def _from_matrix(cls, data, coxeter_type, index_set, coxeter_type_check):
"""
Initiate the Coxeter matrix from a matrix.
TESTS::
sage: CM = CoxeterMatrix([[1,2],[2,1]]); CM
[1 2]
[2 1]
sage: CM = CoxeterMatrix([[1,-1],[-1,1]]); CM
[ 1 -1]
[-1 1]
sage: CM = CoxeterMatrix([[1,-1.5],[-1.5,1]]); CM
[ 1.00000000000000 -1.50000000000000]
[-1.50000000000000 1.00000000000000]
sage: CM = CoxeterMatrix([[1,-3/2],[-3/2,1]]); CM
[ 1 -3/2]
[-3/2 1]
sage: CM = CoxeterMatrix([[1,-3/2,5],[-3/2,1,-1],[5,-1,1]]); CM
[ 1 -3/2 5]
[-3/2 1 -1]
[ 5 -1 1]
sage: CM = CoxeterMatrix([[1,-3/2,5],[-3/2,1,oo],[5,oo,1]]); CM
[ 1 -3/2 5]
[-3/2 1 -1]
[ 5 -1 1]
"""
# Check that the data is valid
check_coxeter_matrix(data)
M = matrix(data)
n = M.ncols()
base_ring = M.base_ring()
if not coxeter_type:
if n == 1:
coxeter_type = CoxeterType(['A', 1])
elif coxeter_type_check:
coxeter_type = recognize_coxeter_type_from_matrix(M, index_set)
else:
coxeter_type = None
raw_data = M.list()
mat = typecall(cls, MatrixSpace(base_ring, n, sparse=False), raw_data,
coxeter_type, index_set)
mat._subdivisions = M._subdivisions
return mat
def _from_matrix(cls, data, coxeter_type, index_set, coxeter_type_check):
"""
Initiate the Coxeter matrix from a matrix.
TESTS::
sage: CM = CoxeterMatrix([[1,2],[2,1]]); CM
[1 2]
[2 1]
sage: CM = CoxeterMatrix([[1,-1],[-1,1]]); CM
[ 1 -1]
[-1 1]
sage: CM = CoxeterMatrix([[1,-1.5],[-1.5,1]]); CM
[ 1.00000000000000 -1.50000000000000]
[-1.50000000000000 1.00000000000000]
sage: CM = CoxeterMatrix([[1,-3/2],[-3/2,1]]); CM
[ 1 -3/2]
[-3/2 1]
sage: CM = CoxeterMatrix([[1,-3/2,5],[-3/2,1,-1],[5,-1,1]]); CM
[ 1 -3/2 5]
[-3/2 1 -1]
[ 5 -1 1]
sage: CM = CoxeterMatrix([[1,-3/2,5],[-3/2,1,oo],[5,oo,1]]); CM
[ 1 -3/2 5]
[-3/2 1 -1]
[ 5 -1 1]
"""
# Check that the data is valid
check_coxeter_matrix(data)
M = matrix(data)
n = M.ncols()
base_ring = M.base_ring()
if not coxeter_type:
if n == 1:
coxeter_type = CoxeterType(['A', 1])
elif coxeter_type_check:
coxeter_type = recognize_coxeter_type_from_matrix(M, index_set)
else:
coxeter_type = None
raw_data = M.list()
mat = typecall(cls, MatrixSpace(base_ring, n, sparse=False), raw_data,
coxeter_type, index_set)
mat._subdivisions = M._subdivisions
return mat
def __classcall__(cls, *args, **options):
"""
Constructs a new object of this class or reuse an existing one.
See also :class:`UniqueRepresentation` for a discussion.
EXAMPLES::
sage: x = UniqueRepresentation()
sage: y = UniqueRepresentation()
sage: x is y
True
"""
instance = typecall(cls, *args, **options)
assert isinstance( instance, cls )
if instance.__class__.__reduce__ == UniqueRepresentation.__reduce__:
instance._reduction = (cls, args, options)
return instance
def __classcall_private__(cls, dynamical_system, domain=None):
"""
Return the approapriate dynamical system on projective Berkovich space over ``Cp``.
EXAMPLES::
sage: P1.<x,y> = ProjectiveSpace(Qp(3), 1)
sage: from sage.dynamics.arithmetic_dynamics.berkovich_ds import DynamicalSystem_Berkovich_projective
sage: DynamicalSystem_Berkovich_projective([y, x])
Dynamical system of Projective Berkovich line over Cp(3) of precision 20
induced by the map
Defn: Defined on coordinates by sending (x : y) to
(y : x)
"""
if not isinstance(dynamical_system, DynamicalSystem):
if not isinstance(dynamical_system, DynamicalSystem_projective):
dynamical_system = DynamicalSystem_projective(dynamical_system)
else:
raise TypeError(
'affine dynamical system passed to projective constructor')
R = dynamical_system.base_ring()
morphism_domain = dynamical_system.domain()
if not is_ProjectiveSpace(morphism_domain):
raise TypeError(
'the domain of dynamical_system must be projective space, not %s'
% morphism_domain)
if morphism_domain.dimension_relative() != 1:
raise ValueError('domain was not relative dimension 1')
if not isinstance(R, pAdicBaseGeneric):
if domain is None:
raise TypeError('dynamical system defined over %s, not p-adic, ' %morphism_domain.base_ring() + \
'and domain is None')
if not isinstance(domain, Berkovich_Cp_Projective):
raise TypeError(
'domain was %s, not a projective Berkovich space over Cp' %
domain)
if domain.base() != morphism_domain:
raise ValueError('base of domain was %s, with coordinate ring %s ' %(domain.base(), \
domain.base().coordinate_ring())+ 'while dynamical_system acts on %s, ' %morphism_domain + \
'with coordinate ring %s' %morphism_domain.coordinate_ring())
else:
domain = Berkovich_Cp_Projective(morphism_domain)
return typecall(cls, dynamical_system, domain)
def factory(self, module, grading):
if module._yacop_grading is grading:
return module
# this code is modelled on the unpickling of unique representation objects:
# we create a clone of the module which we can modify, and we also
# refine the class to SuspendedObjects if that has been requested
cls, args, options = module._reduction
if not hasattr(cls, self._functor_category):
