from sage.all import *
-def rearrangement_cone(p,n):
+def rearrangement_cone(p,n,lattice=None):
r"""
Return the rearrangement cone of order ``p`` in ``n`` dimensions.
INPUT:
- - ``p`` -- The number of components to "rearrange."
+ - ``p`` -- The number of components to "rearrange."
- - ``n`` -- The dimension of the ambient space for the resulting cone.
+ - ``n`` -- The dimension of the ambient space for the resulting cone.
+
+ - ``lattice`` -- (default: ``None``) an ambient lattice of rank ``n``
+ to be passed to the :func:`Cone` constructor.
OUTPUT:
A polyhedral closed convex cone object representing a rearrangement
- cone of order ``p`` in ``n`` dimensions.
+ cone of order ``p`` in ``n`` dimensions. Each generating ray will
+ have the integer ring as its base ring.
- REFERENCES:
+ If a ``lattice`` was specified, then the resulting cone will live in
+ that lattice unless its rank is incompatible with the dimension
+ ``n`` (in which case a ``ValueError`` is raised).
- .. [HenrionSeeger] Rene Henrion and Alberto Seeger.
- Inradius and Circumradius of Various Convex Cones Arising in
- Applications. Set-Valued and Variational Analysis, 18(3-4),
- 483-511, 2010. doi:10.1007/s11228-010-0150-z
+ ALGORITHM:
+
+ The generators for the rearrangement cone are given by [Jeong]_
+ Theorem 5.2.3.
+
+ REFERENCES:
.. [GowdaJeong] Muddappa Seetharama Gowda and Juyoung Jeong.
Spectral cones in Euclidean Jordan algebras.
Linear Algebra and its Applications, 509, 286-305.
doi:10.1016/j.laa.2016.08.004
+ .. [HenrionSeeger] Rene Henrion and Alberto Seeger.
+ Inradius and Circumradius of Various Convex Cones Arising in
+ Applications. Set-Valued and Variational Analysis, 18(3-4),
+ 483-511, 2010. doi:10.1007/s11228-010-0150-z
+
.. [Jeong] Juyoung Jeong.
Spectral sets and functions on Euclidean Jordan algebras.
+ University of Maryland, Baltimore County, Ph.D. thesis, 2017.
SETUP::
sage: rearrangement_cone(5,5).lineality()
4
- All rearrangement cones are proper::
+ All rearrangement cones are proper when ``p`` is less than ``n`` by
+ [Jeong]_ Proposition 5.2.1::
sage: all( rearrangement_cone(p,n).is_proper()
- ....: for n in xrange(10)
- ....: for p in xrange(1, n) )
+ ....: for n in range(10)
+ ....: for p in range(1, 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::
+ dimensions is ``n`` for ``p == 1`` or ``p == n`` and one otherwise,
+ by [Jeong]_ Corollary 5.2.4::
sage: all( rearrangement_cone(p,n).lyapunov_rank() == n
- ....: for n in xrange(2, 10)
+ ....: 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 xrange(3, 10)
- ....: for p in xrange(2, n-1) )
+ ....: for n in range(3, 10)
+ ....: for p in range(2, n-1) )
True
TESTS:
- The rearrangement cone is permutation-invariant::
+ All rearrangement cones are permutation-invariant by [Jeong]_
+ Proposition 5.2.1::
sage: n = ZZ.random_element(2,10).abs()
sage: p = ZZ.random_element(1,n)
....: rearrangement_cone(p,n).random_element(),
....: p
....: )
- ....: for n in xrange(2, 10)
- ....: for p in xrange(1, n-1)
+ ....: for n in range(2, 10)
+ ....: for p in range(1, n-1)
....: )
True
- The rearrangenent cone of order ``p`` is contained in the
- rearrangement cone of order ``p + 1``::
+ The rearrangenent cone of order ``p`` is contained in the rearrangement
+ cone of order ``p + 1`` by [Jeong]_ Proposition 5.2.1::
sage: set_random_seed()
sage: n = ZZ.random_element(2,10)
sage: all( x in K2 for x in K1 )
True
+ The rearrangement cone of order ``p`` is linearly isomorphic to the
+ rearrangement cone of order ``n - p`` when ``p`` is less than ``n``,
+ by [Jeong]_ Proposition 5.2.1::
+
+ sage: set_random_seed()
+ sage: n = ZZ.random_element(2,10)
+ sage: p = ZZ.random_element(1,n)
+ sage: K1 = rearrangement_cone(p,n)
+ sage: K2 = rearrangement_cone(n-p, n)
+ sage: Mp = (1/p)*matrix.ones(QQ,n) - identity_matrix(QQ,n)
+ sage: Cone( (Mp*K2.rays()).columns() ).is_equivalent(K1)
+ True
+
The order ``p`` should be between ``1`` and ``n``, inclusive::
sage: rearrangement_cone(0,3)
...
ValueError: order p=5 should be between 1 and n=3, inclusive
+ If a ``lattice`` was given, it is actually used::
+
+ sage: L = ToricLattice(3, 'M')
+ sage: rearrangement_cone(2, 3, lattice=L)
+ 3-d cone in 3-d lattice M
+
+ Unless the rank of the lattice disagrees with ``n``::
+
+ sage: L = ToricLattice(1, 'M')
+ sage: rearrangement_cone(2, 3, lattice=L)
+ Traceback (most recent call last):
+ ...
+ ValueError: lattice rank=1 and dimension n=3 are incompatible
+
"""
if p < 1 or p > n:
raise ValueError('order p=%d should be between 1 and n=%d, inclusive'
%
(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
+ if lattice is None:
+ lattice = ToricLattice(n)
+
+ if lattice.rank() != n:
+ raise ValueError('lattice rank=%d and dimension n=%d are incompatible'
+ %
+ (lattice.rank(), n))
- G = identity_matrix(ZZ,n).rows() + [ d(j) for j in xrange(n) ]
- return Cone(G)
+ I = identity_matrix(ZZ,n)
+ M = matrix.ones(ZZ,n) - p*I
+ G = identity_matrix(ZZ,n).rows() + M.rows()
+ return Cone(G, lattice=lattice)