def line_vertices(self):
r"""
Return the vertices for each line in a 3-d acyclic vector_configuration.
NOTE:
An acyclic vector configuration corresponds to the normal vectors of
a hyperplane arrangement with a selected base region.
OUTPUT:
A list of pairs.
EXAMPLES::
sage: from cn_hyperarr.vector_classes import *
sage: vc = VectorConfiguration([[1,0,0],[0,1,0],[0,0,1],[1,1,0],[0,1,1],[1,1,1]])
sage: vc.line_vertices()
[(0, 1), (1, 2), (2, 3), (0, 4)]
The vectors [-1,0,1] and [1,0,1] are the vertices of the line containing
[0,0,1]::
sage: vc = VectorConfiguration([[0,0,1],[-1,0,1],[1,0,1]])
sage: vc.line_vertices()
[(1, 2)]
"""
assert self.is_acyclic(), "The vector configuration should be acyclic"
nb_pts = self.n_vectors()
cocircuits = self.three_dim_cocircuits()
list_verts = []
for coc in cocircuits:
zero_indices = coc[1]
if len(zero_indices) > 2:
the_cone = Polyhedron(rays=[self[j] for j in zero_indices],
backend=self._backend)
the_verts = tuple([
_ for _ in zero_indices
if not the_cone.relative_interior_contains(self[_])
])
assert len(the_verts) == 2, "problem with the cocircuits"
if the_verts not in list_verts:
list_verts += [the_verts]
return list_verts
def dominating_pairs(self):
r"""
Return the dominating pairs for each point in a 3-d acyclic vector_configuration.
A point p has a dominating pair (a,b) when p lies in the positive
linear span of a and b.
OUTPUT:
A dictionary where keys are the indices of the vectors and the values
are the associated dominating pairs.
EXAMPLES::
sage: from cn_hyperarr.vector_classes import *
sage: vc = VectorConfiguration([[1,0,0],[0,1,0],[0,0,1],[1,1,0],[0,1,1],[1,1,1]])
sage: vc.dominating_pairs()
{0: set(), 1: set(), 2: set(), 3: {(0, 1)}, 4: {(1, 2)}, 5: {(0, 4), (2, 3)}}
sage: tau = AA((1+sqrt(5))/2)
sage: ncn = [[2*tau+1,2*tau,tau],[2*tau+2,2*tau+1,tau+1],[1,1,1],[tau+1,tau+1,tau],[2*tau,2*tau,tau],[tau+1,tau+1,1],[1,1,0],[0,1,0],[1,0,0],[tau+1,tau,tau]]
sage: ncn_conf = VectorConfiguration(ncn);
sage: ncn_conf.dominating_pairs()
{0: {(4, 8), (6, 9)},
1: {(0, 2), (3, 8), (5, 9)},
2: set(),
3: {(2, 6), (7, 9)},
4: {(1, 7), (2, 6)},
5: {(0, 7), (2, 6)},
6: {(7, 8)},
7: set(),
8: set(),
9: {(2, 8)}}
The vector [0,0,1] is in the positive linear span of [-1,0,1] and
[1,0,1]::
sage: vc = VectorConfiguration([[0,0,1],[-1,0,1],[1,0,1]])
sage: vc.dominating_pairs()
{0: {(1, 2)},
1: set(),
2: set()}
"""
assert self.is_acyclic(), "The vector configuration should be acyclic"
nb_pts = self.n_vectors()
cocircuits = self.three_dim_cocircuits()
dom_dict = {}
for index in range(nb_pts):
the_dominating_pairs = set()
my_v = self[index]
# We only consider the cocircuits that are zero at given index and contains at least 3 zero elements.
relevant_cocircuits = (coc for coc in cocircuits
if index in coc[1] and len(coc[1]) >= 3)
for r_coc in relevant_cocircuits:
the_cone = Polyhedron(rays=[self[j] for j in r_coc[1]],
backend=self._backend)
if the_cone.relative_interior_contains(my_v):
# This means that my_v is dominated.
# Now we catch the dominating rays:
# We are interested in the index of
# the rays and not just the ray (which
# may not be equal to the vector we started with
dom_rays = [
j for j in r_coc[1]
if not the_cone.relative_interior_contains(self[j])
]
assert len(dom_rays) == 2, "problem with the cocircuits"
the_dominating_pairs.add(tuple(dom_rays))
dom_dict[index] = the_dominating_pairs
return dom_dict
def is_acyclic(self):
r"""
Return whether ``self`` is acyclic.
A vector configuration is acyclic if the origin is not in the relative
interior of the cone spanned by the vectors.
OUTPUT:
Boolean. Whether ``self`` is acyclic.
.. SEEALSO::
:meth:`is_totally_cyclic`.
EXAMPLES::
sage: from cn_hyperarr.vector_classes import *
sage: vc = VectorConfiguration([[1,0,0],[0,1,0],[0,0,1],[1,1,0],[0,1,1],[1,1,1]])
sage: vc.is_acyclic()
True
sage: v = [[1, -1, -1],
....: [1, 1, -1],
....: [1, 1, 1],
....: [1, -1, 1],
....: [-1, -1, 1],
....: [-1, -1, -1],
....: [-1, 1, -1],
....: [-1, 1, 1]]
sage: vc = VectorConfiguration(v)
sage: vc.is_acyclic()
False
TESTS::
sage: vc = VectorConfiguration([])
sage: vc.is_acyclic()
True
sage: vc2 = VectorConfiguration([[1,0,0],[-1,0,0]])
sage: vc2.is_acyclic()
False
sage: vc3 = VectorConfiguration([[0,0,0]])
sage: vc3.is_acyclic()
Traceback (most recent call last):
...
ValueError: PPL::ray(e):
e == 0, but the origin cannot be a ray.
sage: vc = VectorConfiguration([[1,0,0],[0,1,0],[0,0,1],[1,1,0],[0,1,1],[1,1,1],[-1,0,0]])
sage: vc.is_acyclic()
False
"""
acyc_vc = Polyhedron(rays=self, backend=self.backend())
if acyc_vc.relative_interior_contains(
vector([0] * acyc_vc.ambient_dim())):
return False
elif not acyc_vc.lines() == ():
return False
else:
return True