# create a dummy SuspendedObjects class - is this a good idea?
# setattr(cls, self._functor_category,cls)
pass
else:
cls = getattr(cls, self._functor_category)
from copy import copy
dct = copy(options)
dct["category"] = module.category().SuspendedObjects()
N = typecall(cls, *args, **dct)
# set the grading and the new name
N._yacop_grading = grading
N._yacop_ref = module
N.rename(grading._format_(module))
# a couple of hacks follow:
# - fix variable injection
if hasattr(N, "gens"):
def gens():
return module.gens()
setattr(N, "gens", gens)
if hasattr(N, "variables"):
def variables():
return module.variables()
setattr(N, "variables", variables)
return N
def __classcall_private__(cls, dynamical_system, domain=None):
"""
Return the appropriate dynamical system on affine Berkovich space over ``Cp``.
EXAMPLES::
sage: A.<x> = AffineSpace(Qp(3), 1)
sage: from sage.dynamics.arithmetic_dynamics.berkovich_ds import DynamicalSystem_Berkovich_affine
sage: DynamicalSystem_Berkovich_affine(DynamicalSystem_affine(x^2))
Dynamical system of Affine Berkovich line over Cp(3) of precision 20 induced by the map
Defn: Defined on coordinates by sending (x) to
(x^2)
"""
if not isinstance(dynamical_system, DynamicalSystem):
if not isinstance(dynamical_system, DynamicalSystem_affine):
dynamical_system = DynamicalSystem_projective(dynamical_system)
else:
raise TypeError(
'projective dynamical system passed to affine constructor')
R = dynamical_system.base_ring()
morphism_domain = dynamical_system.domain()
if not is_AffineSpace(morphism_domain):
raise TypeError(
'the domain of dynamical_system must be affine space, not %s' %
morphism_domain)
if morphism_domain.dimension_relative() != 1:
raise ValueError('domain not relative dimension 1')
if not isinstance(R, pAdicBaseGeneric):
if domain is None:
raise TypeError('dynamical system defined over %s, not padic, ' %morphism_domain.base_ring() + \
'and domain was not specified')
if not isinstance(domain, Berkovich_Cp_Affine):
raise TypeError(
'domain was %s, not an affine Berkovich space over Cp' %
domain)
else:
domain = Berkovich_Cp_Affine(morphism_domain.base_ring())
return typecall(cls, dynamical_system, domain)
def __classcall_private__(cls,
data=None,
index_set=None,
coxeter_type=None,
cartan_type=None,
coxeter_type_check=True):
r"""
A Coxeter matrix can we created via a graph, a Coxeter type, or
a matrix.
.. NOTE::
To disable the Coxeter type check, use the optional argument
``coxeter_type_check = False``.
EXAMPLES::
sage: C = CoxeterMatrix(['A',1,1],['a','b'])
sage: C2 = CoxeterMatrix([[1, -1], [-1, 1]])
sage: C3 = CoxeterMatrix(matrix([[1, -1], [-1, 1]]), [0, 1])
sage: C == C2 and C == C3
True
Check with `\infty` because of the hack of using `-1` to represent
`\infty` in the Coxeter matrix::
sage: G = Graph([(0, 1, 3), (1, 2, oo)])
sage: W1 = CoxeterMatrix([[1, 3, 2], [3, 1, -1], [2, -1, 1]])
sage: W2 = CoxeterMatrix(G)
sage: W1 == W2
True
sage: CoxeterMatrix(W1.coxeter_graph()) == W1
True
The base ring of the matrix depends on the entries given::
sage: CoxeterMatrix([[1,-1],[-1,1]])._matrix.base_ring()
Integer Ring
sage: CoxeterMatrix([[1,-3/2],[-3/2,1]])._matrix.base_ring()
Rational Field
sage: CoxeterMatrix([[1,-1.5],[-1.5,1]])._matrix.base_ring()
Real Field with 53 bits of precision
"""
if not data:
if coxeter_type:
data = CoxeterType(coxeter_type)
elif cartan_type:
data = CoxeterType(CartanType(cartan_type))
# Special cases with no arguments passed
if not data:
data = []
n = 0
index_set = tuple()
coxeter_type = None
base_ring = ZZ
mat = typecall(cls, MatrixSpace(base_ring, n, sparse=False), data,
coxeter_type, index_set)
mat._subdivisions = None
return mat
if isinstance(data, CoxeterMatrix): # Initiate from itself
return data
# Initiate from a graph:
# TODO: Check if a CoxeterDiagram once implemented
if isinstance(data, Graph):
return cls._from_graph(data, coxeter_type_check)
# Get the Coxeter type
coxeter_type = None
from sage.combinat.root_system.cartan_type import CartanType_abstract
if isinstance(data, CartanType_abstract):
coxeter_type = data.coxeter_type()
else:
try:
coxeter_type = CoxeterType(data)
except (TypeError, ValueError, NotImplementedError):
pass
# Initiate from a Coxeter type
if coxeter_type:
return cls._from_coxetertype(coxeter_type)
# TODO:: remove when oo is possible in matrices.
n = len(data[0])
data = [x if x != infinity else -1 for r in data for x in r]
data = matrix(n, n, data)
# until here
# Get the index set
if index_set:
index_set = tuple(index_set)
else:
index_set = tuple(range(1, n + 1))
if len(set(index_set)) != n:
raise ValueError("the given index set is not valid")
return cls._from_matrix(data, coxeter_type, index_set,
coxeter_type_check)
def __classcall_private__(cls, *args, **kwds):
"""
Normalize input so we can inherit from spare integer matrix.
