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Re: Infrequent Mathematica User
*To*: mathgroup at smc.vnet.net
*Subject*: [mg47244] Re: Infrequent Mathematica User
*From*: drbob at bigfoot.com (Bobby R. Treat)
*Date*: Wed, 31 Mar 2004 02:59:52 -0500 (EST)
*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> <c4bdvq$6vn$1@smc.vnet.net>
*Sender*: owner-wri-mathgroup at wolfram.com
OK, as I suspected, there's apparently no standard inequality such as
the Cauchy Sum Inequality that applies (at least, not directly). There
couldn't be, because they're all "tight" for every n and all can be
made strict equalities for some choice of the variables, while neither
is true for the problem inequality, which is never an equality and is
tight only in the limit as n->Infinity.
Meanwhile, the "proof" at the link is a proof only if we blindly trust
Reduce. (I doubt that we always can.) I really liked the
"straightforward" comment before the last use of Reduce. I had no
trouble replacing the first use with an actual proof, but the second
is trickier.
Bobby
Paul Abbott <paul at physics.uwa.edu.au> wrote in message news:<c4bdvq$6vn$1 at smc.vnet.net>...
> In article <c4390g$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
> >
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