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In [[mathematics]], the '''rearrangement inequality'''<ref>{{Citation | last1 = Hardy | first1 = G.H. | authorlink =  G. H. Hardy | last2 = Littlewood | first2 = J.E. | author2-link = John Edensor Littlewood | last3 = Pólya | first3 = G. | author3-link = George Pólya | title = Inequalities | publisher = [[Cambridge University Press]] | series = Cambridge Mathematical Library | edition = 2. | year = 1952 | location = [[Cambridge]] | isbn = 0-521-05206-8 | mr = 0046395 | zbl = 0047.05302}}, Section&nbsp;10.2, Theorem&nbsp;368</ref> states that
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


:<math>x_ny_1 + \cdots + x_1y_n
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]]
\le x_{\sigma (1)}y_1 + \cdots + x_{\sigma (n)}y_n
* Only registered users will be able to execute this rendering mode.
\le x_1y_1 + \cdots + x_ny_n</math>
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.


for every choice of [[real number]]s
Registered users will be able to choose between the following three rendering modes:


:<math>x_1\le\cdots\le x_n\quad\text{and}\quad y_1\le\cdots\le y_n</math>
'''MathML'''
:<math forcemathmode="mathml">E=mc^2</math>


and every [[permutation]]
<!--'''PNG'''  (currently default in production)
:<math forcemathmode="png">E=mc^2</math>


:<math>x_{\sigma(1)},\dots,x_{\sigma(n)}\,</math>
'''source'''
:<math forcemathmode="source">E=mc^2</math> -->


of ''x''<sub>1</sub>, .&nbsp;.&nbsp;., ''x<sub>n</sub>''. If the numbers are different, meaning that
<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].


:<math>x_1<\cdots<x_n\quad\text{and}\quad y_1<\cdots<y_n,</math>
==Demos==


then the lower bound is attained only for the permutation which reverses the order, i.e. σ(''i'')&nbsp;= ''n''&nbsp;&minus;&nbsp;''i''&nbsp;+&nbsp;1 for all ''i''&nbsp;= 1,&nbsp;...,&nbsp;''n'', and the upper bound is attained only for the identity, i.e. σ(''i'')&nbsp;=&nbsp;''i'' for all ''i''&nbsp;= 1,&nbsp;...,&nbsp;''n''.
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:


Note that the rearrangement inequality makes no assumptions on the signs of the real numbers.


==Applications==
* 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.


Many famous inequalities can be proved by the rearrangement inequality, such as the [[inequality of arithmetic and geometric means|arithmetic mean – geometric mean inequality]], the [[Cauchy–Schwarz inequality]], and [[Chebyshev's sum inequality]].
==Test pages ==


==Proof==
To test the '''MathML''', '''PNG''', and '''source''' rendering modes, please go to one of the following test pages:
*[[Displaystyle]]
*[[MathAxisAlignment]]
*[[Styling]]
*[[Linebreaking]]
*[[Unique Ids]]
*[[Help:Formula]]


The lower bound follows by applying the upper bound to
*[[Inputtypes|Inputtypes (private Wikis only)]]
 
*[[Url2Image|Url2Image (private Wikis only)]]
:<math>-x_n\le\cdots\le-x_1.</math>
==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 .
Therefore, it suffices to prove the upper bound. Since there are only finitely many permutations, there exists at least one for which
 
:<math>x_{\sigma (1)}y_1 + \cdots + x_{\sigma (n)}y_n</math>
 
is maximal. In case there are several permutations with this property, let σ denote one with the highest number of [[fixed point (mathematics)|fixed points]].
 
We will now [[reductio ad absurdum|prove by contradiction]], that σ has to be the identity (then we are done). Assume that σ is NOT the identity. Then there exists a ''j'' in {1,&nbsp;...,&nbsp;''n''&nbsp;&minus;&nbsp;1} such that σ(''j'')&nbsp;≠&nbsp;''j'' and σ(''i'')&nbsp;=&nbsp;''i'' for all ''i'' in {1,&nbsp;...,&nbsp;''j''&nbsp;&minus;&nbsp;1}. Hence σ(''j'')&nbsp;>&nbsp;''j'' and there exists a ''k'' in {''j''&nbsp;+&nbsp;1,&nbsp;...,&nbsp;''n''} with σ(''k'')&nbsp;=&nbsp;''j''. Now
 
:<math>j<k\Rightarrow y_j\le y_k
\qquad\text{and}\qquad
j<\sigma(j)\Rightarrow x_j\le x_{\sigma(j)}.\quad(1)</math>
Therefore,
 
:<math>0\le(x_{\sigma(j)}-x_j)(y_k-y_j). \quad(2)</math>
 
Expanding this product and rearranging gives
 
:<math>x_{\sigma(j)}y_j+x_jy_k\le x_jy_j+x_{\sigma(j)}y_k\,, \quad(3)</math>
 
hence the permutation
 
:<math>\tau(i):=\begin{cases}i&\text{for }i\in\{1,\ldots,j\},\\
\sigma(j)&\text{for }i=k,\\
\sigma(i)&\text{for }i\in\{j+1,\ldots,n\}\setminus\{k\},\end{cases}</math>
 
which arises from σ by exchanging the values σ(''j'') and σ(''k''), has at least one additional fixed point compared to σ, namely at ''j'', and also attains the maximum. This contradicts the choice of σ.
 
If
 
:<math>x_1<\cdots<x_n\quad\text{and}\quad y_1<\cdots<y_n,</math>
 
then we have strict inequalities at (1), (2), and (3), hence the maximum can only be attained by the identity, any other permutation σ cannot be optimal.
 
==See also==
* [[Hardy–Littlewood inequality]]
* [[Chebyshev's sum inequality]]
 
==References==
 
<references/>
 
[[Category:Inequalities]]
[[Category:Articles containing proofs]]

Latest revision as of 23:52, 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


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 .

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