.. NOTE::
To disable the Cartan type check, use the optional argument
``cartan_type_check = False``.
EXAMPLES::
sage: C = CartanMatrix(['A',1,1])
sage: C2 = CartanMatrix([[2, -2], [-2, 2]])
sage: C3 = CartanMatrix(matrix([[2, -2], [-2, 2]]), [0, 1])
sage: C == C2 and C == C3
True
TESTS:
Check that :trac:`15740` is fixed::
sage: d = DynkinDiagram()
sage: d.add_edge('a', 'b', 2)
sage: d.index_set()
('a', 'b')
sage: cm = CartanMatrix(d)
sage: cm.index_set()
('a', 'b')
"""
# Special case with 0 args and kwds has Cartan type
if "cartan_type" in kwds and len(args) == 0:
args = (CartanType(kwds["cartan_type"]),)
if len(args) == 0:
data = []
n = 0
index_set = tuple()
cartan_type = None
subdivisions = None
elif len(args) == 4 and isinstance(args[0], MatrixSpace): # For pickling
return typecall(cls, args[0], args[1], args[2], args[3])
elif isinstance(args[0], CartanMatrix):
return args[0]
else:
cartan_type = None
dynkin_diagram = None
subdivisions = None
from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class
if isinstance(args[0], DynkinDiagram_class):
dynkin_diagram = args[0]
cartan_type = args[0]._cartan_type
else:
try:
cartan_type = CartanType(args[0])
dynkin_diagram = cartan_type.dynkin_diagram()
except (TypeError, ValueError):
pass
if dynkin_diagram is not None:
n = dynkin_diagram.rank()
index_set = dynkin_diagram.index_set()
reverse = dict((index_set[i], i) for i in range(len(index_set)))
data = {(i, i): 2 for i in range(n)}
for (i,j,l) in dynkin_diagram.edge_iterator():
data[(reverse[j], reverse[i])] = -l
else:
M = matrix(args[0])
if not is_generalized_cartan_matrix(M):
raise ValueError("the input matrix is not a generalized Cartan matrix")
n = M.ncols()
if "cartan_type" in kwds:
cartan_type = CartanType(kwds["cartan_type"])
elif n == 1:
cartan_type = CartanType(['A', 1])
elif kwds.get("cartan_type_check", True):
cartan_type = find_cartan_type_from_matrix(M)
data = M.dict()
subdivisions = M._subdivisions
if len(args) == 1:
if cartan_type is not None:
index_set = tuple(cartan_type.index_set())
elif dynkin_diagram is None:
index_set = tuple(range(n))
elif len(args) == 2:
index_set = tuple(args[1])
if len(index_set) != n and len(set(index_set)) != n:
raise ValueError("the given index set is not valid")
else:
raise ValueError("too many arguments")
mat = typecall(cls, MatrixSpace(ZZ, n, sparse=True), data, cartan_type, index_set)
mat._subdivisions = subdivisions
return mat
def __classcall_private__(
cls, data=None, index_set=None, coxeter_type=None, cartan_type=None, coxeter_type_check=True
):
r"""
A Coxeter matrix can we created via a graph, a Coxeter type, or
a matrix.
.. NOTE::
To disable the Coxeter type check, use the optional argument
``coxeter_type_check = False``.
EXAMPLES::
sage: C = CoxeterMatrix(['A',1,1],['a','b'])
sage: C2 = CoxeterMatrix([[1, -1], [-1, 1]])
sage: C3 = CoxeterMatrix(matrix([[1, -1], [-1, 1]]), [0, 1])
sage: C == C2 and C == C3
True
Check with `\infty` because of the hack of using `-1` to represent
`\infty` in the Coxeter matrix::
sage: G = Graph([(0, 1, 3), (1, 2, oo)])
sage: W1 = CoxeterMatrix([[1, 3, 2], [3, 1, -1], [2, -1, 1]])
sage: W2 = CoxeterMatrix(G)
sage: W1 == W2
True
sage: CoxeterMatrix(W1.coxeter_graph()) == W1
True
The base ring of the matrix depends on the entries given::
sage: CoxeterMatrix([[1,-1],[-1,1]])._matrix.base_ring()
Integer Ring
sage: CoxeterMatrix([[1,-3/2],[-3/2,1]])._matrix.base_ring()
Rational Field
sage: CoxeterMatrix([[1,-1.5],[-1.5,1]])._matrix.base_ring()
Real Field with 53 bits of precision
"""
if not data:
if coxeter_type:
data = CoxeterType(coxeter_type)
elif cartan_type:
data = CoxeterType(CartanType(cartan_type))
# Special cases with no arguments passed
if not data:
data = []
n = 0
index_set = tuple()
coxeter_type = None
base_ring = ZZ
mat = typecall(cls, MatrixSpace(base_ring, n, sparse=False), data, coxeter_type, index_set)
mat._subdivisions = None
return mat
if isinstance(data, CoxeterMatrix): # Initiate from itself
return data
# Initiate from a graph:
# TODO: Check if a CoxeterDiagram once implemented
if isinstance(data, Graph):
return cls._from_graph(data, coxeter_type_check)
# Get the Coxeter type
coxeter_type = None
from sage.combinat.root_system.cartan_type import CartanType_abstract
if isinstance(data, CartanType_abstract):
coxeter_type = data.coxeter_type()
else:
try:
coxeter_type = CoxeterType(data)
except (TypeError, ValueError, NotImplementedError):
pass
# Initiate from a Coxeter type
if coxeter_type:
return cls._from_coxetertype(coxeter_type)
# TODO:: remove when oo is possible in matrices.
n = len(data[0])
data = [x if x != infinity else -1 for r in data for x in r]
data = matrix(n, n, data)
# until here
# Get the index set
if index_set:
index_set = tuple(index_set)
else:
index_set = tuple(range(1, n + 1))
if len(set(index_set)) != n:
raise ValueError("the given index set is not valid")
return cls._from_matrix(data, coxeter_type, index_set, coxeter_type_check)
def __classcall_private__(cls,
X,
phi,
cache_orbits=False,
create_tuple=False,
inverse=None,
is_finite=None):
"""
Return the correct object based on input.
