Re: [MLUG - DISCUSSION] the Rearrangement Inequality

[Mike Miller <EMAIL:PROTECTED>]

On Tue, 1 Aug 2006, Jonathan King wrote:

> I just felt a need to point out that the Rearrangement Inequality is the 
> most beautiful mathematical idea I have blundered across in several 
> months:
>
> http://en.wikipedia.org/wiki/Rearrangement_inequality
>
> So maybe I'm a shallow person, or maybe I just don't get to look at as 
> much math as I should, but this seems like a very pretty idea to me.


That's pretty fascinating because it is so simple and elegant.  I am 
surprised that I didn't know about it.  This web page does a nice job of 
explaining it:

http://www.artofproblemsolving.com/Wiki/index.php/Rearrangement_Inequality

Wikipedia mentions that the inequality of arithmetic and geometric means 
can be proved from the rearrangement inequality.  Strangely, I was shown a 
proof of this by a professor in population genetics that used Jensen's 
inequality instead.  Jensen's inequality is also pretty cool:

http://en.wikipedia.org/wiki/Jensen%27s_inequality

I think he said it was the nicest proof he knew.  There was probably more 
to that story but I can't remember -- like what does this have to do with 
genetics?!  I'll look at my old notes someday and tell you.  It was about 
19 years ago that I took that course, but I saved the notes.

Mike

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