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__index_level_0__
int64
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def _latex_(self): """ Return Latex representation of this Maxima object. This calls the tex command in Maxima, then does a little post-processing to fix bugs in the resulting Maxima output. EXAMPLES:: sage: maxima('sqrt(2) + 1/3 + asin(5)')._latex_() '\\sin^{-1}\\cdot5+\\sqrt{2}+{{1}\\over{3}}'
def _latex_(self): """ Return Latex representation of this Maxima object. This calls the tex command in Maxima, then does a little post-processing to fix bugs in the resulting Maxima output. EXAMPLES:: sage: maxima('sqrt(2) + 1/3 + asin(5)')._latex_() '\\sin^{-1}\\cdot5+\\sqrt{2}+{{1}\\over{3}}'
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def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the match...
def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the match...
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def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the match...
def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the match...
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def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the match...
def matching_polynomial(self, complement=True, name=None): """ Computes the matching polynomial of the graph G. The algorithm used is a recursive one, based on the following observation: - If e is an edge of G, G' is the result of deleting the edge e, and G'' is the result of deleting each vertex in e, then the match...
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def blocks_and_cut_vertices(self): """ Computes the blocks and cut vertices of the graph. In the case of a digraph, this computation is done on the underlying graph. A cut vertex is one whose deletion increases the number of connected components. A block is a maximal induced subgraph which itself has no cut vertices. ...
def blocks_and_cut_vertices(self): """ Computes the blocks and cut vertices of the graph. In the case of a digraph, this computation is done on the underlying graph. A cut vertex is one whose deletion increases the number of connected components. A block is a maximal induced subgraph which itself has no cut vertices. ...
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def blocks_and_cut_vertices(self): """ Computes the blocks and cut vertices of the graph. In the case of a digraph, this computation is done on the underlying graph. A cut vertex is one whose deletion increases the number of connected components. A block is a maximal induced subgraph which itself has no cut vertices. ...
def blocks_and_cut_vertices(self): """ Computes the blocks and cut vertices of the graph. In the case of a digraph, this computation is done on the underlying graph. A cut vertex is one whose deletion increases the number of connected components. A block is a maximal induced subgraph which itself has no cut vertices. ...
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def _is_a(self, x): """ Check if a sage object belongs to self. This methods is a helper for :meth:`__contains__` and the constructor :meth:`_element_constructor_`.
def _is_a(self, x): """ Check if a Sage object belongs to self. This methods is a helper for :meth:`__contains__` and the constructor :meth:`_element_constructor_`.
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def row_stabilizer(self): """ Return the PermutationGroup corresponding to the row stabilizer of self. EXAMPLES:: sage: rs = Tableau([[1,2,3],[4,5]]).row_stabilizer() sage: rs.order() == factorial(3)*factorial(2) True sage: PermutationGroupElement([(1,3,2),(4,5)]) in rs True sage: PermutationGroupElement([(1,4)]) in ...
def row_stabilizer(self): """ Return the PermutationGroup corresponding to the row stabilizer of self. EXAMPLES:: sage: rs = Tableau([[1,2,3],[4,5]]).row_stabilizer() sage: rs.order() == factorial(3)*factorial(2) True sage: PermutationGroupElement([(1,3,2),(4,5)]) in rs True sage: PermutationGroupElement([(1,4)]) in ...
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def super_categories(self): """ Returns a list of the immediate super categories of self.
def super_categories(self): """ Returns a list of the immediate super categories of self.
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def one(self): r""" Returns the one of the monoid, that is the unique neutral element for `*`.
def one(self): r""" Returns the one of the monoid, that is the unique neutral element for `*`.
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def one_element(self): r""" Backward compatibility alias for :meth:`self.one()`.
def one_element(self): r""" Backward compatibility alias for :meth:`self.one()`.
