[Date Index]
[Thread Index]
[Author Index]
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
Prev by Date:
**NDSolve Repeated convergence test failure**
Next by Date:
**Re: Manipulating the Front End**
Previous by thread:
**Re: Infrequent Mathematica User**
Next by thread:
**Re: Infrequent Mathematica User**
| |