def __call__(self, x):
"""
Coerce x into a boundary symbol space.
If x is a modular symbol (with the same group, weight, character,
sign, and base field), this returns the image of that modular
symbol under the boundary map.
EXAMPLES::
sage: M = ModularSymbols(Gamma0(15), 2) ; B = M.boundary_space()
sage: B(M.0)
[Infinity] - [0]
sage: B(Cusp(1))
[0]
sage: B(Cusp(oo))
[Infinity]
sage: B(7)
Traceback (most recent call last):
...
TypeError: Coercion of 7 (of type <type 'sage.rings.integer.Integer'>) into Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(15) of weight 2 and over Rational Field not (yet) defined.
"""
from .ambient import ModularSymbolsAmbient
if isinstance(x, int) and x == 0:
return BoundarySpaceElement(self, {})
elif isinstance(x, cusps.Cusp):
return self._coerce_cusp(x)
elif isinstance(x, ManinSymbol):
return self._coerce_in_manin_symbol(x)
elif element.is_ModularSymbolsElement(x):
M = x.parent()
if not isinstance(M, ModularSymbolsAmbient):
raise TypeError(
"x (=%s) must be an element of a space of modular symbols of type ModularSymbolsAmbient"
% x)
if M.level() != self.level():
raise TypeError("x (=%s) must have level %s but has level %s" %
(x, self.level(), M.level()))
S = x.manin_symbol_rep()
if len(S) == 0:
return self(0)
return sum([c * self._coerce_in_manin_symbol(v) for c, v in S])
elif is_FreeModuleElement(x):
y = dict([(i, x[i]) for i in xrange(len(x))])
return BoundarySpaceElement(self, y)
raise TypeError(
"Coercion of %s (of type %s) into %s not (yet) defined." %
(x, type(x), self))
def __call__(self, x):
"""
Coerce x into a boundary symbol space.
If x is a modular symbol (with the same group, weight, character,
sign, and base field), this returns the image of that modular
symbol under the boundary map.
EXAMPLES::
sage: M = ModularSymbols(Gamma0(15), 2) ; B = M.boundary_space()
sage: B(M.0)
[Infinity] - [0]
sage: B(Cusp(1))
[0]
sage: B(Cusp(oo))
[Infinity]
sage: B(7)
Traceback (most recent call last):
...
TypeError: Coercion of 7 (of type <type 'sage.rings.integer.Integer'>) into Space of Boundary Modular Symbols for Congruence Subgroup Gamma0(15) of weight 2 and over Rational Field not (yet) defined.
"""
from .ambient import ModularSymbolsAmbient
if isinstance(x, int) and x == 0:
return BoundarySpaceElement(self, {})
elif isinstance(x, cusps.Cusp):
return self._coerce_cusp(x)
elif isinstance(x, ManinSymbol):
return self._coerce_in_manin_symbol(x)
elif element.is_ModularSymbolsElement(x):
M = x.parent()
if not isinstance(M, ModularSymbolsAmbient):
raise TypeError("x (=%s) must be an element of a space of modular symbols of type ModularSymbolsAmbient"%x)
if M.level() != self.level():
raise TypeError("x (=%s) must have level %s but has level %s"%(
x, self.level(), M.level()))
S = x.manin_symbol_rep()
if len(S) == 0:
return self(0)
return sum([c*self._coerce_in_manin_symbol(v) for c, v in S])
elif is_FreeModuleElement(x):
y = dict([(i,x[i]) for i in range(len(x))])
return BoundarySpaceElement(self, y)
raise TypeError("Coercion of %s (of type %s) into %s not (yet) defined."%(x, type(x), self))
def _element_constructor_(self, x):
r"""
Return ``x`` coerced into this forms space.