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def prod(self, args): r""" n-ary product
def prod(self, args): r""" n-ary product
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def prod(self, args): r""" n-ary product
defprod(self,args):r"""n-aryproduct
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def is_one(self): r""" Returns whether self is the one of the monoid
def is_one(self): r""" Returns whether self is the one of the monoid
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def __pow__(self, n): r""" INPUTS: - n: a non negative integer
def __pow__(self, n): r""" INPUTS: - n: a non negative integer
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def _pow_naive(self, n): r""" A naive implementation of __pow__
def _pow_naive(self, n): r""" A naive implementation of __pow__
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def _pow_naive(self, n): r""" A naive implementation of __pow__
def _pow_naive(self, n): r""" A naive implementation of __pow__
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def _pow_naive(self, n): r""" A naive implementation of __pow__
def _pow_naive(self, n): r""" A naive implementation of __pow__
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def is_cyclic_ordered(x1,x2,x3): return ( (x1 < x2 and x2 < x3) or (x2 < x3 and x3 < x1) or (x3 < x1 and x1 < x2))
def is_cyclic_ordered(x1,x2,x3): return ( (x1 < x2 and x2 < x3) or (x2 < x3 and x3 < x1) or (x3 < x1 and x1 < x2))
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def is_cyclic_ordered(x1,x2,x3): return ( (x1 < x2 and x2 < x3) or (x2 < x3 and x3 < x1) or (x3 < x1 and x1 < x2))
def is_cyclic_ordered(x1,x2,x3): return ( (x1 < x2 and x2 < x3) or (x2 < x3 and x3 < x1) or (x3 < x1 and x1 < x2))
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def is_cyclic_ordered(x1,x2,x3): return ( (x1 < x2 and x2 < x3) or (x2 < x3 and x3 < x1) or (x3 < x1 and x1 < x2))
def is_cyclic_ordered(x1,x2,x3): return ( (x1 < x2 and x2 < x3) or (x2 < x3 and x3 < x1) or (x3 < x1 and x1 < x2))
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def __init__(self, x, y, r1, r2, angle, s1, s2, options): """ Initializes base class Arc.
def __init__(self, x, y, r1, r2, angle, s1, s2, options): """ Initializes base class Arc.
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def get_minmax_data(self): """ Returns a dictionary with the bounding box data.
def get_minmax_data(self): """ Returns a dictionary with the bounding box data.
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def _allowed_options(self): """ Return the allowed options for the Arc class.
def _allowed_options(self): """ Return the allowed options for the Arc class.
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def _repr_(self): """ String representation of Arc primitive.
def _repr_(self): """ String representation of Arc primitive.
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def plot3d(self): r""" TESTS:
def plot3d(self): r""" TESTS:
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def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
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def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
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def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
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def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
def if not ZZ(len(S)).is_power_of(2): raise TypeError("Lookup table length is not a power of 2.") __init__(self, if not ZZ(len(S)).is_power_of(2): raise TypeError("Lookup table length is not a power of 2.") *args, if not ZZ(len(S)).is_power_of(2): raise TypeError("Lookup table length is not a power of 2.") if not ZZ(l...
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def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
def __init__(self, *args, **kwargs): """ Construct a substitution box (S-box) for a given lookup table `S`.
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def random_prime(n, proof=None, lbound=2): """ Returns a random prime p between `lbound` and n (i.e. `lbound <= p <= n`). The returned prime is chosen uniformly at random from the set of prime numbers less than or equal to n. INPUT: - ``n`` - an integer >= 2. - ``proof`` - bool or None (default: None) If False, th...
def random_prime(n, proof=None, lbound=2): """ Returns a random prime p between `lbound` and n (i.e. `lbound <= p <= n`). The returned prime is chosen uniformly at random from the set of prime numbers less than or equal to n. INPUT: - ``n`` - an integer >= 2. - ``proof`` - bool or None (default: None) If False, th...
