def compute_xsi(self, u):
r"""
EXAMPLES::
sage: from slabbe.word_morphisms import compute_xsi
sage: s = WordMorphism({0:[0,1],1:[1,0]})
sage: compute_xsi(s, Word([0]))
sigma_u= 0->012, 1->02, 2->1
theta_u= 0->011, 1->01, 2->0
psi= 0->(0, 0),(0, 1),(0, 2), 1->(1, 0),(1, 1), 2->(2, 0)
psi*sigma_u= 0->(0, 0),(0, 1),(0, 2),(1, 0),(1, 1),(2, 0), 1->(0, 0),(0, 1),(0, 2),(2, 0), 2->(1, 0),(1, 1)
Finite words over {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)}
[1 0 0]
[1 0 0]
[1 0 0]
[0 1 0]
[0 1 0]
[0 0 1]
We want zeta such that:
zeta((0, 0),(0, 1),(0, 2)) = (0, 0),(0, 1),(0, 2),(1, 0),(1, 1),(2, 0)
zeta((1, 0),(1, 1)) = (0, 0),(0, 1),(0, 2),(2, 0)
zeta((2, 0)) = (1, 0),(1, 1)
"""
sigma_u, theta_u = return_substitution(self, u, coding=True)
assert theta_u * sigma_u == self * theta_u, "identity is not verified"
print "sigma_u=", sigma_u
print "theta_u=", theta_u
d = {
k: [(k, i) for i in range(len(v))]
for k, v in theta_u._morph.iteritems()
}
psi = WordMorphism(d)
print "psi=", psi
print "psi*sigma_u=", psi * sigma_u
print psi.codomain()
print psi.incidence_matrix()
print "We want zeta such that:"
for k, v in psi._morph.iteritems():
print "zeta({}) = {}".format(v, psi(sigma_u(k)))
def compute_xsi(self, u):
r"""
EXAMPLES::
sage: from slabbe.word_morphisms import compute_xsi
sage: s = WordMorphism({0:[0,1],1:[1,0]})
sage: compute_xsi(s, Word([0]))
sigma_u= 0->012, 1->02, 2->1
theta_u= 0->011, 1->01, 2->0
psi= 0->(0, 0),(0, 1),(0, 2), 1->(1, 0),(1, 1), 2->(2, 0)
psi*sigma_u= 0->(0, 0),(0, 1),(0, 2),(1, 0),(1, 1),(2, 0), 1->(0, 0),(0, 1),(0, 2),(2, 0), 2->(1, 0),(1, 1)
Finite words over {(0, 0), (0, 1), (0, 2), (1, 0), (1, 1), (2, 0)}
[1 0 0]
[1 0 0]
[1 0 0]
[0 1 0]
[0 1 0]
[0 0 1]
We want zeta such that:
zeta((0, 0),(0, 1),(0, 2)) = (0, 0),(0, 1),(0, 2),(1, 0),(1, 1),(2, 0)
zeta((1, 0),(1, 1)) = (0, 0),(0, 1),(0, 2),(2, 0)
zeta((2, 0)) = (1, 0),(1, 1)
"""
sigma_u, theta_u = return_substitution(self, u, coding=True)
assert theta_u*sigma_u == self*theta_u, "identity is not verified"
print("sigma_u=", sigma_u)
print("theta_u=", theta_u)
d = {k:[(k,i) for i in range(len(v))] for k,v in theta_u._morph.iteritems()}
psi = WordMorphism(d)
print("psi=", psi)
print("psi*sigma_u=", psi*sigma_u)
print(psi.codomain())
print(psi.incidence_matrix())
print("We want zeta such that:")
for k,v in psi._morph.iteritems():
print("zeta({}) = {}".format(v, psi(sigma_u(k))))
class GeoSub(SageObject):
r"""
INPUT:
- ``sigma`` -- dict, substitution
- ``k`` -- integer
- ``presuf`` -- string (default: ``"prefix"``), ``"prefix"`` or ``"suffix"``
- ``dual`` -- bool (default: ``False``)
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub, 2)
sage: E
E_2(1->12, 2->13, 3->1)
"""
def __init__(self, sigma, k, presuf='prefix', dual=False):
self._sigma_dict = sigma
self._sigma = WordMorphism(sigma)
self._k = k
if not presuf in ['prefix', 'suffix']:
raise ValueError('Input presuf(={}) should be "prefix" or'
' "suffix"'.format(presuf))
self._presuf = presuf
self._dual = dual
def is_dual(self):
return self._dual
@cached_method
def field(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.field()
