mjo/**/*.py: drop obsolete set_random_seed().

[sage.d.git] / mjo / eja / eja_element.py

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index 47f6ff080c1d48a1a32b4903991aa0ec5bd0a502..0fd1c5f2032151de7d752494b4117825af3109a3 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -43,14 +43,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The definition of `x^2` is the unambiguous `x*x`::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: x*x == (x^2)
             True
 
         A few examples of power-associativity::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: x*(x*x)*(x*x) == x^5
             True
@@ -60,7 +58,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         We also know that powers operator-commute (Koecher, Chapter
         III, Corollary 1)::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: m = ZZ.random_element(0,10)
             sage: n = ZZ.random_element(0,10)
@@ -107,7 +104,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         We should always get back an element of the algebra::
 
-            sage: set_random_seed()
             sage: p = PolynomialRing(AA, 't').random_element()
             sage: J = random_eja()
             sage: x = J.random_element()
@@ -157,7 +153,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The characteristic polynomial of an element should evaluate
         to zero on that element::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: p = x.characteristic_polynomial()
             sage: x.apply_univariate_polynomial(p).is_zero()
@@ -239,7 +234,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         Ensure that we can always compute an inner product, and that
         it gives us back a real number::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x,y = J.random_elements(2)
             sage: x.inner_product(y) in RLF
@@ -267,7 +261,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The definition of a Jordan algebra says that any element
         operator-commutes with its square::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: x.operator_commutes_with(x^2)
             True
@@ -276,7 +269,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         Test Lemma 1 from Chapter III of Koecher::
 
-            sage: set_random_seed()
             sage: u,v = random_eja().random_elements(2)
             sage: lhs = u.operator_commutes_with(u*v)
             sage: rhs = v.operator_commutes_with(u^2)
@@ -286,7 +278,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         Test the first polarization identity from my notes, Koecher
         Chapter III, or from Baes (2.3)::
 
-            sage: set_random_seed()
             sage: x,y = random_eja().random_elements(2)
             sage: Lx = x.operator()
             sage: Ly = y.operator()
@@ -298,7 +289,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         Test the second polarization identity from my notes or from
         Baes (2.4)::
 
-            sage: set_random_seed()
             sage: x,y,z = random_eja().random_elements(3)
             sage: Lx = x.operator()
             sage: Ly = y.operator()
@@ -312,7 +302,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         Test the third polarization identity from my notes or from
         Baes (2.5)::
 
-            sage: set_random_seed()
             sage: u,y,z = random_eja().random_elements(3)
             sage: Lu = u.operator()
             sage: Ly = y.operator()
@@ -377,7 +366,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         An element is invertible if and only if its determinant is
         non-zero::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: x.is_invertible() == (x.det() != 0)
             True
@@ -385,7 +373,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         Ensure that the determinant is multiplicative on an associative
         subalgebra as in Faraut and Korányi's Proposition II.2.2::
 
-            sage: set_random_seed()
             sage: J = random_eja().random_element().subalgebra_generated_by()
             sage: x,y = J.random_elements(2)
             sage: (x*y).det() == x.det()*y.det()
@@ -393,7 +380,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The determinant in real matrix algebras is the usual determinant::
 
-            sage: set_random_seed()
             sage: X = matrix.random(QQ,3)
             sage: X = X + X.T
             sage: J1 = RealSymmetricEJA(3)
@@ -450,7 +436,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The inverse in the spin factor algebra is given in Alizadeh's
         Example 11.11::
 
-            sage: set_random_seed()
             sage: J = JordanSpinEJA.random_instance()
             sage: x = J.random_element()
             sage: while not x.is_invertible():
@@ -475,14 +460,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The identity element is its own inverse::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: J.one().inverse() == J.one()
             True
 
         If an element has an inverse, it acts like one::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x = J.random_element()
             sage: (not x.is_invertible()) or (x.inverse()*x == J.one())
@@ -490,7 +473,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The inverse of the inverse is what we started with::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x = J.random_element()
             sage: (not x.is_invertible()) or (x.inverse().inverse() == x)
@@ -500,7 +482,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         of an element is the inverse of its left-multiplication operator
         applied to the algebra's identity, when that inverse exists::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x = J.random_element()
             sage: (not x.operator().is_invertible()) or (
@@ -510,7 +491,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         Check that the fast (cached) and slow algorithms give the same
         answer::
 
-            sage: set_random_seed()                              # long time
             sage: J = random_eja(field=QQ, orthonormalize=False) # long time
             sage: x = J.random_element()                         # long time
             sage: while not x.is_invertible():                   # long time
@@ -562,14 +542,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The identity element is always invertible::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: J.one().is_invertible()
             True
 
         The zero element is never invertible in a non-trivial algebra::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: (not J.is_trivial()) and J.zero().is_invertible()
             False
@@ -577,7 +555,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         Test that the fast (cached) and slow algorithms give the same
         answer::
 
-            sage: set_random_seed()                              # long time
             sage: J = random_eja(field=QQ, orthonormalize=False) # long time
             sage: x = J.random_element()                         # long time
             sage: slow = x.is_invertible()                       # long time
@@ -660,14 +637,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The identity element is minimal only in an EJA of rank one::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: J.rank() == 1 or not J.one().is_primitive_idempotent()
             True
 
         A non-idempotent cannot be a minimal idempotent::
 
