eja: use "b" as the default prefix.

diff --git a/mjo/eja/eja_algebra.py b/mjo/eja/eja_algebra.py
index da008d0cf70249de27ac1a52b1d53ae4be9ef4f2..c55061e34fb2b7a4d04f127d74148984ce21f9fa 100644 (file)
--- a/mjo/eja/eja_algebra.py
+++ b/mjo/eja/eja_algebra.py
@@ -141,7 +141,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
                  cartesian_product=False,
                  check_field=True,
                  check_axioms=True,
-                 prefix='e'):
+                 prefix="b"):
 
         n = len(basis)
 
@@ -210,7 +210,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
         # ambient vector space V that our (vectorized) basis lives in,
         # as well as a subspace W of V spanned by those (vectorized)
         # basis elements. The W-coordinates are the coefficients that
-        # we see in things like x = 1*e1 + 2*e2.
+        # we see in things like x = 1*b1 + 2*b2.
         vector_basis = basis
 
         degree = 0
@@ -337,16 +337,16 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             sage: set_random_seed()
             sage: J = random_eja()
             sage: n = J.dimension()
-            sage: ei = J.zero()
-            sage: ej = J.zero()
-            sage: ei_ej = J.zero()*J.zero()
+            sage: bi = J.zero()
+            sage: bj = J.zero()
+            sage: bi_bj = J.zero()*J.zero()
             sage: if n > 0:
             ....:     i = ZZ.random_element(n)
             ....:     j = ZZ.random_element(n)
-            ....:     ei = J.monomial(i)
-            ....:     ej = J.monomial(j)
-            ....:     ei_ej = J.product_on_basis(i,j)
-            sage: ei*ej == ei_ej
+            ....:     bi = J.monomial(i)
+            ....:     bj = J.monomial(j)
+            ....:     bi_bj = J.product_on_basis(i,j)
+            sage: bi*bj == bi_bj
             True
 
         """
@@ -618,7 +618,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             sage: J2 = RealSymmetricEJA(2)
             sage: J = cartesian_product([J1,J2])
             sage: J( (J1.matrix_basis()[1], J2.matrix_basis()[2]) )
-            e1 + e5
+            b1 + b5
 
         TESTS:
 
@@ -893,15 +893,15 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
             sage: J = JordanSpinEJA(4)
             sage: J.multiplication_table()
             +----++----+----+----+----+
-            | *  || e0 | e1 | e2 | e3 |
+            | *  || b0 | b1 | b2 | b3 |
             +====++====+====+====+====+
-            | e0 || e0 | e1 | e2 | e3 |
+            | b0 || b0 | b1 | b2 | b3 |
             +----++----+----+----+----+
-            | e1 || e1 | e0 | 0  | 0  |
+            | b1 || b1 | b0 | 0  | 0  |
             +----++----+----+----+----+
-            | e2 || e2 | 0  | e0 | 0  |
+            | b2 || b2 | 0  | b0 | 0  |
             +----++----+----+----+----+
-            | e3 || e3 | 0  | 0  | e0 |
+            | b3 || b3 | 0  | 0  | b0 |
             +----++----+----+----+----+
 
         """
@@ -955,7 +955,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
             sage: J = RealSymmetricEJA(2)
             sage: J.basis()
-            Finite family {0: e0, 1: e1, 2: e2}
+            Finite family {0: b0, 1: b1, 2: b2}
             sage: J.matrix_basis()
             (
             [1 0]  [                  0 0.7071067811865475?]  [0 0]
@@ -966,7 +966,7 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
             sage: J = JordanSpinEJA(2)
             sage: J.basis()
-            Finite family {0: e0, 1: e1}
+            Finite family {0: b0, 1: b1}
             sage: J.matrix_basis()
             (
             [1]  [0]
@@ -1048,20 +1048,20 @@ class FiniteDimensionalEJA(CombinatorialFreeModule):
 
             sage: J = HadamardEJA(5)
             sage: J.one()
-            e0 + e1 + e2 + e3 + e4
+            b0 + b1 + b2 + b3 + b4
 