EXAMPLES::
sage: from graded_ring import MModularFormsRing
sage: from space import ModularForms
sage: MF = ModularForms(k=12, ep=1)
sage: (x,y,z,d) = MF.pol_ring().gens()
sage: Delta = MModularFormsRing().Delta()
sage: Delta.parent()
MeromorphicModularFormsRing(n=3) over Integer Ring
sage: MF(Delta)
q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5)
sage: MF(Delta).parent() == MF
True
sage: MF(x^3)
1 + 720*q + 179280*q^2 + 16954560*q^3 + 396974160*q^4 + O(q^5)
sage: MF(x^3).parent() == MF
True
sage: qexp = Delta.q_expansion(prec=2)
sage: qexp
q + O(q^2)
sage: qexp.parent()
Power Series Ring in q over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: MF(qexp)
q - 24*q^2 + 252*q^3 - 1472*q^4 + O(q^5)
sage: MF(qexp) == MF(Delta)
True
sage: MF([0,1]) == MF(Delta)
True
sage: MF([1,0]) == MF(x^3) - 720*MF(Delta) # todo: this should give True, the result is False because d!=1/1728 but a formal parameter
False
sage: vec = MF(Delta).coordinate_vector()
sage: vec
(0, 1)
sage: vec.parent()
Vector space of dimension 2 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: vec in MF.module()
True
sage: MF(vec) == MF(Delta)
True
sage: subspace = MF.subspace([MF(Delta)])
sage: subspace
Subspace of dimension 1 of ModularForms(n=3, k=12, ep=1) over Integer Ring
sage: subspace(MF(Delta)) == subspace(d*(x^3-y^2)) == subspace(qexp) == subspace([0,1]) == subspace(vec) == subspace.gen()
True
sage: subspace(MF(Delta)).parent() == subspace(d*(x^3-y^2)).parent() == subspace(qexp).parent() == subspace([0,1]).parent() == subspace(vec).parent()
True
sage: subspace([1]) == subspace.gen()
True
sage: ssvec = subspace(vec).coordinate_vector()
sage: ssvec
(1)
sage: ssvec.parent()
Vector space of dimension 1 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
sage: ambvec = subspace(vec).ambient_coordinate_vector()
sage: ambvec
(0, 1)
sage: ambvec.parent()
Vector space of degree 2 and dimension 1 over Fraction Field of Univariate Polynomial Ring in d over Integer Ring
Basis matrix:
[0 1]
sage: subspace(ambvec) == subspace(vec) and subspace(ambvec).parent() == subspace(vec).parent()
True
"""
from graded_ring_element import FormsRingElement
if isinstance(x, FormsRingElement):
return self.element_class(self, x._rat)
if hasattr(x, 'parent') and (is_LaurentSeriesRing(x.parent()) or is_PowerSeriesRing(x.parent())):
# This assumes that the series corresponds to a weakly holomorphic modular form!
# But the construction method (with the assumption) may also be used for more general form spaces...
return self.construct_form(x)
if is_FreeModuleElement(x) and (self.module() == x.parent() or self.ambient_module() == x.parent()):
return self.element_from_ambient_coordinates(x)
if (not self.is_ambient()) and (isinstance(x, list) or isinstance(x, tuple) or is_FreeModuleElement(x)) and len(x) == self.rank():
try:
return self.element_from_coordinates(x)
except ArithmeticError, TypeError:
pass
def symbolic_expression(x):
"""
Create a symbolic expression or vector of symbolic expressions from x.
INPUT:
- ``x`` - an object
OUTPUT:
- a symbolic expression.
EXAMPLES::
sage: a = symbolic_expression(3/2); a
3/2
sage: type(a)
<class 'sage.symbolic.expression.Expression'>
sage: R.<x> = QQ[]; type(x)
<class 'sage.rings.polynomial.polynomial_rational_flint.Polynomial_rational_flint'>
sage: a = symbolic_expression(2*x^2 + 3); a
2*x^2 + 3
sage: type(a)
<class 'sage.symbolic.expression.Expression'>
sage: from sage.structure.element import Expression
sage: isinstance(a, Expression)
True
sage: a in SR
True
sage: a.parent()
Symbolic Ring
Note that equations exist in the symbolic ring::
sage: E = EllipticCurve('15a'); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 10*x - 10 over Rational Field
sage: symbolic_expression(E)
x*y + y^2 + y == x^3 + x^2 - 10*x - 10
sage: symbolic_expression(E) in SR
True
If ``x`` is a list or tuple, create a vector of symbolic expressions::
sage: v=symbolic_expression([x,1]); v
(x, 1)
sage: v.base_ring()
Symbolic Ring
sage: v=symbolic_expression((x,1)); v
(x, 1)
sage: v.base_ring()
Symbolic Ring
sage: v=symbolic_expression((3,1)); v
(3, 1)
sage: v.base_ring()
Symbolic Ring
sage: E = EllipticCurve('15a'); E
Elliptic Curve defined by y^2 + x*y + y = x^3 + x^2 - 10*x - 10 over Rational Field
sage: v=symbolic_expression([E,E]); v
