# The lines() method only returns one generator per line. For a true
# line, we also need a generator pointing in the opposite direction.
S_gens = [ direction*gen for direction in [1,-1] for gen in K.lines() ]
- S = Cone(S_gens, K.lattice())
+ S = Cone(S_gens, K.lattice(), check=False)
# Since ``S`` is a subspace, the rays of its dual generate its
# orthogonal complement.
- S_perp = Cone(S.dual(), K.lattice())
+ S_perp = Cone(S.dual(), K.lattice(), check=False)
P = K.intersection(S_perp)
return (P,S)
sage: K.is_full_space()
True
sage: pi_of_K = positive_operator_gens(K)
- sage: actual = Cone([p.list() for p in pi_of_K], lattice=L).lineality()
- sage: actual == n^2
+ sage: pi_cone = Cone([p.list() for p in pi_of_K], lattice=L)
+ sage: pi_cone.lineality() == n^2
True
sage: K = Cone([(1,0),(0,1),(0,-1)])
sage: pi_of_K = positive_operator_gens(K)
....: check=False)
sage: K.is_proper() == pi_cone.is_proper()
True
+
+ The positive operators of a permuted cone can be obtained by
+ conjugation::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
+ sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
+ sage: pi_of_pK = positive_operator_gens(pK)
+ sage: actual = Cone([t.list() for t in pi_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: pi_of_K = positive_operator_gens(K)
+ sage: expected = Cone([(p*t*p.inverse()).list() for t in pi_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(expected)
+ True
"""
# Matrices are not vectors in Sage, so we have to convert them
# to vectors explicitly before we can find a basis. We need these
sage: Z_transformation_gens(K)
[]
+ Every operator is a Z-transformation on the ambient vector space::
+
+ sage: K = Cone([(1,),(-1,)])
+ sage: K.is_full_space()
+ True
+ sage: Z_transformation_gens(K)
+ [[-1], [1]]
+
+ sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
+ sage: K.is_full_space()
+ True
+ sage: Z_transformation_gens(K)
+ [
+ [-1 0] [1 0] [ 0 -1] [0 1] [ 0 0] [0 0] [ 0 0] [0 0]
+ [ 0 0], [0 0], [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1]
+ ]
+
+ A non-obvious application is to find the Z-transformations on the
+ right half-plane::
+
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Z_transformation_gens(K)
+ [
+ [-1 0] [1 0] [ 0 0] [0 0] [ 0 0] [0 0]
+ [ 0 0], [0 0], [-1 0], [1 0], [ 0 -1], [0 1]
+ ]
+
Z-transformations on a subspace are Lyapunov-like and vice-versa::
sage: K = Cone([(1,0),(-1,0),(0,1),(0,-1)])
....: for (x,s) in dcs])
True
- The lineality space of Z is LL::
+ The lineality space of the cone of Z-transformations is the space of
+ Lyapunov-like transformations::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=4)
sage: L = ToricLattice(K.lattice_dim()**2)
- sage: z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ],
+ sage: Z_cone = Cone([ z.list() for z in Z_transformation_gens(K) ],
....: lattice=L,
....: check=False)
sage: ll_basis = [ vector(l.list()) for l in K.lyapunov_like_basis() ]
sage: lls = L.vector_space().span(ll_basis)
- sage: z_cone.linear_subspace() == lls
+ sage: Z_cone.linear_subspace() == lls
True
- And thus, the lineality of Z is the Lyapunov rank::
+ The lineality of the Z-transformations on a cone is the Lyapunov
+ rank of that cone::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=4)
sage: Z_of_K = Z_transformation_gens(K)
sage: L = ToricLattice(K.lattice_dim()**2)
- sage: z_cone = Cone([ z.list() for z in Z_of_K ],
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K ],
....: lattice=L,
....: check=False)
- sage: z_cone.lineality() == K.lyapunov_rank()
+ sage: Z_cone.lineality() == K.lyapunov_rank()
True
- The lineality spaces of pi-star and Z-star are equal:
+ The lineality spaces of the duals of the positive operator and
+ Z-transformation cones are equal. From this it follows that the
+ dimensions of the Z-transformation cone and positive operator cone
+ are equal::
sage: set_random_seed()
sage: K = random_cone(max_ambient_dim=4)
sage: pi_cone = Cone([p.list() for p in pi_of_K],
....: lattice=L,
....: check=False)
- sage: pi_star = pi_cone.dual()
- sage: z_cone = Cone([ z.list() for z in Z_of_K],
+ sage: Z_cone = Cone([ z.list() for z in Z_of_K],
....: lattice=L,
....: check=False)
- sage: z_star = z_cone.dual()
+ sage: pi_cone.dim() == Z_cone.dim()
+ True
+ sage: pi_star = pi_cone.dual()
+ sage: z_star = Z_cone.dual()
sage: pi_star.linear_subspace() == z_star.linear_subspace()
True
+
+ The trivial cone, full space, and half-plane all give rise to the
+ expected dimensions::
+
+ sage: n = ZZ.random_element().abs()
+ sage: K = Cone([[0] * n], ToricLattice(n))
+ sage: K.is_trivial()
+ True
+ sage: L = ToricLattice(n^2)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Z_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = K.dual()
+ sage: K.is_full_space()
+ True
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual = Z_cone.dim()
+ sage: actual == n^2
+ True
+ sage: K = Cone([(1,0),(0,1),(0,-1)])
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: Z_cone = Cone([z.list() for z in Z_of_K], check=False)
+ sage: Z_cone.dim() == 3
+ True
+
+ The Z-transformations of a permuted cone can be obtained by
+ conjugation::
+
+ sage: set_random_seed()
+ sage: K = random_cone(max_ambient_dim=4)
+ sage: L = ToricLattice(K.lattice_dim()**2)
+ sage: p = SymmetricGroup(K.lattice_dim()).random_element().matrix()
+ sage: pK = Cone([ p*k for k in K ], K.lattice(), check=False)
+ sage: Z_of_pK = Z_transformation_gens(pK)
+ sage: actual = Cone([t.list() for t in Z_of_pK],
+ ....: lattice=L,
+ ....: check=False)
+ sage: Z_of_K = Z_transformation_gens(K)
+ sage: expected = Cone([(p*t*p.inverse()).list() for t in Z_of_K],
+ ....: lattice=L,
+ ....: check=False)
+ sage: actual.is_equivalent(expected)
+ True
"""
# Matrices are not vectors in Sage, so we have to convert them
# to vectors explicitly before we can find a basis. We need these
projects / sage.d.git / blobdiff