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