cduston's profile - activity

I have: https://ask.sagemath.org/question/69124/call-sagemath-from-c/ and still no good answers.

Added MWE...how about this - can I call Py_Initialize(), do those import calls, and then proceed with both C and Python

call SageMath from C I wrote a little Graph Theory / Group Theory function in SageMath (using stuff from GAP, mostly), a

call SageMath from C I wrote a little Graph Theory / Group Theory function in SageMath (using stuff from GAP, mostly), a

call SageMath from C I wrote a little Graph Theory / Group Theory function in SageMath (using stuff from GAP, mostly), a

Unfortunately, it does help demonstrating that Mathematica does no better. I guess this is the answer ("Sage can't do what you want"), but I'll give a few days to see if anyone knows more. Thanks.

One of the frustrations I'm always having with Sage is how it tries to "solve" inequalities. For a random example:

sage: var('a b x')
sage: f = (a - b * x^2) / (x-1)
sage: solve(f > 0, x)
[[x < 1, b*x^2 - a > 0], [1 < x, -b*x^2 + a > 0]]

Which I knew already (since it's just the numerator). Ok, so various signs and things matter, but the fact that it can't even tell me

x^2 > a/b

is frustrating. Is there a reason, or a way to convince Sage to actually "solve" these in some way?

Ok that does seem to be a partial solution, but it doesn't allow the metric to be represented with abstract indices. For example, the answer to this problem is generically $C^c_{ab}=2\delta ^c_{(a}\nabla_{b)}\ln \Omega-g_{ab}g^{cd}\nabla_d \ln \Omega$.

Thanks: I will have to work out getting the latest version on my system before testing/marking correct, but it does seem that you've got a solution here.

I am wondering if there are ways to use abstract index notation in sage. For example, could I define the tensor:

$$C^c_{ab}=\frac{1}{2}g^{cd}(\tilde{\nabla_a} g_{bd}+\tilde{\nabla_b} g_{ad} - \tilde{\nabla_d} g_{ab})$$

this particular object describes the difference between two connections, $\nabla_a$ and $\tilde{\nabla}_b$. Can we define objects like this an manipulate them in Sage? My confusion comes from the common definition of the connection, a la

nabla = g.connection()

Which is not directly a 1-tensor. Specifically, I would like to define a metric, $g_{ab}$, and a conformal transformation $\tilde{g_{ab}}=\Omega^2 g_{ab}$, the corresponding connections, and determine the tensor $C^a_{bc}$ for this particular case. (and of course, we know the answer because this the standard approach to conformal transformations in GR).

Well, since this is my major frustration with Sage, I know that it is not a simple operation, but I can't for the life of me understand why it should be. Although your answer does the job, I'm going to select the other one as it doesn't require a new class definition. Although please, disagree and convince me!

Hello all, I'm working with the Friedman equations, and I've gotten them down to the common form presented in terms of the scale factor. MWE incoming:

M = Manifold(4, 'M', structure='Lorentzian')
fr.<t,r,th,ph> = M.chart(r't r:[0,+oo) th:[0,pi]:\theta ph:[0,2*pi):\phi')
var('G, Lambda, k', domain='real')
a = M.scalar_field(function('a')(t), name='a')
rho = M.scalar_field(function('rho')(t), name='rho')
p = M.scalar_field(function('p')(t), name='p')
g = M.metric()
g[0,0] = -1
g[1,1] = a*a/(1 - k*r^2)
g[2,2] = a*a*r^2
g[3,3] = a*a*(r*sin(th))^2
nabla = g.connection()
Ricci = nabla.ricci()
Ricci_scalar = g.ricci_scalar()
E1 = Ricci - Ricci_scalar/2*g + Lambda*g
print("First Friedmann equation:\n")
E1[0,0].expr().expand() == 0

With the result being

-Lambda + 3*diff(a(t), t)^2/a(t)^2 + 3*k/a(t)^2 == 0

Which is great, but I really want this to be in terms of the Hubble paramter $H=\dot{a}/a$. I can't make that substitution happen:

var('h')
eq1.substitute(diff(a.expr(),t)/a.expr()==h)

just spits out the same thing,

-Lambda + 3*diff(a(t), t)^2/a(t)^2 + 3*k/a(t)^2 == 0

Any ideas?

