Hi there, I am Alyson Pomerleau and I believe it sounds fairly great when you say it. To perform lacross is the thing I love most of all. He is an information officer. Ohio is where her home is.<br><br>my web page good psychic, [http://mybrandcp.com/xe/board_XmDx25/107997 mybrandcp.com],

	~~== Topological description ==~~

	~~Let <math>(X, \omega)</math> be any symplectic manifold and~~

	~~:<math>\mu : X \to \mathbb{R}</math>~~

	a [[Hamiltonian vector field\|Hamiltonian]] on <math>X</math>. Let <math>\epsilon</math> be any regular value of <math>\mu</math>, so that the level set <math>\mu^{-1}(\epsilon)</math> is a smooth manifold. Assume furthermore that <math>\mu^{-1}(\epsilon)</math> is fibered in circles, each of which is an integral curve of the induced [[Hamiltonian vector field]].

	~~Under these assumptions, <math>\mu^{-1}([\epsilon, \infty))</math> is a manifold with boundary <math>\mu^{-1}(\epsilon)</math>, and one can form a manifold~~

	~~:<math>\overline{X}_{\mu \geq \epsilon}</math>~~

	by collapsing each circle fiber to a point. In other words, <math>\overline{X}_{\mu \geq \epsilon}</math> is <math>X</math> with the subset <math>\mu^{-1}((-\infty, \epsilon))</math> removed and the boundary collapsed along each circle fiber. The quotient of the boundary is a submanifold of <math>\overline{X}_{\mu \geq \epsilon}</math> of [[codimension]] two, denoted <math>V</math>.

	Similarly, one may form from <math>\mu^{-1}((-\infty, \epsilon])</math> a manifold <math>\overline{X}_{\mu \leq \epsilon}</math>, which also contains a copy of <math>V</math>. The '''symplectic cut''' is the pair of manifolds <math>\overline{X}_{\mu \leq \epsilon}</math> and <math>\overline{X}_{\mu \geq \epsilon}</math>.

	~~Sometimes~~ it ~~is useful to view the two halves of the symplectic cut as being joined along their shared submanifold <math>V</math> to produce a singular space~~

	~~:<math>\overline{X}_{\mu \leq \epsilon} \cup_V \overline{X}_{\mu \geq \epsilon}~~.~~</math>~~

	~~For example, this singular space~~ is the ~~central fiber in the symplectic sum regarded as a deformation.~~

	~~== Symplectic description ==~~

	~~The preceding description is rather crude; more care is required to keep track~~ of the symplectic structure on the symplectic cut. For this, let <math>(X, \omega)</math> be any symplectic manifold. Assume that the [[circle group]] <math>U(1)</math> [[group action\|acts]] on <math>X</math> in a [[moment map\|Hamiltonian]] way with [[moment map]]

	~~:<math>\mu : X \to \mathbb{R}.</math>~~

	This moment map can be viewed as a Hamiltonian function that generates the circle action. The product space <math>X \times \mathbb{C}</math>, with coordinate <math>z</math> on <math>\mathbb{C}</math>, comes with an induced symplectic form

	~~:<math>\omega \oplus (-i dz \wedge d\bar{z}).</math>~~

	~~The group <math>U(1)</math> acts on the product in a Hamiltonian way by~~

	~~:<math>e^{i\theta} \cdot (x, z) = (e^{i \theta} \cdot x, e^{-i \theta} z)</math>~~

	~~with moment map~~

	~~:<math>\nu(x, z) = \mu(x) - \|z\|^2.</math>~~

	Let <math>\epsilon</math> be any real number such that the circle action is free on <math>\mu^{-1}(\epsilon)</math>. Then <math>\epsilon</math> is a regular value of <math>\nu</math>, and <math>\nu^{-1}(\epsilon)</math> is a manifold.