The main purpose of this method is to decide which
subclass the object will belong to based on the
``inverse`` and ``is_finite`` arguments.
EXAMPLES::
sage: D = DiscreteDynamicalSystem(NN, lambda x: x + 1)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.DiscreteDynamicalSystem'>
sage: f1 = lambda x: (x + 2 if x % 6 < 4 else x - 4)
sage: D = DiscreteDynamicalSystem(NN, f1, inverse=False)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.DiscreteDynamicalSystem'>
sage: D = DiscreteDynamicalSystem(NN, f1, inverse=None)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.DiscreteDynamicalSystem'>
sage: D = DiscreteDynamicalSystem(NN, f1, inverse=True)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.InvertibleDiscreteDynamicalSystem'>
sage: D = DiscreteDynamicalSystem(NN, lambda x: x + 1, is_finite=False)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.DiscreteDynamicalSystem'>
sage: f2 = lambda x: (x + 1 if x < 3 else 1)
sage: f3 = lambda x: (x - 1 if x > 3 else 3)
sage: D = DiscreteDynamicalSystem([1, 2, 3], f2)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.FiniteDynamicalSystem'>
sage: D = DiscreteDynamicalSystem([1, 2, 3], f2, inverse=True)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.InvertibleFiniteDynamicalSystem'>
sage: D = DiscreteDynamicalSystem([1, 2, 3], f2, inverse=f3)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.InvertibleFiniteDynamicalSystem'>
sage: D = DiscreteDynamicalSystem([1, 2, 3], f2, inverse=False)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.FiniteDynamicalSystem'>
sage: D = DiscreteDynamicalSystem([1, 2, 3], f2, inverse=None)
sage: parent(D)
<class 'sage.dynamics.finite_dynamical_system.FiniteDynamicalSystem'>
"""
if is_finite is None:
is_finite = (X in Sets().Finite()
or isinstance(X, (list, tuple, set, frozenset)))
if inverse:
if inverse is True: # invertibility claimed, but inverse not provided
# This is how the input for these subclasses work
inverse = None
ret_cls = (InvertibleFiniteDynamicalSystem
if is_finite else InvertibleDiscreteDynamicalSystem)
return ret_cls(X,
phi,
cache_orbits=cache_orbits,
create_tuple=create_tuple,
inverse=inverse)
if is_finite:
return FiniteDynamicalSystem(X,
phi,
cache_orbits=cache_orbits,
create_tuple=create_tuple)
return typecall(cls,
X,
phi,
cache_orbits=cache_orbits,
create_tuple=create_tuple)
def __classcall__(cls, id=None, **kwds):
r"""
Construct a new texture by id.
INPUT:
- ``id`` - a texture (optional, default: None), a dict, a color, a
str, a tuple, None or any other type acting as an ID. If ``id`` is
None and keyword ``texture`` is empty, then it returns a unique texture object.
- ``texture`` - a texture
- ``color`` - tuple or str, (optional, default: (.4, .4, 1))
- ``opacity`` - number between 0 and 1 (optional, default: 1)
- ``ambient`` - number (optional, default: 0.5)
- ``diffuse`` - number (optional, default: 1)
- ``specular`` - number (optional, default: 0)
- ``shininess`` - number (optional, default: 1)
- ``name`` - str (optional, default: None)
- ``**kwds`` - other valid keywords
OUTPUT:
A texture object.
EXAMPLES:
Texture from integer ``id``::
sage: from sage.plot.plot3d.texture import Texture
sage: Texture(17)
Texture(17, 6666ff)
Texture from rational ``id``::
sage: Texture(3/4)
Texture(3/4, 6666ff)
Texture from a dict::
sage: Texture({'color':'orange','opacity':0.5})
Texture(texture..., orange, ffa500)
Texture from a color::
sage: c = Color('red')
sage: Texture(c)
Texture(texture..., ff0000)
Texture from a valid string color::
sage: Texture('red')
Texture(texture..., red, ff0000)
Texture from a non valid string color::
sage: Texture('redd')
Texture(redd, 6666ff)
Texture from a tuple::
sage: Texture((.2,.3,.4))
Texture(texture..., 334c66)
Now accepting negative arguments, reduced modulo 1::
sage: Texture((-3/8, 1/2, 3/8))
Texture(texture..., 9f7f5f)
Textures using other keywords::
sage: Texture(specular=0.4)
Texture(texture..., 6666ff)
sage: Texture(diffuse=0.4)
Texture(texture..., 6666ff)
sage: Texture(shininess=0.3)
Texture(texture..., 6666ff)
sage: Texture(ambient=0.7)
Texture(texture..., 6666ff)
"""
if isinstance(id, Texture):
return id
if 'texture' in kwds:
t = kwds['texture']
if isinstance(t, Texture):
return t
else:
raise TypeError("texture keyword must be a texture object")
if isinstance(id, dict):
kwds = id
if 'rgbcolor' in kwds:
kwds['color'] = kwds['rgbcolor']
id = None
elif isinstance(id, Color):
kwds['color'] = id.rgb()
id = None
elif isinstance(id, str) and id in colors:
kwds['color'] = id
id = None
elif isinstance(id, tuple):
kwds['color'] = id
id = None
if id is None:
id = _new_global_texture_id()
return typecall(cls, id, **kwds)
def __classcall_private__(cls, *args, **kwds):
"""
Normalize input so we can inherit from spare integer matrix.
