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MathGroup Archive 2004

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Re: Infrequent Mathematica User

  • To: mathgroup at smc.vnet.net
  • Subject: [mg47090] Re: Infrequent Mathematica User
  • From: Paul Abbott <paul at physics.uwa.edu.au>
  • Date: Thu, 25 Mar 2004 05:48:21 -0500 (EST)
  • References: <163.2d33e8ae.2d90396b@aol.com> <001f01c41166$1d4ae340$58868218@we1.client2.attbi.com>
  • Sender: owner-wri-mathgroup at wolfram.com

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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