Magnet - Revision history

en>Materialscientist: Reverted edits by 176.250.67.26 (talk) to last version by Materialscientist

2015-01-02T03:28:23Z

Reverted edits by 176.250.67.26 (talk) to last version by Materialscientist

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-The '''Nash embedding theorems''' (or '''imbedding theorems'''), named after [[John Forbes Nash]], state that every [[Riemannian manifold]] can be isometrically [[embedding|embedded]] into some [[Euclidean space]].  Isometric means preserving the length of every [[rectifiable path|path]].  For instance, bending without stretching or tearing a page of paper gives an [[isometric embedding]] of the page into Euclidean space because curves drawn on the page retain the same [[arclength]] however the page is bent.
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-The first theorem is for [[continuously differentiable]] (''C''<sup>1</sup>) embeddings and the second for [[analytic function|analytic]] embeddings or embeddings that are [[smooth function|smooth]] of class ''C<sup>k</sup>'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other; the first one has a very simple proof and leads to some very counterintuitive conclusions, while the proof of the second one is very technical but the result is not that surprising.
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-The ''C''<sup>1</sup> theorem was published in 1954, the ''C<sup>k</sup>''-theorem in 1956.  The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by {{harvtxt|Greene|Jacobowitz|1971}}. (A local version of this result was proved by [[Élie Cartan]] and [[Maurice Janet]] in the 1920s.)  In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates.  Nash's proof of the ''C<sup>k</sup>''- case was later extrapolated into the [[h-principle]] and [[Nash–Moser theorem|Nash–Moser implicit function theorem]].  A simplified proof of the second Nash embedding theorem was obtained by {{harvtxt|Günther|1989}} who reduced the set of nonlinear [[partial differential equation]]s to an elliptic system, to which the [[contraction mapping theorem]] could be applied.
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en>Materialscientist: Reverted edits by Ganeshwatve (talk) to last version by Magioladitis

2014-01-18T12:49:58Z

Reverted edits by Ganeshwatve (talk) to last version by Magioladitis

← Older revision		Revision as of 14:49, 18 January 2014
Line 1:		Line 1:
			The '''Nash embedding theorems''' (or '''imbedding theorems'''), named after [[John Forbes Nash]], state that every [[Riemannian manifold]] can be isometrically [[embedding\|embedded]] into some [[Euclidean space]]. Isometric means preserving the length of every [[rectifiable path\|path]]. For instance, bending without stretching or tearing a page of paper gives an [[isometric embedding]] of the page into Euclidean space because curves drawn on the page retain the same [[arclength]] however the page is bent.

			The first theorem is for [[continuously differentiable]] (''C''<sup>1</sup>) embeddings and the second for [[analytic function\|analytic]] embeddings or embeddings that are [[smooth function\|smooth]] of class ''C<sup>k</sup>'', 3 ≤ ''k'' ≤ ∞. These two theorems are very different from each other; the first one has a very simple proof and leads to some very counterintuitive conclusions, while the proof of the second one is very technical but the result is not that surprising.