         The unit element in the Hadamard EJA is inherited in the
         subalgebras generated by its elements::
 
             sage: J = HadamardEJA(5)
             sage: J.one()
-            e0 + e1 + e2 + e3 + e4
+            b0 + b1 + b2 + b3 + b4
             sage: x = sum(J.gens())
             sage: A = x.subalgebra_generated_by(orthonormalize=False)
             sage: A.one()
-            f0
+            c0
             sage: A.one().superalgebra_element()
-            e0 + e1 + e2 + e3 + e4
+            b0 + b1 + b2 + b3 + b4
 
         TESTS:
 
@@ -1864,13 +1864,13 @@ class RealSymmetricEJA(ConcreteEJA, RealMatrixEJA):
     EXAMPLES::
 
         sage: J = RealSymmetricEJA(2)
-        sage: e0, e1, e2 = J.gens()
-        sage: e0*e0
-        e0
-        sage: e1*e1
-        1/2*e0 + 1/2*e2
-        sage: e2*e2
-        e2
+        sage: b0, b1, b2 = J.gens()
+        sage: b0*b0
+        b0
+        sage: b1*b1
+        1/2*b0 + 1/2*b2
+        sage: b2*b2
+        b2
 
     In theory, our "field" can be any subfield of the reals::
 
@@ -2621,19 +2621,19 @@ class HadamardEJA(ConcreteEJA):
     This multiplication table can be verified by hand::
 
         sage: J = HadamardEJA(3)
-        sage: e0,e1,e2 = J.gens()
-        sage: e0*e0
-        e0
-        sage: e0*e1
+        sage: b0,b1,b2 = J.gens()
+        sage: b0*b0
+        b0
+        sage: b0*b1
         0
-        sage: e0*e2
+        sage: b0*b2
         0
-        sage: e1*e1
-        e1
-        sage: e1*e2
+        sage: b1*b1
+        b1
+        sage: b1*b2
         0
-        sage: e2*e2
-        e2
+        sage: b2*b2
+        b2
 
     TESTS:
 
@@ -2874,20 +2874,20 @@ class JordanSpinEJA(BilinearFormEJA):
     This multiplication table can be verified by hand::
 
         sage: J = JordanSpinEJA(4)
-        sage: e0,e1,e2,e3 = J.gens()
-        sage: e0*e0
-        e0
-        sage: e0*e1
-        e1
-        sage: e0*e2
-        e2
-        sage: e0*e3
-        e3
-        sage: e1*e2
+        sage: b0,b1,b2,b3 = J.gens()
+        sage: b0*b0
+        b0
+        sage: b0*b1
+        b1
+        sage: b0*b2
+        b2
+        sage: b0*b3
+        b3
+        sage: b1*b2
         0
-        sage: e1*e3
+        sage: b1*b3
         0
-        sage: e2*e3
+        sage: b2*b3
         0
 
     We can change the generator prefix::
@@ -3099,23 +3099,23 @@ class CartesianProductEJA(FiniteDimensionalEJA):
         sage: J = cartesian_product([J1,cartesian_product([J2,J3])])
         sage: J.multiplication_table()
         +----++----+----+----+
-        | *  || e0 | e1 | e2 |
+        | *  || b0 | b1 | b2 |
         +====++====+====+====+
-        | e0 || e0 | 0  | 0  |
+        | b0 || b0 | 0  | 0  |
         +----++----+----+----+
-        | e1 || 0  | e1 | 0  |
+        | b1 || 0  | b1 | 0  |
         +----++----+----+----+
-        | e2 || 0  | 0  | e2 |
+        | b2 || 0  | 0  | b2 |
         +----++----+----+----+
         sage: HadamardEJA(3).multiplication_table()
         +----++----+----+----+
-        | *  || e0 | e1 | e2 |
+        | *  || b0 | b1 | b2 |
         +====++====+====+====+
-        | e0 || e0 | 0  | 0  |
+        | b0 || b0 | 0  | 0  |
         +----++----+----+----+
-        | e1 || 0  | e1 | 0  |
+        | b1 || 0  | b1 | 0  |
         +----++----+----+----+
-        | e2 || 0  | 0  | e2 |
+        | b2 || 0  | 0  | b2 |
         +----++----+----+----+
 
