def _element_constructor_(self, x):
r"""
Create an element in this group from ``x``.
INPUT:
- ``x`` -- a rational number
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(0)(0)
0
sage: DiscreteValueGroup(0)(1)
Traceback (most recent call last):
...
ValueError: `1` is not in Trivial Additive Abelian Group.
sage: DiscreteValueGroup(1)(1)
1
sage: DiscreteValueGroup(1)(1/2)
Traceback (most recent call last):
...
ValueError: `1/2` is not in Additive Abelian Group generated by 1.
"""
x = QQ.coerce(x)
if x == 0 or (self._generator != 0 and x / self._generator in ZZ):
return x
raise ValueError("`{0}` is not in {1}.".format(x, self))
def deformation_space(lengths):
r"""
Deformation space of the given ``lengths``
This is the smallest vector space defined over `\QQ` that contains
the vector ``lengths``. Its dimension is `rank`.
EXAMPLES::
sage: from surface_dynamics.misc.linalg import deformation_space
sage: K.<sqrt2> = QuadraticField(2)
sage: v3 = vector([sqrt2, 1, 1+sqrt2])
sage: deformation_space(v3)
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[1 0 1]
[0 1 1]
sage: v4 = vector([sqrt2, 1, 1+sqrt2, 1-sqrt2])
sage: deformation_space(v4)
Vector space of degree 4 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 1 -1]
[ 0 1 1 1]
sage: v = vector([1, 5, 2, 9])
sage: deformation_space(v)
Vector space of degree 4 and dimension 1 over Rational Field
Basis matrix:
[1 5 2 9]
The deformation space has some covariance relation with respect to matrix
actions::
sage: m3 = matrix(3, [1,-1,0,2,-3,4,5,-2,2])
sage: deformation_space(v3 * m3) == deformation_space(v3) * m3
True
sage: deformation_space(m3 * v3) == deformation_space(v3) * m3.transpose()
True
sage: m4 = matrix(4, [1,-1,0,1,2,-3,0,4,5,3,-2,2,1,1,1,1])
sage: deformation_space(v4 * m4) == deformation_space(v4) * m4
True
sage: deformation_space(m4 * v4) == deformation_space(v4) * m4.transpose()
True
"""
from sage.matrix.constructor import matrix
try:
m_lengths = matrix([u.vector() for u in lengths])
except AttributeError:
from sage.rings.rational_field import QQ
lengths = [QQ.coerce(i) for i in lengths]
m_lengths = matrix([[i] for i in lengths])
return m_lengths.column_space()
def relation_space(v):
r"""
Relation space of the given vector ``v``
This is the sub vector space of `\QQ^d` given as the kernel of the map
`n \mapsto n \cdot \lambda`. The dimension is `d - rank`.
EXAMPLES::
sage: from surface_dynamics.misc.linalg import relation_space
sage: K.<sqrt2> = QuadraticField(2)
sage: v3 = vector([sqrt2, 1, 1+sqrt2])
sage: relation_space(v3)
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 1 -1]
sage: v4 = vector([sqrt2, 1, 1+sqrt2, 1-sqrt2])
sage: relation_space(v4)
Vector space of degree 4 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 -1/2 1/2]
[ 0 1 -1/2 -1/2]
sage: v = vector([1,2,5,3])
sage: relation_space(v)
Vector space of degree 4 and dimension 3 over Rational Field
Basis matrix:
[ 1 0 0 -1/3]
[ 0 1 0 -2/3]
[ 0 0 1 -5/3]
The relation space has some covariance relation with respect to matrix
actions::
sage: m3 = matrix(3, [1,-1,0,2,-3,4,5,-2,2])
sage: relation_space(v3 * m3) == relation_space(v3) * ~m3.transpose()
True
sage: relation_space(m3 * v3) == relation_space(v3) * ~m3
True
sage: m4 = matrix(4, [1,-1,0,1,2,-3,0,4,5,3,-2,2,1,1,1,1])
sage: relation_space(v4 * m4) == relation_space(v4) * ~m4.transpose()
True
sage: relation_space(m4 * v4) == relation_space(v4) * ~m4
True
"""
from sage.matrix.constructor import matrix
try:
m_lengths = matrix([u.vector() for u in v])
except AttributeError:
from sage.rings.rational_field import QQ
v = [QQ.coerce(i) for i in v]
m_lengths = matrix([[i] for i in v])
return m_lengths.left_kernel()
def deformation_space(lengths):
r"""
Deformation space of the given ``lengths``
This is the smallest vector space defined over `\QQ` that contains
the vector ``lengths``. Its dimension is `rank`.
