eja: remove the (completed) Cartesian product TODO.

[sage.d.git] / mjo / eja / TODO

diff --git a/mjo/eja/TODO b/mjo/eja/TODO
index 280811e41761ed62d61a7b54886e88b7795b3d7b..0624ddb1d74a13c1d0bc2cb1679c8b81fbfc13e1 100644 (file)
--- a/mjo/eja/TODO
+++ b/mjo/eja/TODO
@@ -1,29 +1,41 @@
-1. Add cartesian products to random_eja().
+1. Add references and start citing them.

 
 

-2. Add references and start citing them.
-
-3. Implement the octonion simple EJA. We don't actually need octonions
+2. Implement the octonion simple EJA. We don't actually need octonions

    for this to work, only their real embedding (some 8x8 monstrosity).
 
    for this to work, only their real embedding (some 8x8 monstrosity).
 

-4. Pre-cache charpoly for some small algebras?
+3. Pre-cache charpoly for some small algebras?

 
 RealSymmetricEJA(4):
 
 sage: F = J.base_ring()
 sage: a0 = (1/4)*X[4]**2*X[6]**2 - (1/2)*X[2]*X[5]*X[6]**2 - (1/2)*X[3]*X[4]*X[6]*X[7] + (F(2).sqrt()/2)*X[1]*X[5]*X[6]*X[7] + (1/4)*X[3]**2*X[7]**2 - (1/2)*X[0]*X[5]*X[7]**2 + (F(2).sqrt()/2)*X[2]*X[3]*X[6]*X[8] - (1/2)*X[1]*X[4]*X[6*X[8] - (1/2)*X[1]*X[3]*X[7]*X[8] + (F(2).sqrt()/2)*X[0]*X[4]*X[7]*X[8] + (1/4)*X[1]**2*X[8]**2 - (1/2)*X[0]*X[2]*X[8]**2 - (1/2)*X[2]*X[3]**2*X[9] + (F(2).sqrt()/2)*X[1]*X[3]*X[4]*X[9] - (1/2)*X[0]*X[4]**2*X[9] - (1/2)*X[1]**2*X[5]*X[9] + X[0]*X[2]*X[5]*X[9]
 
 
 RealSymmetricEJA(4):
 
 sage: F = J.base_ring()
 sage: a0 = (1/4)*X[4]**2*X[6]**2 - (1/2)*X[2]*X[5]*X[6]**2 - (1/2)*X[3]*X[4]*X[6]*X[7] + (F(2).sqrt()/2)*X[1]*X[5]*X[6]*X[7] + (1/4)*X[3]**2*X[7]**2 - (1/2)*X[0]*X[5]*X[7]**2 + (F(2).sqrt()/2)*X[2]*X[3]*X[6]*X[8] - (1/2)*X[1]*X[4]*X[6*X[8] - (1/2)*X[1]*X[3]*X[7]*X[8] + (F(2).sqrt()/2)*X[0]*X[4]*X[7]*X[8] + (1/4)*X[1]**2*X[8]**2 - (1/2)*X[0]*X[2]*X[8]**2 - (1/2)*X[2]*X[3]**2*X[9] + (F(2).sqrt()/2)*X[1]*X[3]*X[4]*X[9] - (1/2)*X[0]*X[4]**2*X[9] - (1/2)*X[1]**2*X[5]*X[9] + X[0]*X[2]*X[5]*X[9]
 

-5. Profile the construction of "large" matrix algebras (like the
+4. Profile the construction of "large" matrix algebras (like the

    15-dimensional QuaternionHermitianAlgebra(3)) to find out why
    they're so slow.
 
    15-dimensional QuaternionHermitianAlgebra(3)) to find out why
    they're so slow.
 

-6. Instead of storing a basis multiplication matrix, just make
+5. Instead of storing a basis multiplication matrix, just make

    product_on_basis() a cached method and manually cache its
    entries. The cython cached method lookup should be faster than a
    product_on_basis() a cached method and manually cache its
    entries. The cython cached method lookup should be faster than a

-   python-based matrix lookup anyway.
+   python-based matrix lookup anyway. NOTE: we should still be able
+   to recompute the table somehow. Is this worth it?

 
 

-7. What the ever-loving fuck is this shit?
+6. What the ever-loving fuck is this shit?

 
        sage: O = Octonions(QQ)
        sage: e0 = O.monomial(0)
        sage: e0*[[[[]]]]
        [[[[]]]]*e0
 
        sage: O = Octonions(QQ)
        sage: e0 = O.monomial(0)
        sage: e0*[[[[]]]]
        [[[[]]]]*e0

+
+7. In fact, could my octonion matrix algebra be generalized for any
+   algebra of matrices over the reals whose entries are not real? Then
+   we wouldn't need real embeddings at all. They might even be fricking
+   vector spaces if I did that...
+
+8. Every once in a long while, the test
+
+       sage: set_random_seed()
+       sage: x = random_eja().random_element()
+       sage: x.is_invertible() == (x.det() != 0)
+
+   in eja_element.py returns False.