.. NOTE::
To disable the Cartan type check, use the optional argument
``cartan_type_check = False``.
EXAMPLES::
sage: C = CartanMatrix(['A',1,1])
sage: C2 = CartanMatrix([[2, -2], [-2, 2]])
sage: C3 = CartanMatrix(matrix([[2, -2], [-2, 2]]), [0, 1])
sage: C == C2 and C == C3
True
"""
# Special case with 0 args and kwds has cartan type
if "cartan_type" in kwds and len(args) == 0:
args = (CartanType(kwds["cartan_type"]),)
if len(args) == 0:
data = []
n = 0
index_set = tuple()
cartan_type = None
subdivisions = None
elif len(args) == 4 and isinstance(args[0], MatrixSpace): # For pickling
return typecall(cls, args[0], args[1], args[2], args[3])
elif isinstance(args[0], CartanMatrix):
return args[0]
else:
cartan_type = None
dynkin_diagram = None
subdivisions = None
try:
cartan_type = CartanType(args[0])
dynkin_diagram = cartan_type.dynkin_diagram()
except (TypeError, ValueError):
pass
if dynkin_diagram is not None:
n = cartan_type.rank()
index_set = dynkin_diagram.index_set()
reverse = dict((index_set[i], i) for i in range(len(index_set)))
data = {(i, i): 2 for i in range(n)}
for (i,j,l) in dynkin_diagram.edge_iterator():
data[(reverse[j], reverse[i])] = -l
else:
M = matrix(args[0])
if not is_generalized_cartan_matrix(M):
raise ValueError("The input matrix is not a generalized Cartan matrix.")
n = M.ncols()
if "cartan_type" in kwds:
cartan_type = CartanType(kwds["cartan_type"])
elif n == 1:
cartan_type = CartanType(['A', 1])
elif kwds.get("cartan_type_check", True):
cartan_type = find_cartan_type_from_matrix(M)
data = M.dict()
subdivisions = M._subdivisions
if len(args) == 1:
if cartan_type is not None:
index_set = tuple(cartan_type.index_set())
else:
index_set = tuple(range(M.ncols()))
elif len(args) == 2:
index_set = tuple(args[1])
if len(index_set) != n and len(set(index_set)) != n:
raise ValueError("The given index set is not valid.")
else:
raise ValueError("Too many arguments.")
mat = typecall(cls, MatrixSpace(ZZ, n, sparse=True), data, cartan_type, index_set)
mat._subdivisions = subdivisions
return mat
def __classcall_private__(cls, morphism_or_polys, domain=None):
r"""
Return the appropriate dynamical system on an affine scheme.
TESTS::
sage: A.<x> = AffineSpace(ZZ,1)
sage: A1.<z> = AffineSpace(CC,1)
sage: H = End(A1)
sage: f2 = H([z^2+1])
sage: f = DynamicalSystem_affine(f2, A)
sage: f.domain() is A
False
::
sage: P1.<x,y> = ProjectiveSpace(QQ,1)
sage: DynamicalSystem_affine([y, 2*x], domain=P1)
Traceback (most recent call last):
...
ValueError: "domain" must be an affine scheme
sage: H = End(P1)
sage: DynamicalSystem_affine(H([y, 2*x]))
Traceback (most recent call last):
...
ValueError: "domain" must be an affine scheme
"""
if isinstance(morphism_or_polys, SchemeMorphism_polynomial):
morphism = morphism_or_polys
R = morphism.base_ring()
polys = list(morphism)
domain = morphism.domain()
if not is_AffineSpace(domain) and not isinstance(
domain, AlgebraicScheme_subscheme_affine):
raise ValueError('"domain" must be an affine scheme')
if domain != morphism_or_polys.codomain():
raise ValueError('domain and codomain do not agree')
if R not in Fields():
return typecall(cls, polys, domain)
if is_FiniteField(R):
return DynamicalSystem_affine_finite_field(polys, domain)
return DynamicalSystem_affine_field(polys, domain)
elif isinstance(morphism_or_polys, (list, tuple)):
polys = list(morphism_or_polys)
else:
polys = [morphism_or_polys]
# We now arrange for all of our list entries to lie in the same ring
# Fraction field case first
fraction_field = False
for poly in polys:
P = poly.parent()
if is_FractionField(P):
fraction_field = True
break
if fraction_field:
K = P.base_ring().fraction_field()
# Replace base ring with its fraction field
P = P.ring().change_ring(K).fraction_field()
polys = [P(poly) for poly in polys]
else:
# If any of the list entries lies in a quotient ring, we try
# to lift all entries to a common polynomial ring.
quotient_ring = False
for poly in polys:
P = poly.parent()
if is_QuotientRing(P):
quotient_ring = True
break
if quotient_ring:
polys = [P(poly).lift() for poly in polys]
else:
poly_ring = False
for poly in polys:
P = poly.parent()
if is_PolynomialRing(P) or is_MPolynomialRing(P):
poly_ring = True
break
if poly_ring:
polys = [P(poly) for poly in polys]
if domain is None:
f = polys[0]
CR = f.parent()
if CR is SR:
raise TypeError("Symbolic Ring cannot be the base ring")
if fraction_field:
CR = CR.ring()
domain = AffineSpace(CR)
R = domain.base_ring()
if R is SR:
raise TypeError("Symbolic Ring cannot be the base ring")
if not is_AffineSpace(domain) and not isinstance(
domain, AlgebraicScheme_subscheme_affine):
raise ValueError('"domain" must be an affine scheme')
if R not in Fields():
return typecall(cls, polys, domain)
if is_FiniteField(R):
return DynamicalSystem_affine_finite_field(polys, domain)
return DynamicalSystem_affine_field(polys, domain)
def __classcall_private__(cls, data=None, index_set=None,
cartan_type=None, cartan_type_check=True,
borcherds=None):
"""
Normalize input so we can inherit from sparse integer matrix.