	This manifold <math>\nu^{-1}(\epsilon)</math> contains as a submanifold the set of points <math>(x, z)</math> with <math>\mu(x) = \epsilon</math> and <math>\|z\|^2 = 0</math>; this submanifold is naturally identified with <math>\mu^{-1}(\epsilon)</math>. The complement of the submanifold, which consists of points <math>(x, z)</math> with <math>\mu(x) > \epsilon</math>, is naturally identified with the product of

	~~:<math>X_{> \epsilon} := \mu^{-1}((\epsilon, \infty))</math>~~

	~~and the circle.~~

	~~The manifold <math>\nu^{-1}(\epsilon)</math> inherits the Hamiltonian circle action, as do its two submanifolds just described. So one may form the symplectic quotient~~

	~~:<math>\overline{X}_{\mu \geq \epsilon} := \nu^{-1}(\epsilon) / U(1)~~.~~</math>~~

	~~By construction, it contains <math>X_{\mu > \epsilon}</math> as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient~~

	~~:<math>V := \mu^{-1}(\epsilon) / U(1),</math>~~

	~~which is a symplectic submanifold of <math>\overline{X}_{\mu \geq \epsilon}</math> of codimension two.~~

	~~If <math>X</math> is [[Kähler manifold\|Kähler]], then so is the cut space <math>\overline{X}_{\mu \geq \epsilon}</math>; however, the embedding of <math>X_{\mu > \epsilon}</math>~~ is ~~not~~ an ~~isometry.~~

	One constructs <math>\overline{X}_{\mu \leq \epsilon}</math>, the other half of the symplectic cut, in a symmetric manner. The [[normal bundle]]s of <math>V</math> in the two halves of the cut are opposite each other (meaning symplectically anti-isomorphic). The symplectic sum of <math>\overline{X}_{\mu \geq \epsilon}</math> and <math>\overline{X}_{\mu \leq \epsilon}</math> along <math>V</math> recovers <math>X</math>.

	The existence of a global Hamiltonian circle action on <math>X</math> appears to be a restrictive assumption. However, it is not actually necessary; the cut can be performed under more general hypotheses, such as a local Hamiltonian circle action near <math>\mu^{-1}(\epsilon)</math> (since the cut is a local operation).

	~~== Blow up as cut ==~~

	~~When a [[complex manifold]] <math>X</math>~~ is ~~blown up along a submanifold <math>Z</math>, the blow up [[locus (mathematics)\|locus]] <math>Z</math>~~ is ~~replaced by an [[exceptional divisor]] <math>E</math> and the rest of the manifold is left undisturbed~~. ~~Topologically, this operation may also be viewed as the removal of an~~ <~~math~~>~~\epsilon~~<~~/math~~>~~-neighborhood of the blow up locus~~, ~~followed by the collapse of the boundary by the~~ [~~[Hopf fibration\|Hopf map]].~~

	Blowing up a symplectic manifold is more subtle, since the symplectic form must be adjusted in a neighborhood of the blow up locus in order to continue smoothly across the exceptional divisor in the blow up. The symplectic cut is an elegant means of making the neighborhood-deletion/boundary-collapse process symplectically rigorous.

	As before, let <math>(X, \omega)</math> be a symplectic manifold with a Hamiltonian <math>U(1)</math>-action with moment map <math>\mu</math>. Assume that the moment map is proper and that it achieves its maximum <math>m</math> exactly along a symplectic submanifold <math>Z</math> of <math>X</math>. Assume furthermore that the weights of the isotropy representation of <math>U(1)</math> on the normal bundle <math>N_X Z</~~math> are all <math>1</math>.~~

	~~Then for small <math>\epsilon</math> the only critical points in <math>X_{\mu > m - \epsilon}</math> are those on <math>Z<~~/~~math>~~. ~~The symplectic cut <math>\overline{X}_{\mu \leq m - \epsilon}<~~/~~math>, which is formed by deleting a symplectic <math>\epsilon<~~/~~math>-neighborhood of <math>Z<~~/~~math> and collapsing the boundary, is then the symplectic blow up of <math>X</math> along <math>Z</math>~~.

	~~== References ==~~

	* Eugene Lerman: Symplectic cuts, ''Mathematical Research Letters'' 2 (1995), 247–258
	* [[Dusa McDuff]~~] and D. Salamon: ''Introduction to Symplectic Topology'' (1998) Oxford Mathematical Monographs~~, ~~ISBN 0-19-850451-9.~~

