Re: [MLUG - DISCUSSION] the Rearrangement Inequality

["Christian M. Cepel" <EMAIL:PROTECTED>]

I know this is a stupid question, but I don't really know the 'why' of 
it being stupid.  The question is, "exactly how does one do mathematical 
research?"

I'm probably over simplifying but how does one research something that 
seems to me to be pretty static.  What are one's goals when doing 
research.  Are you coming up with new and clever ways to see established 
concepts, or actually discovering new concepts, or finding applications 
for existing concepts?

I have this weird image of my head of someone sitting in a chair for 8 
hours a day with a TI-86 in their hand while they ponder and tap their 
head with the eraser on the end of a pencil.   I suppose I can 
substitute Mathematic for the TI-86, but I still have the same image.

Before you jump on me, I barely made it through pre-calc, and only then 
because my wife was a VERY patient tutor.  I guess I've never really 
been shown a broader world than just doing some fundamental calc and 
trig applications.

I've seen Good Will Hunting and A Beautiful Mind (yes, I know.. that was 
economics), but I don't really have a clear idea of motivations and 
goals and such.


Stephen Montgomery-Smith wrote:
> Mike Miller wrote:
> > 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
>
> Both these inequalities are the bread and butter of my research.  They 
> are both great inequalities, but I use them so much that they have 
> become almost like 1+1=2 to me.  I have been interested in extensions 
> of Jensen's inequality.  The rearrangement inequality is fundamental 
> to the study of so called rearrangement invariant spaces, about which 
> I have many papers.
>
> Stephen
>
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