.. NOTE::
To disable the Cartan type check, use the optional argument
``cartan_type_check = False``.
EXAMPLES::
sage: C = CartanMatrix(['A',1,1])
sage: C2 = CartanMatrix([[2, -2], [-2, 2]])
sage: C3 = CartanMatrix(matrix([[2, -2], [-2, 2]]), [0, 1])
sage: C == C2 and C == C3
True
TESTS:
Check that :trac:`15740` is fixed::
sage: d = DynkinDiagram()
sage: d.add_edge('a', 'b', 2)
sage: d.index_set()
('a', 'b')
sage: cm = CartanMatrix(d)
sage: cm.index_set()
('a', 'b')
"""
# Special case with 0 args and kwds has Cartan type
if cartan_type is not None and data is None:
data = CartanType(cartan_type)
if data is None:
data = []
n = 0
index_set = tuple()
cartan_type = None
subdivisions = None
elif isinstance(data, CartanMatrix):
if index_set is not None:
d = {a: index_set[i] for i,a in enumerate(data.index_set())}
return data.relabel(d)
return data
else:
dynkin_diagram = None
subdivisions = None
from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class
if isinstance(data, DynkinDiagram_class):
dynkin_diagram = data
cartan_type = data._cartan_type
else:
try:
cartan_type = CartanType(data)
dynkin_diagram = cartan_type.dynkin_diagram()
except (TypeError, ValueError):
pass
if dynkin_diagram is not None:
n = dynkin_diagram.rank()
index_set = dynkin_diagram.index_set()
oir = dynkin_diagram.odd_isotropic_roots()
reverse = {a: i for i,a in enumerate(index_set)}
if isinstance(borcherds, (list, tuple)):
if (len(borcherds) != len(index_set)
and not all(val in ZZ
and (val == 2 or (val % 2 == 0 and val < 0))
for val in borcherds)):
raise ValueError("the input data is not a Borcherds-Cartan matrix")
data = {(i, i): val if index_set[i] not in oir else 0
for i,val in enumerate(borcherds)}
else:
data = {(i, i): 2 if index_set[i] not in oir else 0
for i in range(n)}
for (i,j,l) in dynkin_diagram.edge_iterator():
data[(reverse[j], reverse[i])] = -l
else:
M = matrix(data)
if borcherds:
if not is_borcherds_cartan_matrix(M):
raise ValueError("the input matrix is not a Borcherds-Cartan matrix")
else:
if not is_generalized_cartan_matrix(M):
raise ValueError("the input matrix is not a generalized Cartan matrix")
n = M.ncols()
data = M.dict()
subdivisions = M._subdivisions
if index_set is None:
index_set = tuple(range(n))
else:
index_set = tuple(index_set)
if len(index_set) != n and len(set(index_set)) != n:
raise ValueError("the given index set is not valid")
# We can do the Cartan type initialization later as this is not
# a unique representation
mat = typecall(cls, MatrixSpace(ZZ, n, sparse=True), data, False, True)
# FIXME: We have to initialize the CartanMatrix part separately because
# of the __cinit__ of the matrix. We should get rid of this workaround
mat._CM_init(cartan_type, index_set, cartan_type_check)
mat._subdivisions = subdivisions
return mat
def __classcall_private__(cls, morphism_or_polys, domain=None):
r"""
Return the appropriate dynamical system on an affine scheme.
TESTS::
sage: A.<x> = AffineSpace(ZZ,1)
sage: A1.<z> = AffineSpace(CC,1)
sage: H = End(A1)
sage: f2 = H([z^2+1])
sage: f = DynamicalSystem_affine(f2, A)
sage: f.domain() is A
False
::
sage: P1.<x,y> = ProjectiveSpace(QQ,1)
sage: DynamicalSystem_affine([y, 2*x], domain=P1)
Traceback (most recent call last):
...
ValueError: "domain" must be an affine scheme
sage: H = End(P1)
sage: DynamicalSystem_affine(H([y, 2*x]))
Traceback (most recent call last):
...
ValueError: "domain" must be an affine scheme
::
sage: R.<x,y,z> = QQ[]
sage: f = DynamicalSystem_affine([x+y+z, y*z])
Traceback (most recent call last):
...
ValueError: Number of polys does not match dimension of Affine Space of dimension 3 over Rational Field
::
sage: A.<x,y> = AffineSpace(QQ,2)
sage: f = DynamicalSystem_affine([CC.0*x^2, y^2], domain=A)
Traceback (most recent call last):
...
TypeError: coefficients of polynomial not in Rational Field
"""
if isinstance(morphism_or_polys, SchemeMorphism_polynomial):
morphism = morphism_or_polys
R = morphism.base_ring()
polys = list(morphism)
domain = morphism.domain()
if not is_AffineSpace(domain) and not isinstance(domain, AlgebraicScheme_subscheme_affine):
raise ValueError('"domain" must be an affine scheme')
if domain != morphism_or_polys.codomain():
raise ValueError('domain and codomain do not agree')
if R not in Fields():
return typecall(cls, polys, domain)
if is_FiniteField(R):
return DynamicalSystem_affine_finite_field(polys, domain)
return DynamicalSystem_affine_field(polys, domain)
elif isinstance(morphism_or_polys,(list, tuple)):
polys = list(morphism_or_polys)
else:
polys = [morphism_or_polys]
PR = get_coercion_model().common_parent(*polys)
fraction_field = any(is_FractionField(poly.parent()) for poly in polys)
if fraction_field:
K = PR.base_ring().fraction_field()
# Replace base ring with its fraction field
PR = PR.ring().change_ring(K).fraction_field()
polys = [PR(poly) for poly in polys]
else:
quotient_ring = any(is_QuotientRing(poly.parent()) for poly in polys)