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def random_prime(n, proof=None, lbound=2): """ Returns a random prime p between `lbound` and n (i.e. `lbound <= p <= n`). The returned prime is chosen uniformly at random from the set of prime numbers less than or equal to n. INPUT: - ``n`` - an integer >= 2. - ``proof`` - bool or None (default: None) If False, th...
def random_prime(n, proof=None, lbound=2): """ Returns a random prime p between `lbound` and n (i.e. `lbound <= p <= n`). The returned prime is chosen uniformly at random from the set of prime numbers less than or equal to n. INPUT: - ``n`` - an integer >= 2. - ``proof`` - bool or None (default: None) If False, th...
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def random_prime(n, proof=None, lbound=2): """ Returns a random prime p between `lbound` and n (i.e. `lbound <= p <= n`). The returned prime is chosen uniformly at random from the set of prime numbers less than or equal to n. INPUT: - ``n`` - an integer >= 2. - ``proof`` - bool or None (default: None) If False, th...
def random_prime(n, proof=None, lbound=2): """ Returns a random prime p between `lbound` and n (i.e. `lbound <= p <= n`). The returned prime is chosen uniformly at random from the set of prime numbers less than or equal to n. INPUT: - ``n`` - an integer >= 2. - ``proof`` - bool or None (default: None) If False, th...
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def change_weierstrass_model(self, *urst): r""" Return a new Weierstrass model of self under the standard transformation `(u,r,s,,t)` .. math:: (x,y) \mapsto (x',y') = (u^2xr , u^3y + su^2x' + t). EXAMPLES:: sage: E = EllipticCurve('15a') sage: F1 = E.change_weierstrass_model([1/2,0,0,0]); F1 Elliptic Curve define...
def change_weierstrass_model(self, *urst): r""" Return a new Weierstrass model of self under the standard transformation `(u,r,s,t)` .. math:: (x,y) \mapsto (x',y') = (u^2xr , u^3y + su^2x' + t). EXAMPLES:: sage: E = EllipticCurve('15a') sage: F1 = E.change_weierstrass_model([1/2,0,0,0]); F1 Elliptic Curve defined...
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def change_weierstrass_model(self, *urst): r""" Return a new Weierstrass model of self under the standard transformation `(u,r,s,,t)` .. math:: (x,y) \mapsto (x',y') = (u^2xr , u^3y + su^2x' + t). EXAMPLES:: sage: E = EllipticCurve('15a') sage: F1 = E.change_weierstrass_model([1/2,0,0,0]); F1 Elliptic Curve define...
def change_weierstrass_model(self, *urst): r""" Return a new Weierstrass model of self under the standard transformation `(u,r,s,,t)` .. math:: (x,y) \mapsto (x',y') = (u^2x + r , u^3y + su^2x + t). EXAMPLES:: sage: E = EllipticCurve('15a') sage: F1 = E.change_weierstrass_model([1/2,0,0,0]); F1 Elliptic Curve defi...
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def _call_(self, x): """ Construct a module with basis from the data in ``x``
def _call_(self, x): """ Construct a module with basis from the data in ``x``
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def is_abelian(self): """ Returns whether this category is abelian
def is_abelian(self): """ Returns whether this category is abelian
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def module_morphism(self, on_basis = None, diagonal = None, triangular = None, **keywords): r""" Constructs morphisms by linearity
def module_morphism(self, on_basis = None, diagonal = None, triangular = None, **keywords): r""" Constructs morphisms by linearity
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def module_morphism(self, on_basis = None, diagonal = None, triangular = None, **keywords): r""" Constructs morphisms by linearity
def module_morphism(self, on_basis = None, diagonal = None, triangular = None, **keywords): r""" Constructs morphisms by linearity
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def associated_primes(self, algorithm='sy'): r""" Return a list of primary ideals (and their associated primes) such that their intersection is `I` = ``self``. An ideal `Q` is called primary if it is a proper ideal of the ring `R` and if whenever `ab \in Q` and `a \not\in Q` then `b^n \in Q` for some `n \in \ZZ`. If ...
def associated_primes(self, algorithm='sy'): r""" Return a list of the associated primes of primary ideals of which the intersection is `I` = ``self``. An ideal `Q` is called primary if it is a proper ideal of the ring `R` and if whenever `ab \in Q` and `a \not\in Q` then `b^n \in Q` for some `n \in \ZZ`. If `Q` is a...
