]>
gitweb.michael.orlitzky.com - sage.d.git/blob - rearrangement.py
3e09c8ad948a8c70135809b6e3c5e5328babfa36
3 def rearrangement_cone(p
,n
,lattice
=None):
5 Return the rearrangement cone of order ``p`` in ``n`` dimensions.
7 The rearrangement cone in ``n`` dimensions has as its elements
8 vectors of length ``n``. For inclusion in the cone, the smallest
9 ``p`` components of a vector must sum to a nonnegative number.
11 For example, the rearrangement cone of order ``p == 1`` has its
12 single smallest component nonnegative. This implies that all
13 components are nonnegative, and that therefore the rearrangement
14 cone of order one is the nonnegative orthant.
16 When ``p == n``, the sum of all components of a vector must be
17 nonnegative for inclusion in the cone. That is, the cone is a
18 half-space in ``n`` dimensions.
22 - ``p`` -- The number of components to "rearrange."
24 - ``n`` -- The dimension of the ambient space for the resulting cone.
26 - ``lattice`` -- (default: ``None``) an ambient lattice of rank ``n``
27 to be passed to the :func:`Cone` constructor.
31 A polyhedral closed convex cone object representing a rearrangement
32 cone of order ``p`` in ``n`` dimensions. Each generating ray will
33 have the integer ring as its base ring.
35 If a ``lattice`` was specified, then the resulting cone will live in
36 that lattice unless its rank is incompatible with the dimension
37 ``n`` (in which case a ``ValueError`` is raised).
41 .. [HenrionSeeger] Rene Henrion and Alberto Seeger.
42 Inradius and Circumradius of Various Convex Cones Arising in
43 Applications. Set-Valued and Variational Analysis, 18(3-4),
44 483-511, 2010. doi:10.1007/s11228-010-0150-z
46 .. [GowdaJeong] Muddappa Seetharama Gowda and Juyoung Jeong.
47 Spectral cones in Euclidean Jordan algebras.
48 Linear Algebra and its Applications, 509, 286-305.
49 doi:10.1016/j.laa.2016.08.004
51 .. [Jeong] Juyoung Jeong.
52 Spectral sets and functions on Euclidean Jordan algebras.
56 sage: from mjo.cone.rearrangement import rearrangement_cone
60 The rearrangement cones of order one are nonnegative orthants::
62 sage: rearrangement_cone(1,1) == Cone([(1,)])
64 sage: rearrangement_cone(1,2) == Cone([(0,1),(1,0)])
66 sage: rearrangement_cone(1,3) == Cone([(0,0,1),(0,1,0),(1,0,0)])
69 When ``p == n``, the resulting cone will be a half-space, so we
70 expect its lineality to be one less than ``n`` because it will
71 contain a hyperplane but is not the entire space::
73 sage: rearrangement_cone(5,5).lineality()
76 All rearrangement cones are proper::
78 sage: all( rearrangement_cone(p,n).is_proper()
79 ....: for n in xrange(10)
80 ....: for p in xrange(1, n) )
83 The Lyapunov rank of the rearrangement cone of order ``p`` in ``n``
84 dimensions is ``n`` for ``p == 1`` or ``p == n`` and one otherwise::
86 sage: all( rearrangement_cone(p,n).lyapunov_rank() == n
87 ....: for n in xrange(2, 10)
88 ....: for p in [1, n-1] )
90 sage: all( rearrangement_cone(p,n).lyapunov_rank() == 1
91 ....: for n in xrange(3, 10)
92 ....: for p in xrange(2, n-1) )
97 The rearrangement cone is permutation-invariant::
99 sage: n = ZZ.random_element(2,10).abs()
100 sage: p = ZZ.random_element(1,n)
101 sage: K = rearrangement_cone(p,n)
102 sage: P = SymmetricGroup(n).random_element().matrix()
103 sage: all( K.contains(P*r) for r in K )
106 The smallest ``p`` components of every element of the rearrangement
107 cone should sum to a nonnegative number (this tests that the
108 generators really are what we think they are)::
110 sage: set_random_seed()
111 sage: def _has_rearrangement_property(v,p):
112 ....: return sum( sorted(v)[0:p] ) >= 0
113 sage: all( _has_rearrangement_property(
114 ....: rearrangement_cone(p,n).random_element(),
117 ....: for n in xrange(2, 10)
118 ....: for p in xrange(1, n-1)
122 The rearrangenent cone of order ``p`` is contained in the
123 rearrangement cone of order ``p + 1``::
125 sage: set_random_seed()
126 sage: n = ZZ.random_element(2,10)
127 sage: p = ZZ.random_element(1,n)
128 sage: K1 = rearrangement_cone(p,n)
129 sage: K2 = rearrangement_cone(p+1,n)
130 sage: all( x in K2 for x in K1 )
133 The order ``p`` should be between ``1`` and ``n``, inclusive::
135 sage: rearrangement_cone(0,3)
136 Traceback (most recent call last):
138 ValueError: order p=0 should be between 1 and n=3, inclusive
139 sage: rearrangement_cone(5,3)
140 Traceback (most recent call last):
142 ValueError: order p=5 should be between 1 and n=3, inclusive
144 If a ``lattice`` was given, it is actually used::
146 sage: L = ToricLattice(3, 'M')
147 sage: rearrangement_cone(2, 3, lattice=L)
148 3-d cone in 3-d lattice M
150 Unless the rank of the lattice disagrees with ``n``::
152 sage: L = ToricLattice(1, 'M')
153 sage: rearrangement_cone(2, 3, lattice=L)
154 Traceback (most recent call last):
156 ValueError: lattice rank=1 and dimension n=3 are incompatible
160 raise ValueError('order p=%d should be between 1 and n=%d, inclusive'
165 lattice
= ToricLattice(n
)
167 if lattice
.rank() != n
:
168 raise ValueError('lattice rank=%d and dimension n=%d are incompatible'
172 I
= identity_matrix(ZZ
,n
)
173 M
= matrix
.ones(ZZ
,n
) - p
*I
174 G
= identity_matrix(ZZ
,n
).rows() + M
.rows()
175 return Cone(G
, lattice
=lattice
)