Number Field in b with defining polynomial x^3 - x^2 - x - 1
"""
M = self._sigma.incidence_matrix()
b1 = max(M.eigenvalues(), key=abs)
f = b1.minpoly()
K = NumberField(f, 'b')
return K
def gen(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.gen()
b^2 - b - 1
"""
b = self.field().gen()
if self.is_dual():
return b
else:
return b**-1
def dominant_left_eigenvector(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.dominant_left_eigenvector()
(1, b - 1, b^2 - b - 1)
"""
M = self._sigma.incidence_matrix() - self.field().gen()
return M.left_kernel().basis()[0]
def dominant_right_eigenvector(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.dominant_right_eigenvector()
(1, b^2 - b - 1, -b^2 + 2*b)
"""
M = self._sigma.incidence_matrix() - self.field().gen()
return M.right_kernel().basis()[0]
def complex_embeddings(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.complex_embeddings()
[-0.419643377607081 - 0.606290729207199*I,
-0.419643377607081 + 0.606290729207199*I,
1.83928675521416]
"""
return self.field().gen().complex_embeddings()
def contracting_eigenvalues_indices(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.contracting_eigenvalues_indices()
[0, 1]
"""
L = self.complex_embeddings()
return [L.index(x) for x in L if abs(x) < 1]
def dilating_eigenvalues_indices(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.dilating_eigenvalues_indices()
[2]
"""
L = self.complex_embeddings()
return [L.index(x) for x in L if abs(x) > 1]
def minkowski_embedding_with_left_eigenvector(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.minkowski_embedding_with_left_eigenvector()
[ -1.00000000000000 -0.839286755214161 -0.543689012692076]
[ -1.00000000000000 1.41964337760708 0.771844506346038]
[ 0.000000000000000 -0.606290729207199 1.11514250803994]
::
sage: E = GeoSub(sub, 2, dual=True)
sage: E.minkowski_embedding_with_left_eigenvector()
[ 1.00000000000000 0.839286755214161 0.543689012692076]
[ 1.00000000000000 -1.41964337760708 -0.771844506346038]
[ 0.000000000000000 0.606290729207199 -1.11514250803994]
"""
K = self.field()
if self.is_dual():
vb = self.dominant_left_eigenvector()
else:
vb = -self.dominant_left_eigenvector()
return Minkowski_embedding_without_sqrt2(K, vb)
def projection(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: GeoSub(sub, 2).projection()
[ -1.000000000000000000000000000000
-0.8392867552141611325518525646713
-0.5436890126920763615708559718500]
sage: GeoSub(sub, 2, dual=True).projection()
[ 1.00000000000000 -1.41964337760708 -0.771844506346038]
[ 0.000000000000000 0.606290729207199 -1.11514250803994]
"""
K = self.field()
vb = self.dominant_left_eigenvector()
P, Q = Minkowski_projection_pair(K, vb)
if self.is_dual():
return Q
else:
return -P
@cached_method
def base_iter(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.base_iter()
{(1, 2): [[(0, 0, 0), (1,)],
[(1, 0, 0), (2,)],
[(0, 0, 0), (1, 1)],
[(1, 0, 0), (1, 3)],
[(1, 0, 0), (2, 1)],
[(2, 0, 0), (2, 3)]],
(1, 3): [[(0, 0, 0), (1,)],
[(1, 0, 0), (2,)],
[(0, 0, 0), (1, 1)],
[(1, 0, 0), (2, 1)]],
(2, 3): [[(0, 0, 0), (1,)],
[(1, 0, 0), (3,)],