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def associated_primes(self, algorithm='sy'): r""" Return a list of primary ideals (and their associated primes) such that their intersection is `I` = ``self``. An ideal `Q` is called primary if it is a proper ideal of the ring `R` and if whenever `ab \in Q` and `a \not\in Q` then `b^n \in Q` for some `n \in \ZZ`. If ...
defassociated_primes(self,algorithm='sy'):r"""Returnalistofprimaryideals(andtheirassociatedprimes)suchthattheirintersectionis`I`=``self``.Anideal`Q`iscalledprimaryifitisaproperidealofthering`R`andifwhenever`ab\inQ`and`a\not\inQ`then`b^n\inQ`forsome`n\in\ZZ`.If`Q`isaprimaryidealofthering`R`,thentheradicalideal`P`of`Q`,i...
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def associated_primes(self, algorithm='sy'): r""" Return a list of primary ideals (and their associated primes) such that their intersection is `I` = ``self``. An ideal `Q` is called primary if it is a proper ideal of the ring `R` and if whenever `ab \in Q` and `a \not\in Q` then `b^n \in Q` for some `n \in \ZZ`. If ...
def associated_primes(self, algorithm='sy'): r""" Return a list of primary ideals (and their associated primes) such that their intersection is `I` = ``self``. An ideal `Q` is called primary if it is a proper ideal of the ring `R` and if whenever `ab \in Q` and `a \not\in Q` then `b^n \in Q` for some `n \in \ZZ`. If ...
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def ChainPoset(self, n): """ Returns a chain (a totally ordered poset) containing ``n`` elements.
def ChainPoset(self, n): """ Returns a chain (a totally ordered poset) containing ``n`` elements.
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def ChainPoset(self, n): """ Returns a chain (a totally ordered poset) containing ``n`` elements.
def ChainPoset(self, n): """ Returns a chain (a totally ordered poset) containing ``n`` elements.
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def AntichainPoset(self, n): """ Returns an antichain (a poset with no comparable elements) containing ``n`` elements.
def AntichainPoset(self, n): """ Returns an antichain (a poset with no comparable elements) containing ``n`` elements.
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def AntichainPoset(self, n): """ Returns an antichain (a poset with no comparable elements) containing ``n`` elements.
def AntichainPoset(self, n): """ Returns an antichain (a poset with no comparable elements) containing ``n`` elements.
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def eval(self, Vobj): r""" Evaluates the left hand side `A\vec{x}+b` on the given vertex/ray/line. NOTES: * Evaluating on a vertex returns `A\vec{x}+b` * Evaluating on a ray returns `A\vec{r}`. Only the sign or whether it is zero is meaningful. * Evaluating on a line returns `A\vec{l}`. Only whether it is zero or not...
def eval(self, Vobj): r""" Evaluates the left hand side `A\vec{x}+b` on the given vertex/ray/line. NOTES: * Evaluating on a vertex returns `A\vec{x}+b` * Evaluating on a ray returns `A\vec{r}`. Only the sign or whether it is zero is meaningful. * Evaluating on a line returns `A\vec{l}`. Only whether it is zero or not...
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def is_inequality(self): """ Returns True since this is, by construction, an inequality.
def is_inequality(self): """ Returns True since this is, by construction, an inequality.
