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MathGroup Archive 2004

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Re: Infrequent Mathematica User

  • To: mathgroup at smc.vnet.net
  • Subject: [mg47149] Re: Infrequent Mathematica User
  • From: drbob at bigfoot.com (Bobby R. Treat)
  • Date: Sat, 27 Mar 2004 01:35:17 -0500 (EST)
  • Organization: http://groups.google.com
  • References: <163.2d33e8ae.2d90396b@aol.com> <001f01c41166$1d4ae340$58868218@we1.client2.attbi.com> <c3ueie$9ti$1@smc.vnet.net>
  • Sender: owner-wri-mathgroup at wolfram.com

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

Bobby

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


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