r"""
Return the largest linearly-independent subset of ``vs``.
"""
- if len(vs) == 0:
- # ...for lazy enough definitions of linearly-independent
- return vs
-
result = []
- old_V = VectorSpace(vs[0].parent().base_field(), 0)
-
- for v in vs:
- new_V = span(result + [v])
- if new_V.dimension() > old_V.dimension():
- result.append(v)
- old_V = new_V
+ m = matrix(vs).echelon_form()
+ for idx in range(0, m.nrows()):
+ if not m[idx].is_zero():
+ result.append(m[idx])
return result
-def iso_space(K):
- r"""
- Construct the space `W \times W^{\perp}` isomorphic to the ambient space
- of ``K`` where `W` is equal to the span of ``K``.
- """
- V = K.lattice().vector_space()
-
- # Create the space W \times W^{\perp} isomorphic to V.
- # First we get an orthogonal (but not normal) basis...
- M = matrix(V.base_field(), K.rays())
- W_basis = drop_dependent(K.rays())
-
- W = V.subspace_with_basis(W_basis)
- W_perp = W.complement()
-
- return W.cartesian_product(W_perp)
-
-
-def ips_iso(K):
- r"""
- Construct the IPS isomorphism and its inverse from our paper.
-
- Given a cone ``K``, the returned isomorphism will split its ambient
- vector space `V` into a cartesian product `W \times W^{\perp}` where
- `W` equals the span of ``K``.
- """
- V = K.lattice().vector_space()
- V_iso = iso_space(K)
- (W, W_perp) = V_iso.cartesian_factors()
-
- # A space equivalent to V, but using our basis.
- V_user = V.subspace_with_basis( W.basis() + W_perp.basis() )
-
- def phi(v):
- # Write v in terms of our custom basis, where the first dim(W)
- # coordinates are for the W-part of the basis.
- cs = V_user.coordinates(v)
-
- w1 = sum([ V_user.basis()[idx]*cs[idx]
- for idx in range(0, W.dimension()) ])
- w2 = sum([ V_user.basis()[idx]*cs[idx]
- for idx in range(W.dimension(), V.dimension()) ])
-
- return V_iso( (w1, w2) )
-
-
- def phi_inv( pair ):
- # Crash if the arguments are in the wrong spaces.
- V_iso(pair)
-
- #w = sum([ sub_w[idx]*W.basis()[idx] for idx in range(0,m) ])
- #w_prime = sum([ sub_w_prime[idx]*W_perp.basis()[idx]
- # for idx in range(0,n-m) ])
-
- return sum( pair.cartesian_factors() )
-
-
- return (phi,phi_inv)
-
-
def rho(K, K2=None):
r"""
Restrict ``K`` into its own span, or the span of another cone.
EXAMPLES::
sage: K = Cone([(1,)])
- sage: restrict_span(K) == K
+ sage: rho(K) == K
True
sage: K2 = Cone([(1,0)])
- sage: restrict_span(K2).rays()
+ sage: rho(K2).rays()
N(1)
in 1-d lattice N
sage: K3 = Cone([(1,0,0)])
- sage: restrict_span(K3).rays()
+ sage: rho(K3).rays()
N(1)
in 1-d lattice N
- sage: restrict_span(K2) == restrict_span(K3)
+ sage: rho(K2) == rho(K3)
True
TESTS:
sage: set_random_seed()
sage: K = random_cone(max_dim = 8)
- sage: K_S = restrict_span(K)
+ sage: K_S = rho(K)
sage: K_S.is_solid()
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8)
- sage: K_S = restrict_span(K, K.dual() )
+ sage: K_S = rho(K, K.dual() )
sage: K_S.lattice_dim() == K.dual().dim()
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8)
- sage: K.dim() == restrict_span(K).dim()
+ sage: K.dim() == rho(K).dim()
True
Nor should it affect the lineality of a cone::
sage: set_random_seed()
sage: K = random_cone(max_dim = 8)
- sage: lineality(K) == lineality(restrict_span(K))
+ sage: lineality(K) == lineality(rho(K))
True
No matter which space we restrict to, the lineality should not
sage: set_random_seed()
sage: K = random_cone(max_dim = 8)
- sage: lineality(K) >= lineality(restrict_span(K))
+ sage: lineality(K) >= lineality(rho(K))
True
- sage: lineality(K) >= lineality(restrict_span(K, K.dual()))
+ sage: lineality(K) >= lineality(rho(K, K.dual()))
True
If we do this according to our paper, then the result is proper::
sage: set_random_seed()
sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=False)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
+ sage: K_S = rho(K)
+ sage: P = rho(K_S.dual()).dual()
sage: P.is_proper()
True
- sage: P = restrict_span(K_S, K_S.dual())
+ sage: P = rho(K_S, K_S.dual())
sage: P.is_proper()
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=False)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
