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MathGroup Archive 2004

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Re: Infrequent Mathematica User

  • To: mathgroup at smc.vnet.net
  • Subject: [mg47207] Re: Infrequent Mathematica User
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Tue, 30 Mar 2004 04:02:07 -0500 (EST)
  • Organization: The University of Western Australia
  • References: <163.2d33e8ae.2d90396b@aol.com> <001f01c41166$1d4ae340$58868218@we1.client2.attbi.com> <c3ueie$9ti$1@smc.vnet.net> <c4390g$en3$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

In article <c4390g$en3$1 at smc.vnet.net>,
 drbob at bigfoot.com (Bobby R. Treat) wrote:

> If you actually see a way to apply one of the standard inequalities to
> this sum, please share. Nothing obvious springs to mind.

For a proof see

  ftp://physics.uwa.edu.au/pub/Mathematica/MathGroup/InequalityProof.nb

Cheers,
Paul


> Paul Abbott <paul at physics.uwa.edu.au> wrote in message 
> news:<c3ueie$9ti$1 at smc.vnet.net>...
> > Jim Dars wrote:
> > 
> > >A Math NG posed the problem:
> > >
> > >  Let x1,x2,...,xn be real numbers. Prove
> > >  x1/(1+x1^2) + x2/(1+x1^2+x2^2) +...+ xn/(1+x1^2+...+xn^2) < sqrt(n)
> > 
> > To prove this, I would do a search for inequalites, e.g,
> > 
> >   http://mathworld.wolfram.com/ChebyshevSumInequality.html
> > 
> > Also, see
> > 
> > Hardy, G. H.; Littlewood, J. E.; and Pólya, G. 
> > Inequalities, 2nd ed. Cambridge, England: 
> > Cambridge University Press, pp. 43-44, 1988.
> > 
> > To investigate using Mathematica, I like to use subscripted variables:
> > 
> >    s[n_] := Sum[Subscript[x,i]/(1 + Sum[Subscript[x,j]^2, {j, i}]), {i, n}]
> > 
> > If you enter this expression into Mathematica and 
> > do Cell | Convert to StandardForm (or 
> > TraditionalForm) you will get a nicely formatted 
> > expression for the n-th left-hand side of the 
> > inequality.
> > 
> > Note that you can prove the inequality on a 
> > case-by-case basis using CylindricalDecomposition 
> > (in Version 5.0):
> > 
> >   CylindricalDecomposition[s[1] > 1, {Subscript[x, 1]}]
> > 
> >   CylindricalDecomposition[s[2] > Sqrt[2], {Subscript[x, 1], Subscript[x, 
> >   2]}]
> > 
> > and so on. This may not seem convincing, but see 
> > what happens if the you change the inequality:
> > 
> > CylindricalDecomposition[s[2] > 1/2, {Subscript[x, 1], Subscript[x, 2]}]
> > 
> > >To get a feel for the problem, and maybe spark 
> > >an idea, I hoped to look at some few early 
> > >maximum values.  However, these proved difficult 
> > >to come by.
> > 
> > NMaximize is the way to go:
> > 
> >   Table[NMaximize[s[n], Table[{Subscript[x, i], -5, 5}, {i, n}]], {n, 6}]
> > 
> > Cheers,
> > Paul
>

-- 
Paul Abbott                                   Phone: +61 8 9380 2734
School of Physics, M013                         Fax: +61 8 9380 1014
The University of Western Australia      (CRICOS Provider No 00126G)         
35 Stirling Highway
Crawley WA 6009                      mailto:paul at physics.uwa.edu.au 
AUSTRALIA                            http://physics.uwa.edu.au/~paul


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