Tried it both ways and they work. Thanks very much! So the .operator() returns the function, and the .expr() gives the expression of that function?

I would like to utilize the alternating symbol $\epsilon_{abcd}$ in SageManifolds. I've worked out that you can define the volume form like

epsilon = g.volume_form()
print(epsilon) ; epsilon.display()


4-form eps_g on the 4-dimensional differentiable manifold M
eps_g = r^2*sqrt(-f(t))*abs(a(t))^3*sin(th) dt/\dr/\dth/\dph

And I can turn this into "the alternating 1-form" by dividing by the determinant of the metric. However, if I try to find the totally upper symbol $\epsilon^{abcd}$, which should just have components +1 and -1, I get

epsilon_up=epsilon.up(g)*det*det
print(epsilon_up);epsilon_up.display()

Tensor field of type (4,0) on the 4-dimensional differentiable manifold
M
abs(a(t))^6/a(t)^6 d/dt*d/dr*d/dth*d/dph - abs(a(t))^6/a(t)^6
d/dt*d/dr*d/dph*d/dth - abs(a(t))^6/a(t)^6 d/dt*d/dth*d/dr*d/dph +
abs(a(t))^6/a(t)^6 d/dt*d/dth*d/dph*d/dr + abs(a(t))^6/a(t)^6
d/dt*d/dph*d/dr*d/dth - abs(a(t))^6/a(t)^6 d/dt*d/dph*d/dth*d/dr -
abs(a(t))^6/a(t)^6 d/dr*d/dt*d/dth*d/dph + abs(a(t))^6/a(t)^6
d/dr*d/dt*d/dph*d/dth + abs(a(t))^6/a(t)^6 d/dr*d/dth*d/dt*d/dph -
abs(a(t))^6/a(t)^6 d/dr*d/dth*d/dph*d/dt - abs(a(t))^6/a(t)^6
d/dr*d/dph*d/dt*d/dth + abs(a(t))^6/a(t)^6 d/dr*d/dph*d/dth*d/dt +
abs(a(t))^6/a(t)^6 d/dth*d/dt*d/dr*d/dph - abs(a(t))^6/a(t)^6
d/dth*d/dt*d/dph*d/dr - abs(a(t))^6/a(t)^6 d/dth*d/dr*d/dt*d/dph +
abs(a(t))^6/a(t)^6 d/dth*d/dr*d/dph*d/dt + abs(a(t))^6/a(t)^6
d/dth*d/dph*d/dt*d/dr - abs(a(t))^6/a(t)^6 d/dth*d/dph*d/dr*d/dt -
abs(a(t))^6/a(t)^6 d/dph*d/dt*d/dr*d/dth + abs(a(t))^6/a(t)^6
d/dph*d/dt*d/dth*d/dr + abs(a(t))^6/a(t)^6 d/dph*d/dr*d/dt*d/dth -
abs(a(t))^6/a(t)^6 d/dph*d/dr*d/dth*d/dt - abs(a(t))^6/a(t)^6
d/dph*d/dth*d/dt*d/dr + abs(a(t))^6/a(t)^6 d/dph*d/dth*d/dr*d/dt

Which is just happening because at some point Sage has figured it out needs to keep track of the sign of a. Is there a direct way to define $\epsilon_{abcd}$ without accessing to the volume form at all? Essentially, I want to directly calculate things like the Euler Characteristic (and signature), which look like

$$R_{abcd}R_{efgh}\epsilon^{abef}\epsilon^{cdgh}$$