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def interior_contains(self, Vobj): """ Tests whether the interior of the halfspace (excluding its boundary) defined by the inequality contains the given vertex/ray/line.
def interior_contains(self, Vobj): If you pass a vector, it is assumed to be the coordinate vector of a point:: sage: P = Polyhedron(vertices=[[1,1],[1,-1],[-1,1],[-1,-1]]) sage: p = vector(ZZ, [1,0] ) sage: [ ieq.interior_contains(p) for ieq in P.inequality_generator() ] [True, True, True, False] """ try: if Vobj.is...
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def is_equation(self): """ Tests if this object is an equation. By construction, it must be.
def is_equation(self): """ Tests if this object is an equation. By construction, it must be.
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def is_vertex(self): """ Tests if this object is a vertex. By construction it always is.
def is_vertex(self): """ Tests if this object is a vertex. By construction it always is.
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def is_ray(self): """ Tests if this object is a ray. Always True by construction.
def is_ray(self): """ Tests if this object is a ray. Always True by construction.
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def is_line(self): """ Tests if the object is a line. By construction it must be.
def is_line(self): """ Tests if the object is a line. By construction it must be.
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def identity(self): """ Returns the identity projection.
def identity(self): """ Returns the identity projection.
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def identity(self): """ Returns the identity projection.
def identity(self): """ Returns the identity projection.
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def run(self, category = None, skip = [], catch = True, raise_on_failure = False, **options): """ Run all the tests from this test suite:
def run(self, category = None, skip = [], catch = True, raise_on_failure = False, **options): """ Run all the tests from this test suite:
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def run(self, category = None, skip = [], catch = True, raise_on_failure = False, **options): """ Run all the tests from this test suite:
def run(self, category = None, skip = [], catch = True, raise_on_failure = False, **options): """ Run all the tests from this test suite:
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def instance_tester(instance, tester = None, **options): """ Returns a gadget attached to ``instance`` providing testing utilities. EXAMPLES:: sage: from sage.misc.sage_unittest import instance_tester sage: tester = instance_tester(ZZ) sage: tester.assert_(1 == 1) sage: tester.assert_(1 == 0) Traceback (most recent ...
def instance_tester(instance, tester = None, **options): """ Returns a gadget attached to ``instance`` providing testing utilities. EXAMPLES:: sage: from sage.misc.sage_unittest import instance_tester sage: tester = instance_tester(ZZ) sage: tester.assert_(1 == 1) sage: tester.assert_(1 == 0) Traceback (most recent ...
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def __init__(self, instance, elements = None, verbose = False, prefix = "", **options): """ A gadget attached to an instance providing it with testing utilities.
def __init__(self, instance, elements = None, verbose = False, prefix = "", **options): """ A gadget attached to an instance providing it with testing utilities.
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def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol, borderstyle, borderwidth): r""" ``region_plot`` takes a boolean function of two variables, `f(x,y)` and plots the region where f is True over the specified ``xrange`` and ``yrange`` as demonstrated below. ``region_plot(f, (xmin, xmax), (ymin, ym...
def region_plot(f, xrange, yrange, plot_points, incol, outcol, bordercol, borderstyle, borderwidth): r""" ``region_plot`` takes a boolean function of two variables, `f(x,y)` and plots the region where f is True over the specified ``xrange`` and ``yrange`` as demonstrated below. ``region_plot(f, (xmin, xmax), (ymin, ym...
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def equify(f): """ Returns the equation rewritten as a symbolic function to give negative values when True, positive when False. EXAMPLES:: sage: from sage.plot.contour_plot import equify sage: var('x, y') (x, y) sage: equify(x^2 < 2) x^2 - 2 sage: equify(x^2 > 2) -x^2 + 2 sage: equify(x*y > 1) -x*y + 1 sage: equify(...
def equify(f): """ Returns the equation rewritten as a symbolic function to give negative values when True, positive when False. EXAMPLES:: sage: from sage.plot.contour_plot import equify sage: var('x, y') (x, y) sage: equify(x^2 < 2) x^2 - 2 sage: equify(x^2 > 2) -x^2 + 2 sage: equify(x*y > 1) -x*y + 1 sage: equify(...
