From edbce26685b7e62d04da8f037e51621551292225 Mon Sep 17 00:00:00 2001 From: Michael Orlitzky Date: Fri, 11 Sep 2015 11:06:50 -0400 Subject: [PATCH] Begin work on the rearrangement cone. --- mjo/cone/rearrangement.py | 72 +++++++++++++++++++++++++++++++++++++++ 1 file changed, 72 insertions(+) create mode 100644 mjo/cone/rearrangement.py diff --git a/mjo/cone/rearrangement.py b/mjo/cone/rearrangement.py new file mode 100644 index 0000000..6ea993c --- /dev/null +++ b/mjo/cone/rearrangement.py @@ -0,0 +1,72 @@ +# Sage doesn't load ~/.sage/init.sage during testing (sage -t), so we +# have to explicitly mangle our sitedir here so that "mjo.cone" +# resolves. +from os.path import abspath +from site import addsitedir +addsitedir(abspath('../../')) + +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. + + 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 + + TESTS: + + todo. + should be permutation invariant. + should have the expected lyapunov rank. + just loop through them all for n <= 10 and p < n? + + """ + + 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) -- 2.44.2