From: Michael Orlitzky <michael@orlitzky.com>
Date: Wed, 13 Jan 2016 14:51:48 +0000 (-0500)
Subject: Only check=False when K is proper (can't prove it otherwise).
X-Git-Url: https://gitweb.michael.orlitzky.com/?a=commitdiff_plain;h=e0acc328048efbded42e1459e766de1884ced651;p=sage.d.git

Only check=False when K is proper (can't prove it otherwise).
---

diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py
index dd28f2f..3d74b5f 100644
--- a/mjo/cone/cone.py
+++ b/mjo/cone/cone.py
@@ -77,6 +77,17 @@ def positive_operator_gens(K):
     ``K``. Moreover, any nonnegative linear combination of these
     matrices shares the same property.
 
+    REFERENCES:
+
+    .. [Orlitzky-Pi-Z]
+       M. Orlitzky.
+       Positive operators and Z-transformations on closed convex cones.
+
+    .. [Tam]
+       B.-S. Tam.
+       Some results of polyhedral cones and simplicial cones.
+       Linear and Multilinear Algebra, 4:4 (1977) 281--284.
+
     EXAMPLES:
 
     Positive operators on the nonnegative orthant are nonnegative matrices::
@@ -393,14 +404,12 @@ def positive_operator_gens(K):
     vectors = [ W(tp.list()) for tp in tensor_products ]
 
     check = True
-    if K.is_solid() or K.is_strictly_convex():
-        # The lineality space of either ``K`` or ``K.dual()`` is
-        # trivial and it's easy to show that our generating set is
-        # minimal. I would love a proof that this works when ``K`` is
-        # neither pointed nor solid.
-        #
-        # Note that in that case we can get *duplicates*, since the
-        # tensor product of (x,s) is the same as that of (-x,-s).
+    if K.is_proper():
+        # All of the generators involved are extreme vectors and
+        # therefore minimal [Tam]_. If this cone is neither solid nor
+        # strictly convex, then the tensor product of ``s`` and ``x``
+        # is the same as that of ``-s`` and ``-x``. However, as a
+        # /set/, ``tensor_products`` may still be minimal.
         check = False
 
     # Create the dual cone of the positive operators, expressed as
@@ -423,9 +432,15 @@ def Z_transformation_gens(K):
 
     A list of `n`-by-``n`` matrices where ``n == K.lattice_dim()``.
     Each matrix ``L`` in the list should have the property that
-    ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element the
-    discrete complementarity set of ``K``. Moreover, any nonnegative
-    linear combination of these matrices shares the same property.
+    ``(L*x).inner_product(s) <= 0`` whenever ``(x,s)`` is an element of
+    this cone's :meth:`discrete_complementarity_set`. Moreover, any
+    conic (nonnegative linear) combination of these matrices shares the
+    same property.
+
+    REFERENCES:
+
+    M. Orlitzky.
+    Positive operators and Z-transformations on closed convex cones.
 
     EXAMPLES:
 
@@ -643,14 +658,12 @@ def Z_transformation_gens(K):
     vectors = [ W(m.list()) for m in tensor_products ]
 
     check = True
-    if K.is_solid() or K.is_strictly_convex():
-        # The lineality space of either ``K`` or ``K.dual()`` is
-        # trivial and it's easy to show that our generating set is
-        # minimal. I would love a proof that this works when ``K`` is
-        # neither pointed nor solid.
-        #
-        # Note that in that case we can get *duplicates*, since the
-        # tensor product of (x,s) is the same as that of (-x,-s).
+    if K.is_proper():
+        # All of the generators involved are extreme vectors and
+        # therefore minimal. If this cone is neither solid nor
+        # strictly convex, then the tensor product of ``s`` and ``x``
+        # is the same as that of ``-s`` and ``-x``. However, as a
+        # /set/, ``tensor_products`` may still be minimal.
         check = False
 
     # Create the dual cone of the cross-positive operators,