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