+ sage: K_S = rho(K)
+ sage: P = rho(K_S.dual()).dual()
sage: P.is_proper()
True
- sage: P = restrict_span(K_S, K_S.dual())
+ sage: P = rho(K_S, K_S.dual())
sage: P.is_proper()
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8, strictly_convex=False, solid=True)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
+ sage: K_S = rho(K)
+ sage: P = rho(K_S.dual()).dual()
sage: P.is_proper()
True
- sage: P = restrict_span(K_S, K_S.dual())
+ sage: P = rho(K_S, K_S.dual())
sage: P.is_proper()
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8, strictly_convex=True, solid=True)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
+ sage: K_S = rho(K)
+ sage: P = rho(K_S.dual()).dual()
sage: P.is_proper()
True
- sage: P = restrict_span(K_S, K_S.dual())
+ sage: P = rho(K_S, K_S.dual())
sage: P.is_proper()
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=False)
- sage: K_W = restrict_span(K, K.dual())
- sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: K_W = rho(K, K.dual())
+ sage: K_star_W_star = rho(K.dual()).dual()
sage: basically_the_same(K_W, K_star_W_star)
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=False)
- sage: K_W = restrict_span(K, K.dual())
- sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: K_W = rho(K, K.dual())
+ sage: K_star_W_star = rho(K.dual()).dual()
sage: basically_the_same(K_W, K_star_W_star)
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8, solid=False, strictly_convex=True)
- sage: K_W = restrict_span(K, K.dual())
- sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: K_W = rho(K, K.dual())
+ sage: K_star_W_star = rho(K.dual()).dual()
sage: basically_the_same(K_W, K_star_W_star)
True
sage: set_random_seed()
sage: K = random_cone(max_dim = 8, solid=True, strictly_convex=True)
- sage: K_W = restrict_span(K, K.dual())
- sage: K_star_W_star = restrict_span(K.dual()).dual()
+ sage: K_W = rho(K, K.dual())
+ sage: K_star_W_star = rho(K.dual()).dual()
sage: basically_the_same(K_W, K_star_W_star)
True
if K2 is None:
K2 = K
- phi,_ = ips_iso(K2)
- (W, W_perp) = iso_space(K2).cartesian_factors()
+ # First we project K onto the span of K2. This can be done with
+ # cones (i.e. without converting to vector spaces), but it's
+ # annoying to deal with lattice mismatches.
+ span_K2 = Cone(K2.rays() + (-K2).rays(), lattice=K.lattice())
+ K = K.intersection(span_K2)
- ray_pairs = [ phi(r) for r in K.rays() ]
-
- # Shouldn't matter?
- #
- #if any([ w2 != W_perp.zero() for (_, w2) in ray_pairs ]):
- # msg = 'Cone has nonzero components in W-perp!'
- # raise ValueError(msg)
+ V = K.lattice().vector_space()
- # Represent the cone in terms of a basis for W, i.e. with smaller
- # vectors.
- ws = [ W.coordinate_vector(w1) for (w1, _) in ray_pairs ]
+ # Create the space W \times W^{\perp} isomorphic to V.
+ # First we get an orthogonal (but not normal) basis...
+ W_basis = drop_dependent(K2.rays())
+ W = V.subspace_with_basis(W_basis)
- L = ToricLattice(W.dimension())
+ # We've already intersected K with the span of K2, so every
+ # generator of K should belong to W now.
+ W_rays = [ W.coordinate_vector(r) for r in K.rays() ]
- return Cone(ws, lattice=L)
+ L = ToricLattice(K2.dim())
+ return Cone(W_rays, lattice=L)
sage: set_random_seed()
sage: K = random_cone(max_dim=8)
sage: actual = lyapunov_rank(K)
- sage: K_S = restrict_span(K)
- sage: P = restrict_span(K_S.dual()).dual()
+ sage: K_S = rho(K)
+ sage: P = rho(K_S.dual()).dual()
sage: l = lineality(K)
sage: c = codim(K)
sage: expected = lyapunov_rank(P) + K.dim()*(l + c) + c**2
if m < n:
# K is not solid, project onto its span.
- K = restrict_span(K)
+ K = rho(K)
# Lemma 2
beta += m*(n - m) + (n - m)**2
if l > 0:
# K is not pointed, project its dual onto its span.
# Uses a proposition from our paper, i.e. this is
- # equivalent to K = restrict_span(K.dual()).dual()
- #K = restrict_span(intersect_span(K,K.dual()), K.dual())
- K = restrict_span(K, K.dual())
+ # equivalent to K = rho(K.dual()).dual()
+ K = rho(K, K.dual())
# Lemma 3
beta += m * l
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