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def order(self, *gens, **kwds): r""" Return the order with given ring generators in the maximal order of this number field. INPUT: - ``gens`` - list of elements of self; if no generators are given, just returns the cardinality of this number field (oo) for consistency. - ``check_is_integral`` - bool (default: Tru...
def order(self, *args, **kwds): r""" Return the order with given ring generators in the maximal order of this number field. INPUT: - ``gens`` - list of elements of self; if no generators are given, just returns the cardinality of this number field (oo) for consistency. - ``check_is_integral`` - bool (default: Tru...
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def order(self, *gens, **kwds): r""" Return the order with given ring generators in the maximal order of this number field. INPUT: - ``gens`` - list of elements of self; if no generators are given, just returns the cardinality of this number field (oo) for consistency. - ``check_is_integral`` - bool (default: Tru...
deforder(self,*gens,**kwds):r"""Returntheorderwithgivenringgeneratorsinthemaximalorderofthisnumberfield.INPUT:-``gens``-listofelementsofself;ifnogeneratorsaregiven,justreturnsthecardinalityofthisnumberfield(oo)forconsistency.-``check_is_integral``-bool(default:True),whethertocheckthateachgeneratorisintegral.-``check_ra...
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def order(self, *gens, **kwds): r""" Return the order with given ring generators in the maximal order of this number field. INPUT: - ``gens`` - list of elements of self; if no generators are given, just returns the cardinality of this number field (oo) for consistency. - ``check_is_integral`` - bool (default: Tru...
def order(self, *gens, **kwds): r sage: K.<a> = NumberField(x^3 - 2) sage: ZZ[a] Order in Number Field in a0 with defining polynomial x^3 - 2 TESTS: We verify that trac sage: K.<a> = NumberField(x^4 + 4*x^2 + 2) sage: B = K.integral_basis() sage: K.order(*B) Order in Number Field in a with defining polynomial x^4 + ...
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def iterated_right_palindromic_closure(self, f=None, algorithm='recursive'): r""" Returns the iterated (`f`-)palindromic closure of self. INPUT:
def iterated_right_palindromic_closure(self, f=None, algorithm='recursive'): r""" Returns the iterated (`f`-)palindromic closure of self. INPUT:
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def find(self, sub, start=0, end=None): r""" Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
def find(self, sub, start=0, end=None): r""" Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
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def find(self, sub, start=0, end=None): r""" Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
def find(self, sub, start=0, end=None): r""" Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
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def find(self, sub, start=0, end=None): r""" Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
def find(self, sub, start=0, end=None): r""" Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
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def find(self, sub, start=0, end=None): r""" Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
def find(self, sub, start=0, end=None): r""" Returns the index of the first occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
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def rfind(self, sub, start=0, end=None): r""" Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
def rfind(self, sub, start=0, end=None): r""" Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
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def rfind(self, sub, start=0, end=None): r""" Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
def rfind(self, sub, start=0, end=None): r""" Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
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def rfind(self, sub, start=0, end=None): r""" Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
def rfind(self, sub, start=0, end=None): r""" Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
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def rfind(self, sub, start=0, end=None): r""" Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
def rfind(self, sub, start=0, end=None): r""" Returns the index of the last occurrence of sub in self, such that sub is contained within self[start:end]. Returns -1 on failure. INPUT:
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def __init__(self): r""" The inverse of the hyperbolic secant function.
def __init__(self): r""" The inverse of the hyperbolic secant function.
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def eliminate_linear_variables(self, maxlength=3, skip=lambda lm,tail: False): """ Return a new system where "linear variables" are eliminated.
def eliminate_linear_variables(self, maxlength=3, skip=lambda lm,tail: False): """ Return a new system where "linear variables" are eliminated.
