Main Page: Difference between revisions

From formulasearchengine
Jump to navigation Jump to search
← Older edit
mNo edit summary
No edit summary
 
(669 intermediate revisions by more than 100 users not shown)
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.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


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 is very counterintuitive, while the proof of the second one is very technical but the result is not at all surprising.
If you would like use the '''MathML''' rendering mode, you need a wikipedia user account that can be registered here [[https://en.wikipedia.org/wiki/Special:UserLogin/signup]]
* Only registered users will be able to execute this rendering mode.
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


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.
Registered users will be able to choose between the following three rendering modes:


==Nash–Kuiper theorem (''C''<sup>1</sup> embedding theorem) {{anchor|Nash–Kuiper theorem}}==
'''MathML'''
'''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
:<math forcemathmode="mathml">E=mc^2</math>
:(i) in class ''C''<sup>1</sup>,
:(ii) isometric: for any two vectors ''v'',''w''&nbsp;&isin;&nbsp;''T<sub>x</sub>''(''M'') in the [[tangent space]] at ''x'' &isin; ''M'',
:::<math>g(v,w)=\langle df_\epsilon(v),df_\epsilon(w)\rangle</math>,
:(iii) &epsilon;-close to ƒ:
:::|ƒ(''x'')&nbsp;&minus;&nbsp;ƒ<sub>&epsilon;</sub>(''x'')|&nbsp;<&nbsp;&epsilon; for all ''x''&nbsp;&isin;&nbsp;''M''.


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.
<!--'''PNG''' (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


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.
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


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)|&epsilon;-ball]] in Euclidean 3-space (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 ≥ &epsilon;<sup>-2</sup>). And, there exist ''C''<sup>1</sup> isometric embeddings of the hyperbolic plane in '''R'''<sup>3</sup>.
<span style="color: red">Follow this [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering link] to change your Math rendering settings.</span> You can also add a [https://en.wikipedia.org/wiki/Special:Preferences#mw-prefsection-rendering-skin Custom CSS] to force the MathML/SVG rendering or select different font families. See [https://www.mediawiki.org/wiki/Extension:Math#CSS_for_the_MathML_with_SVG_fallback_mode these examples].


==''C''<sup>''k''</sup> embedding theorem==
==Demos==
The technical statement 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:
: &lang; ''u'', ''v'' &rang; = dƒ<sub>''p''</sub>(''u'') &middot; dƒ<sub>''p''</sub>(''v'')
for all vectors ''u'', ''v'' in ''T<sub>p</sub>M''. This is an undetermined system of [[partial differential equation]]s (PDEs).


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 iteration]] that uses Newton's method directly (without the introduction of smoothing operators).
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


==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|publisher=The Annals of Mathematics, Vol. 93, No. 1|year=1971|jstor=1970760}}
* {{citation|first=Matthias|last=Günther|title=Zum Einbettungssatz von J. Nash [On the embedding theorem of J. Nash]|
journal=Math. Nachr.|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|publisher=The Annals of Mathematics, Vol. 60, No. 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|publisher=The Annals of Mathematics, Vol. 84, No. 3|jstor=1970448}}.


[[Category:Theorems in Riemannian geometry]]
* accessibility:
** Safari + VoiceOver: [https://commons.wikimedia.org/wiki/File:VoiceOver-Mac-Safari.ogv video only], [[File:Voiceover-mathml-example-1.wav|thumb|Voiceover-mathml-example-1]], [[File:Voiceover-mathml-example-2.wav|thumb|Voiceover-mathml-example-2]], [[File:Voiceover-mathml-example-3.wav|thumb|Voiceover-mathml-example-3]], [[File:Voiceover-mathml-example-4.wav|thumb|Voiceover-mathml-example-4]], [[File:Voiceover-mathml-example-5.wav|thumb|Voiceover-mathml-example-5]], [[File:Voiceover-mathml-example-6.wav|thumb|Voiceover-mathml-example-6]], [[File:Voiceover-mathml-example-7.wav|thumb|Voiceover-mathml-example-7]]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Audio-Windows7-InternetExplorer.ogg Internet Explorer + MathPlayer (audio)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-SynchronizedHighlighting-WIndows7-InternetExplorer.png Internet Explorer + MathPlayer (synchronized highlighting)]
** [https://commons.wikimedia.org/wiki/File:MathPlayer-Braille-Windows7-InternetExplorer.png Internet Explorer + MathPlayer (braille)]
** NVDA+MathPlayer: [[File:Nvda-mathml-example-1.wav|thumb|Nvda-mathml-example-1]], [[File:Nvda-mathml-example-2.wav|thumb|Nvda-mathml-example-2]], [[File:Nvda-mathml-example-3.wav|thumb|Nvda-mathml-example-3]], [[File:Nvda-mathml-example-4.wav|thumb|Nvda-mathml-example-4]], [[File:Nvda-mathml-example-5.wav|thumb|Nvda-mathml-example-5]], [[File:Nvda-mathml-example-6.wav|thumb|Nvda-mathml-example-6]], [[File:Nvda-mathml-example-7.wav|thumb|Nvda-mathml-example-7]].
** Orca: There is ongoing work, but no support at all at the moment [[File:Orca-mathml-example-1.wav|thumb|Orca-mathml-example-1]], [[File:Orca-mathml-example-2.wav|thumb|Orca-mathml-example-2]], [[File:Orca-mathml-example-3.wav|thumb|Orca-mathml-example-3]], [[File:Orca-mathml-example-4.wav|thumb|Orca-mathml-example-4]], [[File:Orca-mathml-example-5.wav|thumb|Orca-mathml-example-5]], [[File:Orca-mathml-example-6.wav|thumb|Orca-mathml-example-6]], [[File:Orca-mathml-example-7.wav|thumb|Orca-mathml-example-7]].
** From our testing, ChromeVox and JAWS are not able to read the formulas generated by the MathML mode.


[[de:Einbettungssatz von Nash]]
==Test pages ==
[[fr:Théorème de plongement de Nash]]
 
[[ru:Теорема Нэша о регулярных вложениях]]
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
[[zh:纳什嵌入定理]]
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]
 
*[[Inputtypes|Inputtypes (private Wikis only)]]
*[[Url2Image|Url2Image (private Wikis only)]]
==Bug reporting==
If you find any bugs, please report them at [https://bugzilla.wikimedia.org/enter_bug.cgi?product=MediaWiki%20extensions&component=Math&version=master&short_desc=Math-preview%20rendering%20problem Bugzilla], or write an email to math_bugs (at) ckurs (dot) de .

Latest revision as of 22:52, 15 September 2019

This is a preview for the new MathML rendering mode (with SVG fallback), which is availble in production for registered users.

If you would like use the MathML rendering mode, you need a wikipedia user account that can be registered here [[1]]

  • Only registered users will be able to execute this rendering mode.
  • Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.

Registered users will be able to choose between the following three rendering modes:

MathML

E=mc2


Follow this link to change your Math rendering settings. You can also add a Custom CSS to force the MathML/SVG rendering or select different font families. See these examples.

Demos

Here are some demos:


Test pages

To test the MathML, PNG, and source rendering modes, please go to one of the following test pages:

Bug reporting

If you find any bugs, please report them at Bugzilla, or write an email to math_bugs (at) ckurs (dot) de .

Retrieved from "https://en.formulasearchengine.com/index.php?title=Main_Page&oldid=329754"