mjo/cone/rearrangement.py

   1 # Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we
   2 # have to explicitly mangle our sitedir here so that "mjo.cone"
   3 # resolves.
   4 from os.path import abspath
   5 from site import addsitedir
   6 addsitedir(abspath('../../'))
   7
   8 from sage.all import *
   9
  10 def rearrangement_cone(p,n):
  11     r"""
  12     Return the rearrangement cone of order ``p`` in ``n`` dimensions.
  13
  14     The rearrangement cone in ``n`` dimensions has as its elements
  15     vectors of length ``n``. For inclusion in the cone, the smallest
  16     ``p`` components of a vector must sum to a nonnegative number.
  17
  18     For example, the rearrangement cone of order ``p == 1`` has its
  19     single smallest component nonnegative. This implies that all
  20     components are nonnegative, and that therefore the rearrangement
  21     cone of order one is the nonnegative orthant.
  22
  23     When ``p == n``, the sum of all components of a vector must be
  24     nonnegative for inclusion in the cone. That is, the cone is a
  25     half-space in ``n`` dimensions.
  26
  27     INPUT:
  28
  29     - ``p`` -- The number of components to "rearrange."
  30
  31     - ``n`` -- The dimension of the ambient space for the resulting cone.
  32
  33     OUTPUT:
  34
  35     A polyhedral closed convex cone object representing a rearrangement
  36     cone of order ``p`` in ``n`` dimensions.
  37
  38     EXAMPLES:
  39
  40     The rearrangement cones of order one are nonnegative orthants::
  41
  42         sage: rearrangement_cone(1,1) == Cone([(1,)])
  43         True
  44         sage: rearrangement_cone(1,2) == Cone([(0,1),(1,0)])
  45         True
  46         sage: rearrangement_cone(1,3) == Cone([(0,0,1),(0,1,0),(1,0,0)])
  47         True
  48
  49     When ``p == n``, the resulting cone will be a half-space, so we
  50     expect its lineality to be one less than ``n`` because it will
  51     contain a hyperplane but is not the entire space::
  52
  53         sage: rearrangement_cone(5,5).lineality()
  54         4
  55
  56     TESTS:
  57
  58     todo.
  59     should be permutation invariant.
  60     should have the expected lyapunov rank.
  61     just loop through them all for n <= 10 and p < n?
  62
  63     """
  64
  65     def d(j):
  66         v = [1]*n    # Create the list of all ones...
  67         v[j] = 1 - p # Now "fix" the ``j``th entry.
  68         return v
  69
  70     V = VectorSpace(QQ, n)
  71     G = V.basis() + [ d(j) for j in range(n) ]
  72     return Cone(G)