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def modular_symbol(self, sign=1, use_eclib = False, normalize = "L_ratio"): r""" Return the modular symbol associated to this elliptic curve, with given sign and base ring. This is the map that sends `r/s` to a fixed multiple of the integral of `2 \pi i f(z) dz` from `\infty` to `r/s`, normalized so that all values of...
def modular_symbol(self, sign=1, use_eclib = False, normalize = "L_ratio"): r""" Return the modular symbol associated to this elliptic curve, with given sign and base ring. This is the map that sends `r/s` to a fixed multiple of the integral of `2 \pi i f(z) dz` from `\infty` to `r/s`, normalized so that all values of...
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def modular_symbol(self, sign=1, use_eclib = False, normalize = "L_ratio"): r""" Return the modular symbol associated to this elliptic curve, with given sign and base ring. This is the map that sends `r/s` to a fixed multiple of the integral of `2 \pi i f(z) dz` from `\infty` to `r/s`, normalized so that all values of...
def modular_symbol(self, sign=1, use_eclib = False, normalize = "L_ratio"): r""" Return the modular symbol associated to this elliptic curve, with given sign and base ring. This is the map that sends `r/s` to a fixed multiple of the integral of `2 \pi i f(z) dz` from `\infty` to `r/s`, normalized so that all values of...
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def __classcall_private__(cls, fam, facade=True, keepkey=False): # was *args, **options): """ Normalization of arguments; see :cls:`UniqueRepresentation`.
def __classcall_private__(cls, fam, facade=True, keepkey=False): # was *args, **options): """ Normalization of arguments; see :cls:`UniqueRepresentation`.
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def derivative(self, ex, operator): """ EXAMPLES::
def derivative(self, ex, operator): """ EXAMPLES::
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def derivative(self, ex, operator): """ EXAMPLES::
def derivative(self, ex, operator): """ EXAMPLES::
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def derivative(self, ex, operator): """ EXAMPLES::
def derivative(self, ex, operator): """ EXAMPLES::
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def derivative(self, ex, operator): """ EXAMPLES::
def derivative(self, ex, operator): """ EXAMPLES::
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def strip_answer(self, s): """ Returns the string s with Matlab's answer prompt removed. EXAMPLES:: sage: s = '\nans =\n\n 2\n' sage: matlab.strip_answer(s) ' 2' """ i = s.find('=') return s[i+1:].strip('\n')
def strip_answer(self, s): r""" Returns the string s with Matlab's answer prompt removed. EXAMPLES:: sage: s = '\nans =\n\n 2\n' sage: matlab.strip_answer(s) ' 2' r""" i = s.find('=') return s[i+1:].strip('\n')
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def pn(self, n): """ Return the number of the `n`-th partial convergent, computed using the recurrence. EXAMPLES:: sage: c = continued_fraction(pi); c [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3] sage: c.pn(0), c.qn(0) (3, 1) sage: len(c) 14 sage: c.pn(13), c.qn(13) (245850922, 78256779) """ if n < -2: raise ValueE...
def pn(self, n): """ Return the numerator of the `n`-th partial convergent, computed using the recurrence. EXAMPLES:: sage: c = continued_fraction(pi); c [3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14, 3] sage: c.pn(0), c.qn(0) (3, 1) sage: len(c) 14 sage: c.pn(13), c.qn(13) (245850922, 78256779) """ if n < -2: raise Val...
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def ContinuedFractionField(): """ Return the (unique) field of all contiued fractions. EXAMPLES:: sage: ContinuedFractionField() Field of all continued fractions """ return CFF
def ContinuedFractionField(): """ Return the (unique) field of all continued fractions. EXAMPLES:: sage: ContinuedFractionField() Field of all continued fractions """ return CFF
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def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): """ Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): r""" Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
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def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): """ Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): """ Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
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def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): """ Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
def cospectral_graphs(self, vertices, matrix_function=lambda g: g.adjacency_matrix(), graphs=None): """ Find all sets of graphs on ``vertices`` vertices (with possible restrictions) which are cospectral with respect to a constructed matrix.
