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The '''Moffat distribution''', named after the [[physicist]] [[Anthony Moffat]], is a [[continuous probability distribution]] based upon the [[Cauchy distribution|Lorentzian distribution]].  Its particular importance in [[astrophysics]] is due to its ability to accurately reconstruct [[point spread function]]s, whose wings cannot be accurately portrayed by either a [[Gaussian function|Gaussian]] or [[Cauchy distribution|Lorentzian]] function.
This is a preview for the new '''MathML rendering mode''' (with SVG fallback), which is availble in production for registered users.


==Characterisation==
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]]
===Probability density function===
* Only registered users will be able to execute this rendering mode.
The Moffat distribution can be described in two ways. Firstly as the distribution of a bivariate random variable (''X'',''Y'') centred at zero, and secondly as the distribution of the corresponding radii
* Note: you need not enter a email address (nor any other private information). Please do not use a password that you use elsewhere.
:<math>R=\sqrt{X^2+Y^2}.</math>
In terms of the random vector (''X'',''Y''), the distribution has the [[probability density function]] (pdf)
: <math>
f(x,y; \alpha,\beta)=\left(\beta-1\right)\left(\pi\alpha^2\right)^{-1}\left[1+\left(\frac{x^2+y^2}{\alpha^2}\right)\right]^{-\beta} , \,
</math>


where <math>\alpha</math> and <math>\beta</math> are [[astronomical seeing|seeing]] dependent parameters. In this form, the distribution is a reparameterisation of a [[bivariate Student distribution]] with zero correlation.
Registered users will be able to choose between the following three rendering modes:


In terms of the random variable ''R'', the distribution has density
'''MathML'''
: <math>
:<math forcemathmode="mathml">E=mc^2</math>
f(r; \alpha,\beta)=2r \frac{\beta-1}{\alpha^2} \left[1+\left(\frac{r^2}{\alpha^2}\right)\right]^{-\beta} . \,
</math>


===Differential equation===
<!--'''PNG'''  (currently default in production)
The pdf of the Moffat distribution is a solution to the following [[differential equation]]:
:<math forcemathmode="png">E=mc^2</math>
:<math>\left\{\begin{array}{l}
\left(r^3+\alpha ^2 r\right) f'(r)+f(r) \left(-\alpha ^2+2 \beta r^2-r^2\right)=0, \\
f(1)=\frac{2 (\beta -1) \left(\frac{1}{\alpha ^2}+1\right)^{-\beta}}{\alpha ^2}
\end{array}\right\}
</math>


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


* [http://adsabs.harvard.edu/abs/1969A%26A.....3..455M A Theoretical Investigation of Focal Stellar Images in the Photographic Emulsion (1969) &ndash; A. F. J. Moffat]
<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].
{{ProbDistributions|multivariate}}
 
[[Category:Continuous distributions]]
==Demos==
[[Category:Probability distributions]]
 
Here are some [https://commons.wikimedia.org/w/index.php?title=Special:ListFiles/Frederic.wang demos]:
 
 
* 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.
 
==Test pages ==
 
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]]
 
*[[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 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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