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