# If any of the list entries lies in a quotient ring, we try
# to lift all entries to a common polynomial ring.
if quotient_ring:
polys = [PR(poly).lift() for poly in polys]
else:
polys = [PR(poly) for poly in polys]
if domain is None:
if PR is SR:
raise TypeError("Symbolic Ring cannot be the base ring")
if fraction_field:
PR = PR.ring()
domain = AffineSpace(PR)
else:
# Check if we can coerce the given polynomials over the given domain
PR = domain.ambient_space().coordinate_ring()
try:
if fraction_field:
PR = PR.fraction_field()
polys = [PR(poly) for poly in polys]
except TypeError:
raise TypeError('coefficients of polynomial not in {}'.format(domain.base_ring()))
if len(polys) != domain.ambient_space().coordinate_ring().ngens():
raise ValueError('Number of polys does not match dimension of {}'.format(domain))
R = domain.base_ring()
if R is SR:
raise TypeError("Symbolic Ring cannot be the base ring")
if not is_AffineSpace(domain) and not isinstance(domain, AlgebraicScheme_subscheme_affine):
raise ValueError('"domain" must be an affine scheme')
if R not in Fields():
return typecall(cls, polys, domain)
if is_FiniteField(R):
return DynamicalSystem_affine_finite_field(polys, domain)
return DynamicalSystem_affine_field(polys, domain)
def __classcall_private__(cls, morphism_or_polys, domain=None):
r"""
Return the appropriate dynamical system on an affine scheme.
TESTS::
sage: A.<x> = AffineSpace(ZZ,1)
sage: A1.<z> = AffineSpace(CC,1)
sage: H = End(A1)
sage: f2 = H([z^2+1])
sage: f = DynamicalSystem_affine(f2, A)
sage: f.domain() is A
False
::
sage: P1.<x,y> = ProjectiveSpace(QQ,1)
sage: DynamicalSystem_affine([y, 2*x], domain=P1)
Traceback (most recent call last):
...
ValueError: "domain" must be an affine scheme
sage: H = End(P1)
sage: DynamicalSystem_affine(H([y, 2*x]))
Traceback (most recent call last):
...
ValueError: "domain" must be an affine scheme
::
sage: R.<x,y,z> = QQ[]
sage: f = DynamicalSystem_affine([x+y+z, y*z])
Traceback (most recent call last):
...
ValueError: Number of polys does not match dimension of Affine Space of dimension 3 over Rational Field
::
sage: A.<x,y> = AffineSpace(QQ,2)
sage: f = DynamicalSystem_affine([CC.0*x^2, y^2], domain=A)
Traceback (most recent call last):
...
TypeError: coefficients of polynomial not in Rational Field
"""
if isinstance(morphism_or_polys, SchemeMorphism_polynomial):
morphism = morphism_or_polys
R = morphism.base_ring()
polys = list(morphism)
domain = morphism.domain()
if not is_AffineSpace(domain) and not isinstance(
domain, AlgebraicScheme_subscheme_affine):
raise ValueError('"domain" must be an affine scheme')
if domain != morphism_or_polys.codomain():
raise ValueError('domain and codomain do not agree')
if R not in Fields():
return typecall(cls, polys, domain)
if is_FiniteField(R):
return DynamicalSystem_affine_finite_field(polys, domain)
return DynamicalSystem_affine_field(polys, domain)
elif isinstance(morphism_or_polys, (list, tuple)):
polys = list(morphism_or_polys)
else:
polys = [morphism_or_polys]
PR = get_coercion_model().common_parent(*polys)
fraction_field = any(is_FractionField(poly.parent()) for poly in polys)
if fraction_field:
K = PR.base_ring().fraction_field()
# Replace base ring with its fraction field
PR = PR.ring().change_ring(K).fraction_field()
polys = [PR(poly) for poly in polys]
else:
quotient_ring = any(
is_QuotientRing(poly.parent()) for poly in polys)
# If any of the list entries lies in a quotient ring, we try
# to lift all entries to a common polynomial ring.
if quotient_ring:
polys = [PR(poly).lift() for poly in polys]
else:
polys = [PR(poly) for poly in polys]
if domain is None:
if PR is SR:
raise TypeError("Symbolic Ring cannot be the base ring")
if fraction_field:
PR = PR.ring()
domain = AffineSpace(PR)
else:
# Check if we can coerce the given polynomials over the given domain
PR = domain.ambient_space().coordinate_ring()
try:
if fraction_field:
PR = PR.fraction_field()
polys = [PR(poly) for poly in polys]
except TypeError:
raise TypeError('coefficients of polynomial not in {}'.format(
domain.base_ring()))
if len(polys) != domain.ambient_space().coordinate_ring().ngens():
raise ValueError(
'Number of polys does not match dimension of {}'.format(
domain))
R = domain.base_ring()
if R is SR:
raise TypeError("Symbolic Ring cannot be the base ring")
if not is_AffineSpace(domain) and not isinstance(
domain, AlgebraicScheme_subscheme_affine):
raise ValueError('"domain" must be an affine scheme')
if R not in Fields():
return typecall(cls, polys, domain)
if is_FiniteField(R):
return DynamicalSystem_affine_finite_field(polys, domain)
return DynamicalSystem_affine_field(polys, domain)
def __classcall__(cls, *args, **options):
instance = typecall(cls, *args, **options)
assert isinstance(instance, cls)
instance._init_args = (cls, args, options)
return instance
def __classcall_private__(cls,
data=None,
index_set=None,
cartan_type=None,
cartan_type_check=True):
"""
Normalize input so we can inherit from sparse integer matrix.
.. NOTE::
To disable the Cartan type check, use the optional argument
``cartan_type_check = False``.