	~~[[Category:Symplectic topology]]~~

141.30.128.137 at 14:55, 13 November 2013

2013-11-13T14:55:51Z

← Older revision		Revision as of 16:55, 13 November 2013
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	~~Greetings~~. The ~~author's title~~ is ~~Phebe~~ and ~~she feels comfortable when people use~~ the ~~complete title~~. ~~She~~ is a ~~librarian but she~~'~~s usually wanted her own business~~. ~~To play baseball~~ is the ~~hobby he will never quit performing~~. ~~Years~~ in ~~over~~ the ~~counter std test~~ ([~~http~~://~~Wixothek~~.~~com~~/~~user~~/~~MBuckmast click~~ to ~~find out~~ more]) ~~past he moved~~ to ~~North Dakota~~ and ~~his family loves~~ it.		In [[mathematics]], specifically in [[symplectic geometry]], the '''symplectic cut''' is a geometric modification on [[symplectic manifold]]s. Its effect is to decompose a given manifold into two pieces. There is an inverse operation, the [[symplectic sum]], that glues two manifolds together into one. The symplectic cut can also be viewed as a generalization of symplectic [[Blowing up\|blow up]]. The cut was introduced in 1995 by Eugene Lerman, who used it to study the [[moment map\|symplectic quotient]] and other operations on manifolds.

			== Topological description ==

			Let <math>(X, \omega)</math> be any symplectic manifold and

			:<math>\mu : X \to \mathbb{R}</math>

			a [[Hamiltonian vector field\|Hamiltonian]] on <math>X</math>. Let <math>\epsilon</math> be any regular value of <math>\mu</math>, so that the level set <math>\mu^{-1}(\epsilon)</math> is a smooth manifold. Assume furthermore that <math>\mu^{-1}(\epsilon)</math> is fibered in circles, each of which is an integral curve of the induced [[Hamiltonian vector field]].

			Under these assumptions, <math>\mu^{-1}([\epsilon, \infty))</math> is a manifold with boundary <math>\mu^{-1}(\epsilon)</math>, and one can form a manifold

			:<math>\overline{X}_{\mu \geq \epsilon}</math>

			by collapsing each circle fiber to a point. In other words, <math>\overline{X}_{\mu \geq \epsilon}</math> is <math>X</math> with the subset <math>\mu^{-1}((-\infty, \epsilon))</math> removed and the boundary collapsed along each circle fiber. The quotient of the boundary is a submanifold of <math>\overline{X}_{\mu \geq \epsilon}</math> of [[codimension]] two, denoted <math>V</math>.

			Similarly, one may form from <math>\mu^{-1}((-\infty, \epsilon])</math> a manifold <math>\overline{X}_{\mu \leq \epsilon}</math>, which also contains a copy of <math>V</math>. The '''symplectic cut''' is the pair of manifolds <math>\overline{X}_{\mu \leq \epsilon}</math> and <math>\overline{X}_{\mu \geq \epsilon}</math>.

			Sometimes it is useful to view the two halves of the symplectic cut as being joined along their shared submanifold <math>V</math> to produce a singular space

			:<math>\overline{X}_{\mu \leq \epsilon} \cup_V \overline{X}_{\mu \geq \epsilon}.</math>

			For example, this singular space is the central fiber in the symplectic sum regarded as a deformation.

			== Symplectic description ==

			The preceding description is rather crude; more care is required to keep track of the symplectic structure on the symplectic cut. For this, let <math>(X, \omega)</math> be any symplectic manifold. Assume that the [[circle group]] <math>U(1)</math> [[group action\|acts]] on <math>X</math> in a [[moment map\|Hamiltonian]] way with [[moment map]]

			:<math>\mu : X \to \mathbb{R}.</math>

			This moment map can be viewed as a Hamiltonian function that generates the circle action. The product space <math>X \times \mathbb{C}</math>, with coordinate <math>z</math> on <math>\mathbb{C}</math>, comes with an induced symplectic form

			:<math>\omega \oplus (-i dz \wedge d\bar{z}).</math>

			The group <math>U(1)</math> acts on the product in a Hamiltonian way by

			:<math>e^{i\theta} \cdot (x, z) = (e^{i \theta} \cdot x, e^{-i \theta} z)</math>

			with moment map

			:<math>\nu(x, z) = \mu(x) - \|z\|^2.</math>

			Let <math>\epsilon</math> be any real number such that the circle action is free on <math>\mu^{-1}(\epsilon)</math>. Then <math>\epsilon</math> is a regular value of <math>\nu</math>, and <math>\nu^{-1}(\epsilon)</math> is a manifold.