(x*y + y^2 + y == x^3 + x^2 - 10*x - 10, x*y + y^2 + y == x^3 + x^2 - 10*x - 10)
sage: v.base_ring()
Symbolic Ring
Likewise, if ``x`` is a vector, create a vector of symbolic expressions::
sage: u = vector([1, 2, 3])
sage: v = symbolic_expression(u); v
(1, 2, 3)
sage: v.parent()
Vector space of dimension 3 over Symbolic Ring
If ``x`` is a list or tuple of lists/tuples/vectors, create a matrix of symbolic expressions::
sage: M = symbolic_expression([[1, x, x^2], (x, x^2, x^3), vector([x^2, x^3, x^4])]); M
[ 1 x x^2]
[ x x^2 x^3]
[x^2 x^3 x^4]
sage: M.parent()
Full MatrixSpace of 3 by 3 dense matrices over Symbolic Ring
If ``x`` is a matrix, create a matrix of symbolic expressions::
sage: A = matrix([[1, 2, 3], [4, 5, 6]])
sage: B = symbolic_expression(A); B
[1 2 3]
[4 5 6]
sage: B.parent()
Full MatrixSpace of 2 by 3 dense matrices over Symbolic Ring
If ``x`` is a function, for example defined by a ``lambda`` expression, create a
symbolic function::
sage: f = symbolic_expression(lambda z: z^2 + 1); f
z |--> z^2 + 1
sage: f.parent()
Callable function ring with argument z
sage: f(7)
50
If ``x`` is a list or tuple of functions, or if ``x`` is a function that returns a list
or tuple, create a callable symbolic vector::
sage: symbolic_expression([lambda mu, nu: mu^2 + nu^2, lambda mu, nu: mu^2 - nu^2])
(mu, nu) |--> (mu^2 + nu^2, mu^2 - nu^2)
sage: f = symbolic_expression(lambda uwu: [1, uwu, uwu^2]); f
uwu |--> (1, uwu, uwu^2)
sage: f.parent()
Vector space of dimension 3 over Callable function ring with argument uwu
sage: f(5)
(1, 5, 25)
sage: f(5).parent()
Vector space of dimension 3 over Symbolic Ring
TESTS:
Lists, tuples, and vectors of length 0 become vectors over a symbolic ring::
sage: symbolic_expression([]).parent()
Vector space of dimension 0 over Symbolic Ring
sage: symbolic_expression(()).parent()
Vector space of dimension 0 over Symbolic Ring
sage: symbolic_expression(vector(QQ, 0)).parent()
Vector space of dimension 0 over Symbolic Ring
If a matrix has dimension 0, the result is still a matrix over a symbolic ring::
sage: symbolic_expression(matrix(QQ, 2, 0)).parent()
Full MatrixSpace of 2 by 0 dense matrices over Symbolic Ring
sage: symbolic_expression(matrix(QQ, 0, 3)).parent()
Full MatrixSpace of 0 by 3 dense matrices over Symbolic Ring
Also functions defined using ``def`` can be used, but we do not advertise it as a use case::
sage: def sos(x, y):
....: return x^2 + y^2
sage: symbolic_expression(sos)
(x, y) |--> x^2 + y^2
Functions that take a varying number of arguments or keyword-only arguments are not accepted::
sage: def variadic(x, *y):
....: return x
sage: symbolic_expression(variadic)
Traceback (most recent call last):
...
TypeError: unable to convert <function variadic at 0x...> to a symbolic expression
sage: def function_with_keyword_only_arg(x, *, sign=1):
....: return sign * x
sage: symbolic_expression(function_with_keyword_only_arg)
Traceback (most recent call last):
...
TypeError: unable to convert <function function_with_keyword_only_arg at 0x...>
to a symbolic expression
"""
from sage.symbolic.expression import Expression
from sage.symbolic.ring import SR
from sage.modules.free_module_element import is_FreeModuleElement
from sage.structure.element import is_Matrix
if isinstance(x, Expression):
return x
elif hasattr(x, '_symbolic_'):
return x._symbolic_(SR)
elif isinstance(x, (tuple, list)) or is_FreeModuleElement(x):
expressions = [symbolic_expression(item) for item in x]
if not expressions:
# Make sure it is symbolic also when length is 0
return vector(SR, 0)
if is_FreeModuleElement(expressions[0]):
return matrix(expressions)
return vector(expressions)
elif is_Matrix(x):
if not x.nrows() or not x.ncols():
# Make sure it is symbolic and of correct dimensions
# also when a matrix dimension is 0
return matrix(SR, x.nrows(), x.ncols())
rows = [symbolic_expression(row) for row in x.rows()]
return matrix(rows)
elif callable(x):
from inspect import signature, Parameter
try:
s = signature(x)
except ValueError:
pass
else:
if all(param.kind in (Parameter.POSITIONAL_ONLY,
Parameter.POSITIONAL_OR_KEYWORD)
for param in s.parameters.values()):
vars = [SR.var(name) for name in s.parameters.keys()]
result = x(*vars)
if isinstance(result, (tuple, list)):
return vector(SR, result).function(*vars)
else:
return SR(result).function(*vars)
return SR(x)
