Think of k0 as one of the symbolic solutions to a cubic
equation. It comes from using "roots()" on some cubic polynomial. k0
turns out to be a function of three parameters: "tl,tr and beta" (you
can see what k0 looks like at the end of this message).
my problem is that i write
assume(tl>0,tr>.5,tl<.5,tr<1,beta>0,beta<1)
and then write
bool(k0.real()>0)
Sage then gives me the following error:
Computation failed since Maxima requested additional constraints (use
assume):
Is 4*(4*(2*(2*tl-2*beta+1)*(2*tr+2*beta+1)+4*(2*beta+1)*tr
+4*(1-2*beta)*tl)^3
-18*(-2*(2*tr+2*beta+1)-2*tr-2*tl+2*(-2*tl+2*beta-1)-2)
*(-2*(1-2*beta)*tl*(2*tr+2*beta+1)-2*(2*beta+1)*(2*tl-2*beta
+1)*tr)
*(2*(2*tl-2*beta+1)*(2*tr+2*beta+1)+4*(2*beta+1)*tr
+4*(1-2*beta)*tl))
-(-2*(2*tr+2*beta+1)-2*tr-2*tl+2*(-2*tl+2*beta-1)-2)^2
*(2*(2*tl-2*beta+1)*(2*tr+2*beta+1)+4*(2*beta+1)*tr
+4*(1-2*beta)*tl)^2
+432*(-2*(1-2*beta)*tl*(2*tr+2*beta+1)-2*(2*beta+1)*(2*tl-2*beta
+1)*tr)^2
+4*(-2*(2*tr+2*beta+1)-2*tr-2*tl+2*(-2*tl+2*beta-1)-2)^3
*(-2*(1-2*beta)*tl*(2*tr+2*beta+1)-2*(2*beta+1)*(2*tl-2*beta
+1)*tr) positive, negative, or zero?
So SUPPOSE I know that such term is indeed negative. So I write
assume(4*(4*(2*(2*tl-2*beta+1)*(2*tr+2*beta+1)+4*(2*beta+1)*tr
+4*(1-2*beta)*tl)^3
-18*(-2*(2*tr+2*beta+1)-2*tr-2*tl+2*(-2*tl+2*beta-1)-2)
*(-2*(1-2*beta)*tl*(2*tr+2*beta+1)-2*(2*beta+1)*(2*tl-2*beta
+1)*tr)
*(2*(2*tl-2*beta+1)*(2*tr+2*beta+1)+4*(2*beta+1)*tr
+4*(1-2*beta)*tl))
-(-2*(2*tr+2*beta+1)-2*tr-2*tl+2*(-2*tl+2*beta-1)-2)^2
*(2*(2*tl-2*beta+1)*(2*tr+2*beta+1)+4*(2*beta+1)*tr
+4*(1-2*beta)*tl)^2
+432*(-2*(1-2*beta)*tl*(2*tr+2*beta+1)-2*(2*beta+1)*(2*tl-2*beta
+1)*tr)^2
+4*(-2*(2*tr+2*beta+1)-2*tr-2*tl+2*(-2*tl+2*beta-1)-2)^3
*(-2*(1-2*beta)*tl*(2*tr+2*beta+1)-2*(2*beta+1)*(2*tl-2*beta
+1)*tr)<0)
and then I write again
bool(k0.real()>0)
Here's the problem: Sage gives me exactly the same error, about the
same expression, as if I had not assumed it negative. And I checked
with assumptions() and verified that the assumption about the
expression being negative was there.
Any insights as to what may be going on?
(to begin with I'm not even sure if writing k0.real() is legal, given
what k0 is)
Should I be doing
bool(k0>0)
instead?
(case in which my problem is that I have tried bool(k0>0), bool(k0<0)
and bool(k0==0) and I always get "False" even though I ran
simulations for 1000 values of the parameters that go in k0 and in all
cases k0>0. Is my problem then that I am expecting too much of
"bool"?)
Thanks!
