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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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