     TESTS:

diff --git a/mjo/eja/eja_element.py b/mjo/eja/eja_element.py
index 52933e2decdcf3d9dc1218a568bf29bcff6cf5f1..ff25b82073a85f9b2f92301d69f37455f231f5ea 100644 (file)
--- a/mjo/eja/eja_element.py
+++ b/mjo/eja/eja_element.py
@@ -167,8 +167,8 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
             sage: J = HadamardEJA(3)
             sage: p1 = J.one().characteristic_polynomial()
             sage: q1 = J.zero().characteristic_polynomial()
-            sage: e0,e1,e2 = J.gens()
-            sage: A = (e0 + 2*e1 + 3*e2).subalgebra_generated_by() # dim 3
+            sage: b0,b1,b2 = J.gens()
+            sage: A = (b0 + 2*b1 + 3*b2).subalgebra_generated_by() # dim 3
             sage: p2 = A.one().characteristic_polynomial()
             sage: q2 = A.zero().characteristic_polynomial()
             sage: p1 == p2
@@ -348,7 +348,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         EXAMPLES::
 
             sage: J = JordanSpinEJA(2)
-            sage: e0,e1 = J.gens()
             sage: x = sum( J.gens() )
             sage: x.det()
             0
@@ -356,7 +355,6 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         ::
 
             sage: J = JordanSpinEJA(3)
-            sage: e0,e1,e2 = J.gens()
             sage: x = sum( J.gens() )
             sage: x.det()
             -1
@@ -793,7 +791,9 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
             sage: J = JordanSpinEJA(5)
             sage: J.one().is_regular()
             False
-            sage: e0, e1, e2, e3, e4 = J.gens() # e0 is the identity
+            sage: b0, b1, b2, b3, b4 = J.gens()
+            sage: b0 == J.one()
+            True
             sage: for x in J.gens():
             ....:     (J.one() + x).is_regular()
             False
@@ -843,8 +843,8 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
             sage: J = JordanSpinEJA(4)
             sage: J.one().degree()
             1
-            sage: e0,e1,e2,e3 = J.gens()
-            sage: (e0 - e1).degree()
+            sage: b0,b1,b2,b3 = J.gens()
+            sage: (b0 - b1).degree()
             2
 
         In the spin factor algebra (of rank two), all elements that
@@ -1096,7 +1096,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
             sage: J = ComplexHermitianEJA(3)
             sage: J.one()
-            e0 + e3 + e8
+            b0 + b3 + b8
             sage: J.one().to_matrix()
             [1 0 0 0 0 0]
             [0 1 0 0 0 0]
@@ -1109,7 +1109,7 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
             sage: J = QuaternionHermitianEJA(2)
             sage: J.one()
-            e0 + e5
+            b0 + b5
             sage: J.one().to_matrix()
             [1 0 0 0 0 0 0 0]
             [0 1 0 0 0 0 0 0]
@@ -1355,11 +1355,11 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
 
             sage: J = RealSymmetricEJA(3)
             sage: J.one()
-            e0 + e2 + e5
+            b0 + b2 + b5
             sage: J.one().spectral_decomposition()
-            [(1, e0 + e2 + e5)]
+            [(1, b0 + b2 + b5)]
             sage: J.zero().spectral_decomposition()
-            [(0, e0 + e2 + e5)]
+            [(0, b0 + b2 + b5)]
 
         TESTS::
 
@@ -1384,13 +1384,13 @@ class FiniteDimensionalEJAElement(IndexedFreeModuleElement):
         The spectral decomposition should work in subalgebras, too::
 
             sage: J = RealSymmetricEJA(4)
-            sage: (e0, e1, e2, e3, e4, e5, e6, e7, e8, e9) = J.gens()
-            sage: A = 2*e5 - 2*e8
+            sage: (b0, b1, b2, b3, b4, b5, b6, b7, b8, b9) = J.gens()
+            sage: A = 2*b5 - 2*b8
             sage: (lambda1, c1) = A.spectral_decomposition()[1]
             sage: (J0, J5, J1) = J.peirce_decomposition(c1)
             sage: (f0, f1, f2) = J1.gens()
             sage: f0.spectral_decomposition()
-            [(0, f2), (1, f0)]
+            [(0, c2), (1, c0)]
 