Alberto
ps-- here's the root i'm evaluating, k0:
(-sqrt(3)*I/2 - 1/2)*(sqrt(4*(4*(2*(2*tl - 2*beta + 1)*(2*tr + 2*beta
+
1) + 4*(2*beta + 1)*tr + 4*(1 - 2*beta)*tl)^3 - 18*(-2*(2*tr + 2*beta
+
1) - 2*tr - 2*tl + 2*(-2*tl + 2*beta - 1) - 2)*(-2*(1 -
2*beta)*tl*(2*tr
+ 2*beta + 1) - 2*(2*beta + 1)*(2*tl - 2*beta + 1)*tr)*(2*(2*tl -
2*beta
+ 1)*(2*tr + 2*beta + 1) + 4*(2*beta + 1)*tr + 4*(1 - 2*beta)*tl)) -
(-2*(2*tr + 2*beta + 1) - 2*tr - 2*tl + 2*(-2*tl + 2*beta - 1) -
2)^2*(2*(2*tl - 2*beta + 1)*(2*tr + 2*beta + 1) + 4*(2*beta + 1)*tr +
4*(1 - 2*beta)*tl)^2 + 432*(-2*(1 - 2*beta)*tl*(2*tr + 2*beta + 1) -
2*(2*beta + 1)*(2*tl - 2*beta + 1)*tr)^2 + 4*(-2*(2*tr + 2*beta + 1) -
2*tr - 2*tl + 2*(-2*tl + 2*beta - 1) - 2)^3*(-2*(1 - 2*beta)*tl*(2*tr
+
2*beta + 1) - 2*(2*beta + 1)*(2*tl - 2*beta + 1)*tr))/(96*sqrt(3)) -
((-36*(-2*(2*tr + 2*beta + 1) - 2*tr - 2*tl + 2*(-2*tl + 2*beta - 1) -
2)*(2*(2*tl - 2*beta + 1)*(2*tr + 2*beta + 1) + 4*(2*beta + 1)*tr +
4*(1
- 2*beta)*tl) + 432*(-2*(1 - 2*beta)*tl*(2*tr + 2*beta + 1) -
2*(2*beta
+ 1)*(2*tl - 2*beta + 1)*tr) + 2*(-2*(2*tr + 2*beta + 1) - 2*tr - 2*tl
+
2*(-2*tl + 2*beta - 1) - 2)^3)/3456))^(1/3) - ((sqrt(3)*I/2 -
1/2)*(12*(2*(2*tl - 2*beta + 1)*(2*tr + 2*beta + 1) + 4*(2*beta +
1)*tr
+ 4*(1 - 2*beta)*tl) - (-2*(2*tr + 2*beta + 1) - 2*tr - 2*tl +
2*(-2*tl
+ 2*beta - 1) - 2)^2)/(144*(sqrt(4*(4*(2*(2*tl - 2*beta + 1)*(2*tr +
2*beta + 1) + 4*(2*beta + 1)*tr + 4*(1 - 2*beta)*tl)^3 - 18*(-2*(2*tr
+
2*beta + 1) - 2*tr - 2*tl + 2*(-2*tl + 2*beta - 1) - 2)*(-2*(1 -
2*beta)*tl*(2*tr + 2*beta + 1) - 2*(2*beta + 1)*(2*tl - 2*beta +
1)*tr)*(2*(2*tl - 2*beta + 1)*(2*tr + 2*beta + 1) + 4*(2*beta + 1)*tr
+
4*(1 - 2*beta)*tl)) - (-2*(2*tr + 2*beta + 1) - 2*tr - 2*tl + 2*(-2*tl
+
2*beta - 1) - 2)^2*(2*(2*tl - 2*beta + 1)*(2*tr + 2*beta + 1) +
4*(2*beta + 1)*tr + 4*(1 - 2*beta)*tl)^2 + 432*(-2*(1 -
2*beta)*tl*(2*tr
+ 2*beta + 1) - 2*(2*beta + 1)*(2*tl - 2*beta + 1)*tr)^2 + 4*(-2*(2*tr
+
2*beta + 1) - 2*tr - 2*tl + 2*(-2*tl + 2*beta - 1) - 2)^3*(-2*(1 -
2*beta)*tl*(2*tr + 2*beta + 1) - 2*(2*beta + 1)*(2*tl - 2*beta +
1)*tr))/(96*sqrt(3)) - ((-36*(-2*(2*tr + 2*beta + 1) - 2*tr - 2*tl +
2*(-2*tl + 2*beta - 1) - 2)*(2*(2*tl - 2*beta + 1)*(2*tr + 2*beta + 1)
+
4*(2*beta + 1)*tr + 4*(1 - 2*beta)*tl) + 432*(-2*(1 - 2*beta)*tl*(2*tr
+
2*beta + 1) - 2*(2*beta + 1)*(2*tl - 2*beta + 1)*tr) + 2*(-2*(2*tr +
2*beta + 1) - 2*tr - 2*tl + 2*(-2*tl + 2*beta - 1) -
2)^3)/3456))^(1/3))) + (2*(2*tr + 2*beta + 1) + 2*tr + 2*tl - 2*(-2*tl
+
2*beta - 1) + 2)/12
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