+++ /dev/null
-from sage.all import *
-
-def rearrangement_cone(p,n):
- r"""
- Return the rearrangement cone of order ``p`` in ``n`` dimensions.
-
- The rearrangement cone in ``n`` dimensions has as its elements
- vectors of length ``n``. For inclusion in the cone, the smallest
- ``p`` components of a vector must sum to a nonnegative number.
-
- For example, the rearrangement cone of order ``p == 1`` has its
- single smallest component nonnegative. This implies that all
- components are nonnegative, and that therefore the rearrangement
- cone of order one is the nonnegative orthant.
-
- When ``p == n``, the sum of all components of a vector must be
- nonnegative for inclusion in the cone. That is, the cone is a
- half-space in ``n`` dimensions.
-
- INPUT:
-
- - ``p`` -- The number of components to "rearrange."
-
- - ``n`` -- The dimension of the ambient space for the resulting cone.
-
- OUTPUT:
-
- A polyhedral closed convex cone object representing a rearrangement
- cone of order ``p`` in ``n`` dimensions.
-
- SETUP::
-
- sage: from mjo.cone.rearrangement import rearrangement_cone
-
- EXAMPLES:
-
- The rearrangement cones of order one are nonnegative orthants::
-
- sage: rearrangement_cone(1,1) == Cone([(1,)])
- True
- sage: rearrangement_cone(1,2) == Cone([(0,1),(1,0)])
- True
- sage: rearrangement_cone(1,3) == Cone([(0,0,1),(0,1,0),(1,0,0)])
- True
-
- When ``p == n``, the resulting cone will be a half-space, so we
- expect its lineality to be one less than ``n`` because it will
- contain a hyperplane but is not the entire space::
-
- sage: rearrangement_cone(5,5).lineality()
- 4
-
- All rearrangement cones are proper::
-
- sage: all([ rearrangement_cone(p,n).is_proper()
- ....: for n in range(10)
- ....: for p in range(n) ])
- True
-
- The Lyapunov rank of the rearrangement cone of order ``p`` in ``n``
- dimensions is ``n`` for ``p == 1`` or ``p == n`` and one otherwise::
-
- sage: all([ rearrangement_cone(p,n).lyapunov_rank() == n
- ....: for n in range(2, 10)
- ....: for p in [1, n-1] ])
- True
- sage: all([ rearrangement_cone(p,n).lyapunov_rank() == 1
- ....: for n in range(3, 10)
- ....: for p in range(2, n-1) ])
- True
-
- TESTS:
-
- The rearrangement cone is permutation-invariant::
-
- sage: n = ZZ.random_element(2,10).abs()
- sage: p = ZZ.random_element(1,n)
- sage: K = rearrangement_cone(p,n)
- sage: P = SymmetricGroup(n).random_element().matrix()
- sage: all([ K.contains(P*r) for r in K.rays() ])
- True
-
- """
-
- def d(j):
- v = [1]*n # Create the list of all ones...
- v[j] = 1 - p # Now "fix" the ``j``th entry.
- return v
-
- V = VectorSpace(QQ, n)
- G = V.basis() + [ d(j) for j in range(n) ]
- return Cone(G)
-
-
-def has_rearrangement_property(v, p):
- r"""
- Test if the vector ``v`` has the "rearrangement property."
-
- The rearrangement cone of order ``p`` in `n` dimensions has its
- members vectors of length `n`. The "rearrangement property,"
- satisfied by its elements, is to have its smallest ``p`` components
- sum to a nonnegative number.
-
- We believe that we have a description of the extreme vectors of the
- rearrangement cone: see ``rearrangement_cone()``. This function is
- used to test that conic combinations of those extreme vectors are in
- fact elements of the rearrangement cone. We can't test all conic
- combinations, obviously, but we can test a random one.
-
- To become more sure of the result, generate a bunch of vectors with
- ``random_element()`` and test them with this function.
-
- INPUT:
-
- - ``v`` -- An element of a cone suspected of being the rearrangement
- cone of order ``p``.
-
- - ``p`` -- The suspected order of the rearrangement cone.
-
- OUTPUT:
-
- If ``v`` has the rearrangement property (that is, if its smallest ``p``
- components sum to a nonnegative number), ``True`` is returned. Otherwise
- ``False`` is returned.
-
- SETUP::
-
- sage: from mjo.cone.rearrangement import (has_rearrangement_property,
- ....: rearrangement_cone)
-
- EXAMPLES:
-
- Every element of a rearrangement cone should have the property::
-
- sage: for n in range(2,10):
- ....: for p in range(1, n-1):
- ....: K = rearrangement_cone(p,n)
- ....: v = K.random_element()
- ....: if not has_rearrangement_property(v,p):
- ....: print(v)
-
- """
- components = sorted(v)[0:p]
- return sum(components) >= 0
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