	~~Modern CCTV for home will allow you to securely access it to remotely check up on your own [http:~~//~~www~~.~~bestwestern~~.~~no/hotellredirect~~.~~aspx?url=cctvdvrreviews~~.~~com cctv avtech dvr software for computer~~] ~~home from anywhere inside~~ the ~~world~~ by ~~having~~ an ~~internet connection~~. ~~An outlet socket for any video recorder was provided~~, ~~although reviewing could~~ be a ~~little tedious if the cameras was set to sequence. Cctv dvr [http~~://~~Www~~.~~goodgreentea.net~~/~~techef1~~/xe/~~?document_srl=241903 software~~] ~~[http~~://~~rebuild.digitalexcellence.net/index~~.~~php?title=The_New_Angle_On_Cctv_Dvr_Review_Just_Released android] However~~, the [~~http:~~/~~/Www~~.~~villaespania~~.~~se/index~~.~~php~~/~~The_Nuiances_Of_Linux_Cctv_Dvr_Software Sling Application~~] for ~~Apple i - Pad~~ is ~~particularly built to set you together [http:~~/~~/deuxrosesdansledesert.trophee~~-~~roses~~-~~des-sables~~.~~com~~/~~out~~.~~php?url~~=~~http~~://~~cctvdvrreviews.com cctv dvr usb] with your TV~~ or ~~DVR along your i~~ - ~~Pad.~~<br><br>~~For males and females~~ that ~~know little or nothing about what~~ a ~~CCTV digicam is, or what it really does. The Swann DVR or Digital Video Recorder provides all~~ from the ~~requirements~~ to ~~defend your own home or business and safeguard your treasures preventing intruders from invading your~~ space.<br><br>~~Writing letters buy one remembered if not together cctv dvr review or telephoning. Then, keep~~ the ~~poster for~~ the ~~wall like a reminder~~, ~~and inspiration~~, of ~~the descriptive words and phrases~~. ~~Mac dvr viewer software for cctv security cameras qv -~~ [http://~~massivecams~~.tv/~~external_link~~/~~?url=http%3A%2F%2Fcctvdvrreviews~~.~~com&domain=cctvdvrreviews~~.~~com massivecams~~.tv] ~~- Second~~, ~~third and fourth paragraph must retain~~ the ~~body~~ in the ~~paper~~ in ~~which it~~ is ~~possible to describe~~ the ~~niche~~, ~~provide proof~~ and ~~arguments~~. ~~Because~~ a ~~ghost writer who reflects your core values~~ is ~~really~~ a ~~ghost writer who looks~~ with ~~the world from a similar perspective~~.		The ''C''<sup>1</sup> theorem was published in 1954, the ''C<sup>k</sup>''-theorem in 1956. The real analytic theorem was first treated by Nash in 1966; his argument was simplified considerably by {{harvtxt\|Greene\|Jacobowitz\|1971}}. (A local version of this result was proved by [[Élie Cartan]] and [[Maurice Janet]] in the 1920s.) In the real analytic case, the smoothing operators (see below) in the Nash inverse function argument can be replaced by Cauchy estimates. Nash's proof of the ''C<sup>k</sup>''- case was later extrapolated into the [[h-principle]] and [[Nash–Moser theorem\|Nash–Moser implicit function theorem]]. A simplified proof of the second Nash embedding theorem was obtained by {{harvtxt\|Günther\|1989}} who reduced the set of nonlinear [[partial differential equation]]s to an elliptic system, to which the [[contraction mapping theorem]] could be applied.

			==Nash–Kuiper theorem (''C''<sup>1</sup> embedding theorem) {{anchor\|Nash–Kuiper theorem}}==
			'''Theorem.''' Let (''M'',''g'') be a Riemannian manifold and ƒ: ''M<sup>m</sup>'' → '''R'''<sup>''n''</sup> a [[short map\|short]] ''C''<sup>∞</sup>-embedding (or [[Immersion (mathematics)\|immersion]]) into Euclidean space '''R'''<sup>''n''</sup>, where ''n'' ≥ ''m''+1. Then for arbitrary ε > 0 there is an embedding (or immersion) ƒ<sub>ε</sub>: ''M<sup>m</sup>'' → '''R'''<sup>''n''</sup> which is
			:(i) in class ''C''<sup>1</sup>,
			:(ii) isometric: for any two vectors ''v'',''w'' ∈ ''T<sub>x</sub>''(''M'') in the [[tangent space]] at ''x'' ∈ ''M'',
			:::<math>g(v,w)=\langle df_\epsilon(v),df_\epsilon(w)\rangle</math>,
			:(iii) ε-close to ƒ:
			:::<math> \| f(x) - f_\epsilon (x) \| < \epsilon ~\forall~ x\in M</math>.

			In particular, as follows from the [[Whitney embedding theorem]], any ''m''-dimensional Riemannian manifold admits an isometric ''C''<sup>1</sup>-embedding into an ''arbitrarily small neighborhood'' in 2''m''-dimensional Euclidean space.

			The theorem was originally proved by John Nash with the condition ''n'' ≥ ''m''+2 instead of ''n'' ≥ ''m''+1 and generalized by [[Nicolaas Kuiper]], by a relatively easy trick.

			The theorem has many counterintuitive implications. For example, it follows that any closed oriented Riemannian surface can be ''C''<sup>1</sup> isometrically embedded into an arbitrarily small [[ball (mathematics)\|ε-ball]] in Euclidean 3-space (for small <math>\epsilon</math> there is no such ''C''<sup>2</sup>-embedding since from the [[Gaussian_curvature#Alternative_Formulas\|formula for the Gauss curvature]] an extremal point of such an embedding would have curvature ≥ ε<sup>-2</sup>). And, there exist ''C''<sup>1</sup> isometric embeddings of the hyperbolic plane in '''R'''<sup>3</sup>.