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def inject_coefficients(self, scope=None, verbose=True): r""" Inject generators of the base field of ``self`` into ``scope``.
def inject_coefficients(self, scope=None, verbose=True): r""" Inject generators of the base field of ``self`` into ``scope``.
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def inject_coefficients(self, scope=None, verbose=True): r""" Inject generators of the base field of ``self`` into ``scope``.
def inject_coefficients(self, scope=None, verbose=True): r""" Inject generators of the base field of ``self`` into ``scope``.
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def inject_coefficients(self, scope=None, verbose=True): r""" Inject generators of the base field of ``self`` into ``scope``.
def inject_coefficients(self, scope=None, verbose=True): r""" Inject generators of the base field of ``self`` into ``scope``.
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def _repr_defn(self): """ This function is used internally for printing.
def _repr_defn(self): """ This function is used internally for printing.
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def __init__(self, parent, polys, check=True): SchemeMorphism_on_points.__init__(self, parent, polys, check) if check: # morphisms from projective space are always given by # homogeneous polynomials of the same degree deg = self.defining_polynomials()[0].degree() for poly in self.defining_polynomials(): if (poly.degree...
def __init__(self, parent, polys, check=True): SchemeMorphism_on_points.__init__(self, parent, polys, check) if check: # morphisms from projective space are always given by # homogeneous polynomials of the same degree polys = self.defining_polynomials() try: d = polys[0].degree() except AttributeError: polys = [f.lift(...
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def identity_matrix(ring, n=0, sparse=False): r""" Return the `n \times n` identity matrix over the given ring. The default ring is the integers. EXAMPLES:: sage: M = identity_matrix(QQ, 2); M [1 0] [0 1] sage: M.parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field sage: M = identity_matrix(2); M [...
def identity_matrix(ring, n=0, sparse=False): r""" Return the `n \times n` identity matrix over the given ring. The default ring is the integers. EXAMPLES:: sage: M = identity_matrix(QQ, 2); M [1 0] [0 1] sage: M.parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field sage: M = identity_matrix(2); M [...
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def zero_matrix(ring, nrows, ncols=None, sparse=False): r""" Return the `nrows \times ncols` zero matrix over the given ring. The default ring is the integers. EXAMPLES:: sage: M = zero_matrix(QQ, 2); M [0 0] [0 0] sage: M.parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field sage: M = zero_matrix(2...
def zero_matrix(ring, nrows, ncols=None, sparse=False): r""" Return the `nrows \times ncols` zero matrix over the given ring. The default ring is the integers. EXAMPLES:: sage: M = zero_matrix(QQ, 2); M [0 0] [0 0] sage: M.parent() Full MatrixSpace of 2 by 2 dense matrices over Rational Field sage: M = zero_matrix(2...
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def __call__(self, obj, output='html', view=True): r""" Return the documentation for ``obj``.
def __call__(self, obj, output='html', view=True): r""" Return the documentation for ``obj``.
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def is_square_free(self): r""" Returns True if self does not contain squares, and False otherwise. EXAMPLES:: sage: W = Words('123') sage: W('12312').is_square_free() True sage: W('31212').is_square_free() False sage: W().is_square_free() True """ l = self.length() if l < 2: return True suff = self for i in xrange(0,...
def is_square_free(self): r""" Returns True if self does not contain squares, and False otherwise. EXAMPLES:: sage: W = Words('123') sage: W('12312').is_square_free() True sage: W('31212').is_square_free() False sage: W().is_square_free() True """ l = self.length() if l < 2: return True suff = self for i in xrange(0,...
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def coerce_field(self, other): """ Return the number type that contains both `self.field()` and `other`.
def coerce_field(self, other): """ Return the number type that contains both `self.field()` and `other`.
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def coerce_field(self, other): """ Return the number type that contains both `self.field()` and `other`.
def coerce_field(self, other): """ Return the number type that contains both `self.field()` and `other`.
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