EXAMPLES::
sage: C = CartanMatrix(['A',1,1])
sage: C2 = CartanMatrix([[2, -2], [-2, 2]])
sage: C3 = CartanMatrix(matrix([[2, -2], [-2, 2]]), [0, 1])
sage: C == C2 and C == C3
True
TESTS:
Check that :trac:`15740` is fixed::
sage: d = DynkinDiagram()
sage: d.add_edge('a', 'b', 2)
sage: d.index_set()
('a', 'b')
sage: cm = CartanMatrix(d)
sage: cm.index_set()
('a', 'b')
"""
# Special case with 0 args and kwds has Cartan type
if cartan_type is not None and data is None:
data = CartanType(cartan_type)
if data is None:
data = []
n = 0
index_set = tuple()
cartan_type = None
subdivisions = None
elif isinstance(data, CartanMatrix):
if index_set is not None:
d = {a: index_set[i] for i, a in enumerate(data.index_set())}
return data.relabel(d)
return data
else:
dynkin_diagram = None
subdivisions = None
from sage.combinat.root_system.dynkin_diagram import DynkinDiagram_class
if isinstance(data, DynkinDiagram_class):
dynkin_diagram = data
cartan_type = data._cartan_type
else:
try:
cartan_type = CartanType(data)
dynkin_diagram = cartan_type.dynkin_diagram()
except (TypeError, ValueError):
pass
if dynkin_diagram is not None:
n = dynkin_diagram.rank()
index_set = dynkin_diagram.index_set()
oir = dynkin_diagram.odd_isotropic_roots()
reverse = {a: i for i, a in enumerate(index_set)}
data = {(i, i): 2 if index_set[i] not in oir else 0
for i in range(n)}
for (i, j, l) in dynkin_diagram.edge_iterator():
data[(reverse[j], reverse[i])] = -l
else:
M = matrix(data)
if not is_generalized_cartan_matrix(M):
raise ValueError(
"the input matrix is not a generalized Cartan matrix")
n = M.ncols()
data = M.dict()
subdivisions = M._subdivisions
if index_set is None:
index_set = tuple(range(n))
else:
index_set = tuple(index_set)
if len(index_set) != n and len(set(index_set)) != n:
raise ValueError("the given index set is not valid")
# We can do the Cartan type initialization later as this is not
# a unique representation
mat = typecall(cls, MatrixSpace(ZZ, n, sparse=True), data, False, True)
# FIXME: We have to initialize the CartanMatrix part separately because
# of the __cinit__ of the matrix. We should get rid of this workaround
mat._CM_init(cartan_type, index_set, cartan_type_check)
mat._subdivisions = subdivisions
return mat
def __classcall_private__(cls, morphism_or_polys, domain=None):
r"""
Return the appropriate dynamical system on an affine scheme.
TESTS::
sage: A.<x> = AffineSpace(ZZ,1)
sage: A1.<z> = AffineSpace(CC,1)
sage: H = End(A1)
sage: f2 = H([z^2+1])
sage: f = DynamicalSystem_affine(f2, A)
sage: f.domain() is A
False
::
sage: P1.<x,y> = ProjectiveSpace(QQ,1)
sage: DynamicalSystem_affine([y, 2*x], domain=P1)
Traceback (most recent call last):
...
ValueError: "domain" must be an affine scheme
sage: H = End(P1)
sage: DynamicalSystem_affine(H([y, 2*x]))
Traceback (most recent call last):
...
ValueError: "domain" must be an affine scheme
"""
if isinstance(morphism_or_polys, SchemeMorphism_polynomial):
morphism = morphism_or_polys
R = morphism.base_ring()
polys = list(morphism)
domain = morphism.domain()
if not is_AffineSpace(domain) and not isinstance(domain, AlgebraicScheme_subscheme_affine):
raise ValueError('"domain" must be an affine scheme')
if domain != morphism_or_polys.codomain():
raise ValueError('domain and codomain do not agree')
if R not in Fields():
return typecall(cls, polys, domain)
if is_FiniteField(R):
return DynamicalSystem_affine_finite_field(polys, domain)
return DynamicalSystem_affine_field(polys, domain)
elif isinstance(morphism_or_polys,(list, tuple)):
polys = list(morphism_or_polys)
else:
polys = [morphism_or_polys]
# We now arrange for all of our list entries to lie in the same ring
# Fraction field case first
fraction_field = False
for poly in polys:
P = poly.parent()
if is_FractionField(P):
fraction_field = True
break
if fraction_field:
K = P.base_ring().fraction_field()
# Replace base ring with its fraction field
P = P.ring().change_ring(K).fraction_field()
polys = [P(poly) for poly in polys]
else:
# If any of the list entries lies in a quotient ring, we try
# to lift all entries to a common polynomial ring.
quotient_ring = False
for poly in polys:
P = poly.parent()
if is_QuotientRing(P):
quotient_ring = True
break
if quotient_ring:
polys = [P(poly).lift() for poly in polys]
else:
poly_ring = False
for poly in polys:
P = poly.parent()
if is_PolynomialRing(P) or is_MPolynomialRing(P):
poly_ring = True
break
if poly_ring:
polys = [P(poly) for poly in polys]
if domain is None:
f = polys[0]
CR = f.parent()
if CR is SR:
raise TypeError("Symbolic Ring cannot be the base ring")
if fraction_field:
CR = CR.ring()
domain = AffineSpace(CR)
R = domain.base_ring()
if R is SR:
raise TypeError("Symbolic Ring cannot be the base ring")
if not is_AffineSpace(domain) and not isinstance(domain, AlgebraicScheme_subscheme_affine):
raise ValueError('"domain" must be an affine scheme')
if R not in Fields():
return typecall(cls, polys, domain)
if is_FiniteField(R):
return DynamicalSystem_affine_finite_field(polys, domain)
return DynamicalSystem_affine_field(polys, domain)