-            sage: set_random_seed()
             sage: J = JordanSpinEJA(4)
             sage: x = J.random_element()
             sage: (not x.is_idempotent()) and x.is_primitive_idempotent()
@@ -677,7 +652,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         idempotent if and only if it's idempotent with trace equal to
         unity::
 
-            sage: set_random_seed()
             sage: J = JordanSpinEJA(4)
             sage: x = J.random_element()
             sage: expected = (x.is_idempotent() and x.trace() == 1)
@@ -687,7 +661,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         Primitive idempotents must be non-zero::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: J.zero().is_idempotent()
             True
@@ -744,14 +717,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The identity element is never nilpotent, except in a trivial EJA::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: J.one().is_nilpotent() and not J.is_trivial()
             False
 
         The additive identity is always nilpotent::
 
-            sage: set_random_seed()
             sage: random_eja().zero().is_nilpotent()
             True
 
@@ -794,7 +765,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The zero element should never be regular, unless the parent
         algebra has dimension less than or equal to one::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: J.dimension() <= 1 or not J.zero().is_regular()
             True
@@ -802,7 +772,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The unit element isn't regular unless the algebra happens to
         consist of only its scalar multiples::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: J.dimension() <= 1 or not J.one().is_regular()
             True
@@ -837,7 +806,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         In the spin factor algebra (of rank two), all elements that
         aren't multiples of the identity are regular::
 
-            sage: set_random_seed()
             sage: J = JordanSpinEJA.random_instance()
             sage: n = J.dimension()
             sage: x = J.random_element()
@@ -849,7 +817,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The zero and unit elements are both of degree one in nontrivial
         algebras::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: d = J.zero().degree()
             sage: (J.is_trivial() and d == 0) or d == 1
@@ -860,7 +827,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         Our implementation agrees with the definition::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: x.degree() == x.minimal_polynomial().degree()
             True
@@ -969,7 +935,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         always the same, except in trivial algebras where the minimal
         polynomial of the unit/zero element is ``1``::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: mu = J.one().minimal_polynomial()
             sage: t = mu.parent().gen()
@@ -983,7 +948,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The degree of an element is (by one definition) the degree
         of its minimal polynomial::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: x.degree() == x.minimal_polynomial().degree()
             True
@@ -994,7 +958,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         identity. We require the dimension of the algebra to be at least
         two here so that said elements actually exist::
 
-            sage: set_random_seed()
             sage: d_max = JordanSpinEJA._max_random_instance_dimension()
             sage: n = ZZ.random_element(2, max(2,d_max))
             sage: J = JordanSpinEJA(n)
@@ -1011,7 +974,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The minimal polynomial should always kill its element::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: p = x.minimal_polynomial()
             sage: x.apply_univariate_polynomial(p)
@@ -1020,7 +982,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The minimal polynomial is invariant under a change of basis,
         and in particular, a re-scaling of the basis::
 
-            sage: set_random_seed()
             sage: d_max = RealSymmetricEJA._max_random_instance_dimension()
             sage: d = ZZ.random_element(1, d_max)
             sage: n = RealSymmetricEJA._max_random_instance_size(d)
@@ -1167,7 +1128,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         TESTS::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x,y = J.random_elements(2)
             sage: x.operator()(y) == x*y
@@ -1196,7 +1156,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The explicit form in the spin factor algebra is given by
         Alizadeh's Example 11.12::
 
-            sage: set_random_seed()
             sage: x = JordanSpinEJA.random_instance().random_element()
             sage: x_vec = x.to_vector()
             sage: Q = matrix.identity(x.base_ring(), 0)
@@ -1216,7 +1175,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         Test all of the properties from Theorem 11.2 in Alizadeh::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x,y = J.random_elements(2)
             sage: Lx = x.operator()
@@ -1410,7 +1368,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         This subalgebra, being composed of only powers, is associative::
 
-            sage: set_random_seed()
             sage: x0 = random_eja().random_element()
             sage: A = x0.subalgebra_generated_by()
             sage: x,y,z = A.random_elements(3)
@@ -1420,7 +1377,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         Squaring in the subalgebra should work the same as in
         the superalgebra::
 
-            sage: set_random_seed()
             sage: x = random_eja().random_element()
             sage: A = x.subalgebra_generated_by()
             sage: A(x^2) == A(x)*A(x)
@@ -1431,7 +1387,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         element... unless the original algebra was trivial, in which
         case the subalgebra is trivial too::
 
-            sage: set_random_seed()
             sage: A = random_eja().zero().subalgebra_generated_by()
             sage: (A.is_trivial() and A.dimension() == 0) or A.dimension() == 1
             True
@@ -1462,7 +1417,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         where there are non-nilpotent elements, or that we get the dumb
         solution in the trivial algebra::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x = J.random_element()
             sage: while x.is_nilpotent() and not J.is_trivial():
@@ -1546,14 +1500,12 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The trace of an element is a real number::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: J.random_element().trace() in RLF
             True
 
         The trace is linear::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x,y = J.random_elements(2)
             sage: alpha = J.base_ring().random_element()
@@ -1589,7 +1541,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
         The trace inner product is commutative, bilinear, and associative::
 
-            sage: set_random_seed()
             sage: J = random_eja()
             sage: x,y,z = J.random_elements(3)
             sage: # commutative