[(0, 0, 0), (1, 1)],
[(1, 0, 0), (3, 1)]]}
"""
X = {}
S = self._sigma_dict.keys()
for x in combinations(S, self._k):
X[x] = []
bigL = []
for y in x:
if self.is_dual():
bigL.append(
ps_automaton_inverted(self._sigma_dict,
self._presuf)[y])
else:
bigL.append(
ps_automaton(self._sigma_dict, self._presuf)[y])
Lpro = list(product(*bigL))
for el in Lpro:
z = []
w = []
for i in range(len(el)):
z += el[i][1]
w.append(el[i][0])
if self.is_dual():
M = self._sigma.incidence_matrix()
X[x].append([-M.inverse() * abelian(z, S), tuple(w)])
else:
X[x].append([abelian(z, S), tuple(w)])
return X
def __call__(self, patch, iterations=1):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub, kPatch, kFace
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: geosub = GeoSub(sub,2, 'prefix',1)
sage: P = kPatch([kFace((0,0,0),(1,2),dual=True),
....: kFace((0,0,1),(1,3),dual=True),
....: kFace((0,1,0),(2,1),dual=True),
....: kFace((0,0,0),(3,1),dual=True)])
sage: Q = geosub(P, 6)
sage: Q
Patch of 47 faces
"""
if iterations == 0:
return kPatch(patch)
elif iterations < 0:
raise ValueError("iterations (=%s) must be >= 0." % iterations)
else:
old_faces = patch
for i in range(iterations):
new_faces = kPatch([])
for f, m in old_faces:
new_faces += m * kPatch(
self._call_on_face(f, color=f.color()))
old_faces = new_faces
return new_faces
def matrix(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
sage: E.matrix()
[1 1 1]
[1 0 0]
[0 1 0]
"""
if self.is_dual():
return self._sigma.incidence_matrix().inverse()
else:
return self._sigma.incidence_matrix()
def _call_on_face(self, face, color=None):
r"""
INPUT:
- ``face`` -- a face
- ``color`` -- a color or None
OUTPUT:
dict of the form ``{face:multiplicity for face in faces}``
EXAMPLES::
sage: from EkEkstar import GeoSub, kFace
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: E = GeoSub(sub,2)
The available face type are::
sage: E.base_iter().keys()
[(1, 2), (1, 3), (2, 3)]
For each we get::
sage: d = E._call_on_face(kFace((10,11,12), (1,2)))
sage: sorted(d.items())
[([(34, 10, 11), (1, 3)], 1),
([(33, 10, 11), (1, 1)], 1),
([(34, 10, 11), (2, 1)], 1),
([(35, 10, 11), (2, 3)], 1)]
sage: sorted(E._call_on_face(kFace((10,11,12), (1,3))).items())
[([(34, 10, 11), (2, 1)], 1), ([(33, 10, 11), (1, 1)], 1)]
sage: E._call_on_face(kFace((10,11,12), (2,3)))
{[(33, 10, 11), (1, 1)]: 1, [(34, 10, 11), (3, 1)]: 1}
"""
x_new = self.matrix() * face.vector()
#t = face.type()
t = face.sorted_type()
if self.is_dual():
return {
kFace(x_new - vv, tt, dual=self.is_dual()):
(-1)**(sum(t) + sum(tt)) * face.sign()
for (vv, tt) in self.base_iter()[t] if len(tt) == self._k
}
else:
return {
kFace(x_new + vv, tt, dual=self.is_dual()): face.sign()
for (vv, tt) in self.base_iter()[t] if len(tt) == self._k
}
def __repr__(self):
r"""
EXAMPLES::
sage: from EkEkstar import GeoSub
sage: sub = {1:[1,2], 2:[1,3], 3:[1]}
sage: GeoSub(sub, 2)
E_2(1->12, 2->13, 3->1)
sage: GeoSub(sub, 2, dual=True)
E*_2(1->12, 2->13, 3->1)
"""
if self.is_dual():
return "E*_%s(%s)" % (self._k, str(self._sigma))
else:
return "E_%s(%s)" % (self._k, str(self._sigma))
