from sage.all import * def rearrangement_cone(p,n,lattice=None): 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. - ``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. Each generating ray will have the integer ring as its base ring. 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). 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: from mjo.cone.rearrangement import rearrangement_cone 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 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 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, by [Jeong]_ Corollary 5.2.4:: sage: all( rearrangement_cone(p,n).lyapunov_rank() == n ....: 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 range(3, 10) ....: for p in range(2, n-1) ) True TESTS: 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) sage: K = rearrangement_cone(p,n) sage: P = SymmetricGroup(n).random_element().matrix() sage: all( K.contains(P*r) for r in K ) True The smallest ``p`` components of every element of the rearrangement cone should sum to a nonnegative number (this tests that the generators really are what we think they are):: sage: def _has_rearrangement_property(v,p): ....: return sum( sorted(v)[0:p] ) >= 0 sage: all( _has_rearrangement_property( ....: rearrangement_cone(p,n).random_element(), ....: p ....: ) ....: 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`` by [Jeong]_ Proposition 5.2.1:: sage: n = ZZ.random_element(2,10) sage: p = ZZ.random_element(1,n) sage: K1 = rearrangement_cone(p,n) sage: K2 = rearrangement_cone(p+1,n) 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: 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) Traceback (most recent call last): ... ValueError: order p=0 should be between 1 and n=3, inclusive sage: rearrangement_cone(5,3) Traceback (most recent call last): ... 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)) 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)) 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)