mjo/cone/schur.py

   1 """
   2 The Schur cone, as described in the "Critical angles..." papers by
   3 Iusem, Seeger, and Sossa. It defines the Schur ordering on `R^{n}`.
   4 """
   5
   6 from sage.all import *
   7
   8 def schur_cone(n):
   9     r"""
  10     Return the Schur cone in ``n`` dimensions.
  11
  12     INPUT:
  13
  14     - ``n`` -- the dimension the ambient space.
  15
  16     OUTPUT:
  17
  18     A rational closed convex Schur cone of dimension ``n``.
  19
  20     REFERENCES:
  21
  22     .. [SeegerSossaI] Alberto Seeger and David Sossa.
  23        Critical angles between two convex cones I. General theory.
  24        TOP, 24(1):44-65, 2016, doi:10.1007/s11750-015-0375-y.
  25
  26     SETUP::
  27
  28         sage: from mjo.cone.nonnegative_orthant import nonnegative_orthant
  29         sage: from mjo.cone.schur import schur_cone
  30
  31     EXAMPLES:
  32
  33     Verify the claim that the maximal angle between any two generators
  34     of the Schur cone and the nonnegative quintant is ``3*pi/4``::
  35
  36         sage: P = schur_cone(5)
  37         sage: Q = nonnegative_orthant(5)
  38         sage: G = [ g.change_ring(QQbar).normalized() for g in P ]
  39         sage: H = [ h.change_ring(QQbar).normalized() for h in Q ]
  40         sage: actual = max([arccos(u.inner_product(v)) for u in G for v in H])
  41         sage: expected = 3*pi/4
  42         sage: abs(actual - expected).n() < 1e-12
  43         True
  44
  45     TESTS:
  46
  47     We get the trivial cone when ``n`` is zero::
  48
  49         sage: schur_cone(0).is_trivial()
  50         True
  51
  52     """
  53
  54     def _f(i,j):
  55         if i == j:
  56             return 1
  57         elif j - i == 1:
  58             return -1
  59         else:
  60             return 0
  61
  62     # The "max" below catches the trivial case where n == 0.
  63     S = matrix(ZZ, max(0,n-1), n, _f)
  64
  65     # Likewise, when n == 0, we need to specify the lattice.
  66     return Cone(S.rows(), ToricLattice(n))