			This manifold <math>\nu^{-1}(\epsilon)</math> contains as a submanifold the set of points <math>(x, z)</math> with <math>\mu(x) = \epsilon</math> and <math>\|z\|^2 = 0</math>; this submanifold is naturally identified with <math>\mu^{-1}(\epsilon)</math>. The complement of the submanifold, which consists of points <math>(x, z)</math> with <math>\mu(x) > \epsilon</math>, is naturally identified with the product of

			:<math>X_{> \epsilon} := \mu^{-1}((\epsilon, \infty))</math>

			and the circle.

			The manifold <math>\nu^{-1}(\epsilon)</math> inherits the Hamiltonian circle action, as do its two submanifolds just described. So one may form the symplectic quotient

			:<math>\overline{X}_{\mu \geq \epsilon} := \nu^{-1}(\epsilon) / U(1).</math>

			By construction, it contains <math>X_{\mu > \epsilon}</math> as a dense open submanifold; essentially, it compactifies this open manifold with the symplectic quotient

			:<math>V := \mu^{-1}(\epsilon) / U(1),</math>

			which is a symplectic submanifold of <math>\overline{X}_{\mu \geq \epsilon}</math> of codimension two.

			If <math>X</math> is [[Kähler manifold\|Kähler]], then so is the cut space <math>\overline{X}_{\mu \geq \epsilon}</math>; however, the embedding of <math>X_{\mu > \epsilon}</math> is not an isometry.

			One constructs <math>\overline{X}_{\mu \leq \epsilon}</math>, the other half of the symplectic cut, in a symmetric manner. The [[normal bundle]]s of <math>V</math> in the two halves of the cut are opposite each other (meaning symplectically anti-isomorphic). The symplectic sum of <math>\overline{X}_{\mu \geq \epsilon}</math> and <math>\overline{X}_{\mu \leq \epsilon}</math> along <math>V</math> recovers <math>X</math>.

			The existence of a global Hamiltonian circle action on <math>X</math> appears to be a restrictive assumption. However, it is not actually necessary; the cut can be performed under more general hypotheses, such as a local Hamiltonian circle action near <math>\mu^{-1}(\epsilon)</math> (since the cut is a local operation).

			== Blow up as cut ==

			When a [[complex manifold]] <math>X</math> is blown up along a submanifold <math>Z</math>, the blow up [[locus (mathematics)\|locus]] <math>Z</math> is replaced by an [[exceptional divisor]] <math>E</math> and the rest of the manifold is left undisturbed. Topologically, this operation may also be viewed as the removal of an <math>\epsilon</math>-neighborhood of the blow up locus, followed by the collapse of the boundary by the [[Hopf fibration\|Hopf map]].

			Blowing up a symplectic manifold is more subtle, since the symplectic form must be adjusted in a neighborhood of the blow up locus in order to continue smoothly across the exceptional divisor in the blow up. The symplectic cut is an elegant means of making the neighborhood-deletion/boundary-collapse process symplectically rigorous.

			As before, let <math>(X, \omega)</math> be a symplectic manifold with a Hamiltonian <math>U(1)</math>-action with moment map <math>\mu</math>. Assume that the moment map is proper and that it achieves its maximum <math>m</math> exactly along a symplectic submanifold <math>Z</math> of <math>X</math>. Assume furthermore that the weights of the isotropy representation of <math>U(1)</math> on the normal bundle <math>N_X Z</math> are all <math>1</math>.

			Then for small <math>\epsilon</math> the only critical points in <math>X_{\mu > m - \epsilon}</math> are those on <math>Z</math>. The symplectic cut <math>\overline{X}_{\mu \leq m - \epsilon}</math>, which is formed by deleting a symplectic <math>\epsilon</math>-neighborhood of <math>Z</math> and collapsing the boundary, is then the symplectic blow up of <math>X</math> along <math>Z</math>.

			== References ==

			* Eugene Lerman: Symplectic cuts, ''Mathematical Research Letters'' 2 (1995), 247–258
			* [[Dusa McDuff]] and D. Salamon: ''Introduction to Symplectic Topology'' (1998) Oxford Mathematical Monographs, ISBN 0-19-850451-9.

			[[Category:Symplectic topology]]

en>BR84: recategorizing

2012-04-10T13:24:24Z

recategorizing

New page

Greetings. The author's title is Phebe and she feels comfortable when people use the complete title. She is a librarian but she's usually wanted her own business. To play baseball is the hobby he will never quit performing. Years in over the counter std test ([http://Wixothek.com/user/MBuckmast click to find out more]) past he moved to North Dakota and his family loves it.