X-Git-Url: https://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Ffaces.py;h=6c5843e860ef9da5151150103223a1d507914fe1;hb=928b7d49fda98ff105c92293b5797bb7a2b9873a;hp=81b8421e94b28e0b9b9a68387a6b67f92c903a78;hpb=064a3c7c3796f931954f8ece0b6e9912163efc1b;p=sage.d.git

diff --git a/mjo/cone/faces.py b/mjo/cone/faces.py
index 81b8421..6c5843e 100644
--- a/mjo/cone/faces.py
+++ b/mjo/cone/faces.py
@@ -51,38 +51,34 @@ def face_generated_by(K,S):
 
     The face generated by <whatever> should be a face::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8, max_rays=10)
-        sage: S = [K.random_element() for idx in range(0,5)]
+        sage: S = ( K.random_element() for idx in range(5) )
         sage: F = face_generated_by(K, S)
         sage: F.is_face_of(K)
         True
 
     The face generated by a set should always contain that set::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8, max_rays=10)
-        sage: S = [K.random_element() for idx in range(0,5)]
+        sage: S = ( K.random_element() for idx in range(5) )
         sage: F = face_generated_by(K, S)
-        sage: all([F.contains(x) for x in S])
+        sage: all(F.contains(x) for x in S)
         True
 
     The generators of a proper cone are all extreme vectors of the cone,
     and therefore generate their own faces::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8,
         ....:                 max_rays=10,
         ....:                 strictly_convex=True,
         ....:                 solid=True)
-        sage: all([face_generated_by(K, [r]) == Cone([r]) for r in K])
+        sage: all(face_generated_by(K, [r]) == Cone([r]) for r in K)
         True
 
     For any point ``x`` in ``K`` and any face ``F`` of ``K``, we have
     that ``x`` is in the relative interior of ``F`` if and only if
     ``F`` is the face generated by ``x`` [Tam]_::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8, max_rays=10)
         sage: x = K.random_element()
         sage: S = [x]
@@ -96,7 +92,6 @@ def face_generated_by(K,S):
     and ``G`` in the face lattice is equal to the face generated by
     ``F + G`` (in the Minkowski sense) [Tam]_::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8, max_rays=10)
         sage: L = K.face_lattice()
         sage: F = L.random_element()
@@ -109,7 +104,6 @@ def face_generated_by(K,S):
     Combining Proposition 3.1 and Corollary 3.9 in [Tam]_ gives the
     following equality for any ``y`` in ``K``::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8, max_rays=10)
         sage: y = K.random_element()
         sage: S = [y]
@@ -123,7 +117,7 @@ def face_generated_by(K,S):
 
     """
     face_lattice = K.face_lattice()
-    candidates = [F for F in face_lattice if all([F.contains(x) for x in S])]
+    candidates = [F for F in face_lattice if all(F.contains(x) for x in S)]
 
     # K itself is a face of K, so unless we were given a set S that
     # isn't a subset of K, the candidates list will be nonempty.
@@ -143,6 +137,10 @@ def dual_face(K,F):
 
     REFERENCES:
 
+    .. [HilgertHofmannLawson] Joachim Hilgert, Karl Heinrich Hofmann, and Jimmie
+       D. Lawson. Lie groups, convex cones and semigroups. Oxford Mathematical
+       Monographs. Clarendon Press, Oxford, 1989. ISBN 9780198535690.
+
     .. [Tam] Bit-Shun Tam. On the duality operator of a convex cone. Linear
        Algebra and its Applications, 64:33-56, 1985, doi:10.1016/0024-3795(85)
        90265-4.
@@ -168,10 +166,9 @@ def dual_face(K,F):
     The dual face of ``K`` with respect to itself should be the
     lineality space of its dual [Tam]_::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8, max_rays=10)
         sage: K_dual = K.dual()
-        sage: lKd_gens = [ dir*l for dir in [1,-1] for l in K_dual.lines() ]
+        sage: lKd_gens = ( dir*l for dir in [1,-1] for l in K_dual.lines() )
         sage: linspace_K_dual = Cone(lKd_gens, K_dual.lattice())
         sage: dual_face(K,K).is_equivalent(linspace_K_dual)
         True
@@ -179,7 +176,6 @@ def dual_face(K,F):
     If ``K`` is proper, then the dual face of its trivial face is the
     dual of ``K`` [Tam]_::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8,
         ....:                 max_rays=10,
         ....:                 strictly_convex=True,
@@ -192,7 +188,6 @@ def dual_face(K,F):
     The dual of the cone of ``K`` at ``y`` is the dual face of the face
     of ``K`` generated by ``y`` ([Tam]_ Corollary 3.2)::
 
-        sage: set_random_seed()
         sage: K = random_cone(max_ambient_dim=8, max_rays=10)
         sage: y = K.random_element()
         sage: S = [y]
@@ -205,11 +200,10 @@ def dual_face(K,F):
     Since all faces of a polyhedral cone are exposed, the dual face of a
     dual face should be the original face [HilgertHofmannLawson]_::
 
-        sage: set_random_seed()
         sage: def check_prop(K,F):
         ....:     return dual_face(K.dual(), dual_face(K,F)).is_equivalent(F)
         sage: K = random_cone(max_ambient_dim=8, max_rays=10)
-        sage: all([check_prop(K,F) for F in K.face_lattice()])
+        sage: all(check_prop(K,F) for F in K.face_lattice())
         True
 
     """
@@ -217,5 +211,5 @@ def dual_face(K,F):
     if not F.is_face_of(K):
         raise ValueError("%s is not a face of %s" % (F,K))
 
-    span_F = Cone([c*g for c in [1,-1] for g in F], F.lattice())
+    span_F = Cone((c*g for c in [1,-1] for g in F), F.lattice())
     return K.dual().intersection(span_F.dual())