EXAMPLES::
sage: from surface_dynamics.misc.linalg import deformation_space
sage: K.<sqrt2> = QuadraticField(2)
sage: v3 = vector([sqrt2, 1, 1+sqrt2])
sage: deformation_space(v3)
Vector space of degree 3 and dimension 2 over Rational Field
Basis matrix:
[1 0 1]
[0 1 1]
sage: v4 = vector([sqrt2, 1, 1+sqrt2, 1-sqrt2])
sage: deformation_space(v4)
Vector space of degree 4 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 1 -1]
[ 0 1 1 1]
sage: v = vector([1, 5, 2, 9])
sage: deformation_space(v)
Vector space of degree 4 and dimension 1 over Rational Field
Basis matrix:
[1 5 2 9]
The deformation space has some covariance relation with respect to matrix
actions::
sage: m3 = matrix(3, [1,-1,0,2,-3,4,5,-2,2])
sage: deformation_space(v3 * m3) == deformation_space(v3) * m3
True
sage: deformation_space(m3 * v3) == deformation_space(v3) * m3.transpose()
True
sage: m4 = matrix(4, [1,-1,0,1,2,-3,0,4,5,3,-2,2,1,1,1,1])
sage: deformation_space(v4 * m4) == deformation_space(v4) * m4
True
sage: deformation_space(m4 * v4) == deformation_space(v4) * m4.transpose()
True
"""
from sage.matrix.constructor import matrix
try:
m_lengths = matrix([u.vector() for u in lengths])
except AttributeError:
from sage.rings.rational_field import QQ
lengths = [QQ.coerce(i) for i in lengths]
m_lengths = matrix([[i] for i in lengths])
return m_lengths.column_space()
def relation_space(v):
r"""
Relation space of the given vector ``v``
This is the sub vector space of `\QQ^d` given as the kernel of the map
`n \mapsto n \cdot \lambda`. The dimension is `d - rank`.
EXAMPLES::
sage: from surface_dynamics.misc.linalg import relation_space
sage: K.<sqrt2> = QuadraticField(2)
sage: v3 = vector([sqrt2, 1, 1+sqrt2])
sage: relation_space(v3)
Vector space of degree 3 and dimension 1 over Rational Field
Basis matrix:
[ 1 1 -1]
sage: v4 = vector([sqrt2, 1, 1+sqrt2, 1-sqrt2])
sage: relation_space(v4)
Vector space of degree 4 and dimension 2 over Rational Field
Basis matrix:
[ 1 0 -1/2 1/2]
[ 0 1 -1/2 -1/2]
sage: v = vector([1,2,5,3])
sage: relation_space(v)
Vector space of degree 4 and dimension 3 over Rational Field
Basis matrix:
[ 1 0 0 -1/3]
[ 0 1 0 -2/3]
[ 0 0 1 -5/3]
The relation space has some covariance relation with respect to matrix
actions::
sage: m3 = matrix(3, [1,-1,0,2,-3,4,5,-2,2])
sage: relation_space(v3 * m3) == relation_space(v3) * ~m3.transpose()
True
sage: relation_space(m3 * v3) == relation_space(v3) * ~m3
True
sage: m4 = matrix(4, [1,-1,0,1,2,-3,0,4,5,3,-2,2,1,1,1,1])
sage: relation_space(v4 * m4) == relation_space(v4) * ~m4.transpose()
True
sage: relation_space(m4 * v4) == relation_space(v4) * ~m4
True
"""
from sage.matrix.constructor import matrix
try:
m_lengths = matrix([u.vector() for u in v])
except AttributeError:
from sage.rings.rational_field import QQ
v = [QQ.coerce(i) for i in v]
m_lengths = matrix([[i] for i in v])
return m_lengths.left_kernel()
def __classcall__(cls, generator):
r"""
Normalizes ``generator`` to a positive rational so that this is a
unique parent.
TESTS::
sage: from sage.rings.valuation.value_group import DiscreteValueGroup
sage: DiscreteValueGroup(1) is DiscreteValueGroup(-1)
True
"""
generator = QQ.coerce(generator)
generator = generator.abs()
return super(DiscreteValueGroup, cls).__classcall__(cls, generator)
def some_elements(self):
r"""
Return some typical elements in this semigroup.
EXAMPLES::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: list(DiscreteValueSemigroup([-3/8,1/2]).some_elements())
[0, -3/8, 1/2, ...]
"""
yield self(0)
if self.is_trivial():
return
for g in self._generators:
yield g
from sage.rings.integer_ring import ZZ
for x in (ZZ**len(self._generators)).some_elements():
yield QQ.coerce(
sum([abs(c) * g for c, g in zip(x, self._generators)]))
def __classcall__(cls, generators):
r"""
Normalize ``generators``.
TESTS:
We do not find minimal generators or something like that but just sort the
generators and drop generators that are trivially contained in the
semigroup generated by the remaining generators::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: DiscreteValueSemigroup([1,2]) is DiscreteValueSemigroup([1])
True
In this case, the normalization is not sufficient to determine that
these are the same semigroup::
sage: DiscreteValueSemigroup([1,-1,1/3]) is DiscreteValueSemigroup([1/3,-1/3])
False
"""
if generators in QQ:
generators = [generators]
generators = list(set([QQ.coerce(g) for g in generators if g != 0]))
generators.sort()
simplified_generators = generators
# this is not very efficient but there should never be more than a
# couple of generators
for g in generators:
for h in generators:
if g == h:
continue
from sage.rings.all import NN
if h / g in NN:
simplified_generators.remove(h)
break
return super(DiscreteValueSemigroup,
cls).__classcall__(cls, tuple(simplified_generators))
def _mul_(self, other, switch_sides=False):
r"""
Return the semigroup generated by ``other`` times the generators of this
semigroup.
INPUT:
- ``other`` -- a rational number
EXAMPLES::
sage: from sage.rings.valuation.value_group import DiscreteValueSemigroup
sage: D = DiscreteValueSemigroup(1/2)
sage: 1/2 * D
Additive Abelian Semigroup generated by 1/4
sage: D * (1/2)
Additive Abelian Semigroup generated by 1/4
sage: D * 0
Trivial Additive Abelian Semigroup
"""
other = QQ.coerce(other)
return DiscreteValueSemigroup([g * other for g in self._generators])