			==''C''<sup>''k''</sup> embedding theorem==
			The technical statement appearing in Nash's original paper is as follows: if ''M'' is a given ''m''-dimensional Riemannian manifold (analytic or of class ''C<sup>k</sup>'', 3 ≤ ''k'' ≤ ∞), then there exists a number ''n'' (with ''n'' ≤ ''m''(3''m''+11)/2 if ''M'' is a compact manifold, or ''n'' ≤ ''m''(''m''+1)(3''m''+11)/2 if ''M'' is a non-compact manifold) and an [[injective]] map ƒ: ''M'' → '''R'''<sup>''n''</sup> (also analytic or of class ''C<sup>k</sup>'') such that for every point ''p'' of ''M'', the [[derivative]] dƒ<sub>''p''</sub> is a [[linear operator\|linear map]] from the [[tangent space]] ''T<sub>p</sub>M'' to '''R'''<sup>''n''</sup> which is compatible with the given [[inner product space\|inner product]] on ''T<sub>p</sub>M'' and the standard [[scalar product\|dot product]] of '''R'''<sup>''n''</sup> in the following sense:
			: <math>\langle u,v \rangle = df_p(u)\cdot df_p(v)</math>
			for all vectors ''u'', ''v'' in ''T<sub>p</sub>M''. This is an undetermined system of [[partial differential equation]]s (PDEs).

			In a later [http://web.math.princeton.edu/jfnj/texts_and_graphics/Main.Content/Erratum.txt conversation with Robert M. Solovay], Nash mentioned of a fault in the original argument in deriving the sufficing value of the dimension of the embedding space for the case of non-compact manifolds.

			The Nash embedding theorem is a global theorem in the sense that the whole manifold is embedded into '''R'''<sup>''n''</sup>. A local embedding theorem is much simpler and can be proved using the [[implicit function theorem]] of advanced calculus.{{Citation needed\|date=November 2011}} The proof of the global embedding theorem relies on Nash's far-reaching generalization of the implicit function theorem, the [[Nash–Moser theorem]] and Newton's method with postconditioning. The basic idea of Nash's solution of the embedding problem is the use of [[Newton's method]] to prove the existence of a solution to the above system of PDEs. The standard Newton's method fails to converge when applied to the system; Nash uses smoothing operators defined by [[convolution]] to make the Newton iteration converge: this is Newton's method with postconditioning. The fact that this technique furnishes a solution is in itself an [[existence theorem]] and of independent interest. There is also an older method called [[Kantorovich theorem\|Kantorovich iteration]] that uses Newton's method directly (without the introduction of smoothing operators).

			==References==
			* {{citation\|last=Greene\|first=Robert E.\|last2 = Jacobowitz\|first2=Howard\|title= Analytic Isometric Embeddings\|journal=[[Annals of Mathematics]]\|volume=93\|pages=189–204\|doi=10.2307/1970760\|issue=1\|year=1971\|jstor=1970760}}
			* {{citation\|first=Matthias\|last=Günther\|title=Zum Einbettungssatz von J. Nash [On the embedding theorem of J. Nash]\|
			journal=[[Mathematische Nachrichten]]\|volume= 144 \|year=1989\|pages= 165–187\|doi=10.1002/mana.19891440113}}
			*{{citation \| last1 = Han\|first1=Qing\|last2=Hong\|first2=Jia-Xing \| title = Isometric Embedding of Riemannian Manifolds in Euclidean Spaces \| publisher = American Mathematical Society \| year = 2006 \| isbn= 0-8218-4071-1}}
			* {{citation\|first=N.H.\|last=Kuiper\|authorlink=Nicolaas Kuiper\|title=On ''C''<sup>1</sup>-isometric imbeddings I\|journal=Nederl. Akad. Wetensch. Proc. Ser. A.\|volume=58\|year=1955\|pages=545–556}}.
			* {{citation\|first=John\|last=Nash\|authorlink=John Forbes Nash, Jr.\|title=''C''<sup>1</sup>-isometric imbeddings\|journal=[[Annals of Mathematics]]\|volume=60\|year=1954\|pages=383–396\|doi=10.2307/1969840\|issue=3\|jstor=1969840}}.
			* {{citation\|first=John\|last=Nash\|authorlink=John Forbes Nash, Jr.\|title=The imbedding problem for Riemannian manifolds\|journal=[[Annals of Mathematics]]\|volume=63\|year=1956\|pages=20–63\|doi=10.2307/1969989\|issue=1\|mr=0075639\|jstor=1969989}}.
			* {{citation\|first=John\|last=Nash\|title=Analyticity of the solutions of implicit function problem with analytic data\|authorlink=John Forbes Nash, Jr.\|journal=[[Annals of Mathematics]]\|volume=84\|year=1966\|pages=345–355\|doi=10.2307/1970448\|issue=3\|jstor=1970448}}.

			[[Category:Theorems in Riemannian geometry]]

en>Mr. Vernon: Reverted edits by 46.21.62.208 (talk) to last revision by Discuss-Dubious (HG)

2012-08-28T01:07:12Z

Reverted edits by 46.21.62.208 (talk) to last revision by Discuss-Dubious (HG)

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