         """
         A = self.subalgebra_generated_by(orthonormalize=True)

diff --git a/mjo/eja/eja_operator.py b/mjo/eja/eja_operator.py
index 4d24b35c5d902f0dbd2824242a88be614d45562e..7100ea68f9e1dd7db7f6bcc60558e4a187668553 100644 (file)
--- a/mjo/eja/eja_operator.py
+++ b/mjo/eja/eja_operator.py
@@ -239,8 +239,8 @@ class FiniteDimensionalEJAOperator(Map):
         We can scale an operator on a rational algebra by a rational number::
 
             sage: J = RealSymmetricEJA(2)
-            sage: e0,e1,e2 = J.gens()
-            sage: x = 2*e0 + 4*e1 + 16*e2
+            sage: b0,b1,b2 = J.gens()
+            sage: x = 2*b0 + 4*b1 + 16*b2
             sage: x.operator()
             Linear operator between finite-dimensional Euclidean Jordan algebras
             represented by the matrix:

diff --git a/mjo/eja/eja_subalgebra.py b/mjo/eja/eja_subalgebra.py
index 92fd296b3003a06e7bafb79f9d9c37d9cb6b13cb..1b86d236c390691939fd84cabb597e6b6159d406 100644 (file)
--- a/mjo/eja/eja_subalgebra.py
+++ b/mjo/eja/eja_subalgebra.py
@@ -51,20 +51,20 @@ class FiniteDimensionalEJASubalgebraElement(FiniteDimensionalEJAElement):
             sage: J = RealSymmetricEJA(3)
             sage: x = sum(J.gens())
             sage: x
-            e0 + e1 + e2 + e3 + e4 + e5
+            b0 + b1 + b2 + b3 + b4 + b5
             sage: A = x.subalgebra_generated_by(orthonormalize=False)
             sage: A(x)
-            f1
+            c1
             sage: A(x).superalgebra_element()
-            e0 + e1 + e2 + e3 + e4 + e5
+            b0 + b1 + b2 + b3 + b4 + b5
             sage: y = sum(A.gens())
             sage: y
-            f0 + f1
+            c0 + c1
             sage: B = y.subalgebra_generated_by(orthonormalize=False)
             sage: B(y)
-            g1
+            d1
             sage: B(y).superalgebra_element()
-            f0 + f1
+            c0 + c1
 
         TESTS:
 
@@ -121,17 +121,18 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA):
 
     TESTS:
 
-    Ensure that our generator names don't conflict with the superalgebra::
+    Ensure that our generator names don't conflict with the
+    superalgebra::
 
         sage: J = JordanSpinEJA(3)
         sage: J.one().subalgebra_generated_by().gens()
-        (f0,)
+        (c0,)
         sage: J = JordanSpinEJA(3, prefix='f')
         sage: J.one().subalgebra_generated_by().gens()
         (g0,)
-        sage: J = JordanSpinEJA(3, prefix='b')
+        sage: J = JordanSpinEJA(3, prefix='a')
         sage: J.one().subalgebra_generated_by().gens()
-        (c0,)
+        (b0,)
 
     Ensure that we can find subalgebras of subalgebras::
 
@@ -139,7 +140,6 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA):
         sage: B = A.one().subalgebra_generated_by()
         sage: B.dimension()
         1
-
     """
     def __init__(self, superalgebra, basis, **kwargs):
         self._superalgebra = superalgebra
@@ -152,7 +152,7 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA):
         # try to "increment" the parent algebra's prefix, although
         # this idea goes out the window fast because some prefixen
         # are off-limits.
-        prefixen = [ 'f', 'g', 'h', 'a', 'b', 'c', 'd' ]
+        prefixen = ["b","c","d","e","f","g","h","l","m"]
         try:
             prefix = prefixen[prefixen.index(self._superalgebra.prefix()) + 1]
         except ValueError:
@@ -200,9 +200,9 @@ class FiniteDimensionalEJASubalgebra(FiniteDimensionalEJA):
             ....:                                    associative=True,
             ....:                                    orthonormalize=False)
             sage: K(J.one())
-            f1
+            c1
             sage: K(J.one() + x)
-            f0 + f1
+            c0 + c1
 
         ::