Re: Solve or Reduce on a monstrosity of an expresssion (and a prize!)

*To*: mathgroup at smc.vnet.net*Subject*: [mg57294] Re: [mg57278] Solve or Reduce on a monstrosity of an expresssion (and a prize!)*From*: Andrzej Kozlowski <andrzej at akikoz.net>*Date*: Tue, 24 May 2005 05:12:31 -0400 (EDT)*References*: <200505230620.CAA04045@smc.vnet.net>*Sender*: owner-wri-mathgroup at wolfram.com

On 23 May 2005, at 15:20, Daniel Reeves wrote: > > Mathemahomies, > I have a beast of a function (though continuously differentiable) > that I > need to prove is strictly decreasing in a certain range (which I > *know* it > is just from plotting it). Every combination I can think of of > Reduce and > Solve and Simplify with assumptions leaves Mathematica spinning its > wheels > indefinitely. > > Do you have any ideas for cajoling Mathematica into crunching through > this? > > Here's the function: > > f[x_,n_] := 9/2/c[x,n]^2*(n+1)b[x,n]^2 (x-d[x,n])(x-x*d[x,n]+d[x,n]+ > d[x,n]^2+n (d[x,n]-1) (x+d[x,n])) > > where > > a[x_,n_] := 9*(n+1)^2 + Sqrt[3(n+1)^3 (x^2 (n-1) + 27(n+1))]; > > b[x_,n_] := (a[x,n](n-1) x^2)^(1/3); > > c[x_,n_] := -3^(2/3) x^2 (n^2-1) + 3^(1/3)(x^2(n^2-1) (9 + 9n + > Sqrt[3(n+1) (x^2(n-1) + 27(n+1))]))^(2/3); > > d[x_,n_] := c[x,n] / (3 b[x,n] (n+1)); > > > Show that f[x,n] is strictly decreasing for x in (0,(n-1)/n) for all > integers n >= 2. > > Note that the limit of f[x,n] as x->0 is (n-1)/(2(n+1)) > 0 > and f[(n-1)/n,n] == 0. So it would suffice to show that f' has no > roots > in (0,(n-1)/n). > > > PS: I have a cool prize for information leading to a solution! > (whether or > not it actually involves Mathematica) > > -- > http://ai.eecs.umich.edu/people/dreeves - - google://"Daniel Reeves" > > Sowmya: Is this guy a mathematician? > Terence: Worse, an economist. At least mathematicians are honest > about > their disdain for the real world. > > Unfortunately I can only manage the most trivial case, n=2, which I do not think deserves any prize but it seems to show that if anyone does get the prize it will be well deserved. We evaluate your definitions and then set n = 2; Even in this simplest of cases trying to use Solve directly Solve[D[f[x,2],x]==0,x] $Aborted takes for ever, so we resort to Groebner basis (actually what I just called "a hack" in another discussion ;-)) We first evaluate z = FullSimplify[D[f[x, 2], x] /. x^2 + 81 -> u^2, u > 0]; And then use GroebnerBasis: GroebnerBasis[{x^2 + 81 - u^2, z}, {x}, {u}] {4*x^2 + 16*x - 9} This we can actually solve (even without Mathematica): r=Solve[4*x^2 + 16*x - 9 == 0, x] {{x -> -(9/2)}, {x -> 1/2}} we can check that the derivative of f[x,2] does indeed vanish at these points: D[f[x,2],x]/.r//FullSimplify {0,0} Assuming that Groebner basis is correctly implemented this also shows that f'[x,2] has no roots in the interval (0,1/2) so we are done. Unfortunately the same method with n=3 already seems to take for ever. For small n>2 it may be possible perhaps to combine in some way numerical methods with Groebner basis to show similarly that D[f [x,n],x] has no roots in the interval (0,(n-1)/n) but obviously this won't work for a general n. At the moment the general problem looks pretty intractable to me, so much that I am almost willing to offer an additional prize to whoever manages to solve it (but I won't make any offers before seeing other people's attempts ;-)) Andrzej Kozlowski

**References**:**Solve or Reduce on a monstrosity of an expresssion (and a prize!)***From:*Daniel Reeves <dreeves@umich.edu>