X-Git-Url: http://gitweb.michael.orlitzky.com/?a=blobdiff_plain;f=mjo%2Fcone%2Fcone.py;h=21f9862c24a9e9e3d9cffc52a1e789018f59a153;hb=c9ab469a53dc691d2646e22bdbc0ea0b3f3961c1;hp=a327720132b3907562f269fc4872a2b829226a59;hpb=48a7ad09084a859da72e39e60312bfffb6b806e9;p=sage.d.git diff --git a/mjo/cone/cone.py b/mjo/cone/cone.py index a327720..21f9862 100644 --- a/mjo/cone/cone.py +++ b/mjo/cone/cone.py @@ -419,8 +419,8 @@ def positive_operator_gens(K): 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) @@ -441,6 +441,25 @@ def 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 @@ -562,32 +581,37 @@ def Z_transformation_gens(K): ....: 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) @@ -597,13 +621,65 @@ def Z_transformation_gens(K): 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