bugged stringlengths 4 228k | fixed stringlengths 0 96.3M | __index_level_0__ int64 0 481k |
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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. | 459,100 |
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... | 459,101 |
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. | 459,102 |
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. | 459,103 |
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. | 459,104 |
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. | 459,105 |
def identity(self): """ Returns the identity projection. | def identity(self): """ Returns the identity projection. | 459,106 |
def identity(self): """ Returns the identity projection. | def identity(self): """ Returns the identity projection. | 459,107 |
def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | 459,108 |
def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | deflocal_coordinates_at_nonweierstrass(self,P,prec=20,name='t'):"""Foranon-WeierstrasspointP=(a,b)onthehyperellipticcurvey^2=f(x),returns(x(t),y(t))suchthat(y(t))^2=f(x(t)),wheret=x-aisthelocalparameter. | 459,109 |
def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | 459,110 |
def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | 459,111 |
def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | def local_coordinates_at_nonweierstrass(self, P, prec = 20, name = 't'): """ For a non-Weierstrass point P = (a,b) on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x - a is the local parameter. | 459,112 |
def local_coordinates_at_weierstrass(self, P, prec = 20, name = 't'): """ For a finite Weierstrass point on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = y is the local parameter. | def local_coordinates_at_weierstrass(self, P, prec = 20, name = 't'): """ For a finite Weierstrass point on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = y is the local parameter. | 459,113 |
def local_coordinates_at_weierstrass(self, P, prec = 20, name = 't'): """ For a finite Weierstrass point on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = y is the local parameter. | def local_coordinates_at_weierstrass(self, P, prec = 20, name = 't'): """ For a finite Weierstrass point on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = y is the local parameter. | 459,114 |
def local_coordinates_at_weierstrass(self, P, prec = 20, name = 't'): """ For a finite Weierstrass point on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = y is the local parameter. | def local_coordinates_at_weierstrass(self, P, prec = 20, name = 't'): """ For a finite Weierstrass point on the hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = y is the local parameter. | 459,115 |
def local_coordinates_at_infinity(self, prec = 20, name = 't'): """ For the genus g hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x^g/y is the local parameter at infinity | deflocal_coordinates_at_infinity(self,prec=20,name='t'):"""Forthegenusghyperellipticcurvey^2=f(x),returns(x(t),y(t))suchthat(y(t))^2=f(x(t)),wheret=x^g/yisthelocalparameteratinfinity | 459,116 |
def local_coordinates_at_infinity(self, prec = 20, name = 't'): """ For the genus g hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x^g/y is the local parameter at infinity | deflocal_coordinates_at_infinity(self,prec=20,name='t'):"""Forthegenusghyperellipticcurvey^2=f(x),returns(x(t),y(t))suchthat(y(t))^2=f(x(t)),wheret=x^g/yisthelocalparameteratinfinity | 459,117 |
def local_coordinates_at_infinity(self, prec = 20, name = 't'): """ For the genus g hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x^g/y is the local parameter at infinity | def local_coordinates_at_infinity(self, prec = 20, name = 't'): """ For the genus g hyperelliptic curve y^2 = f(x), returns (x(t), y(t)) such that (y(t))^2 = f(x(t)), where t = x^g/y is the local parameter at infinity | 459,118 |
def local_coord(self, P, prec = 20, name = 't'): """ If P is not infinity, calls the appropriate local_coordinates function. | def local_coord(self, P, prec = 20, name = 't'): """ If P is not infinity, calls the appropriate local_coordinates function. | 459,119 |
def local_coord(self, P, prec = 20, name = 't'): """ If P is not infinity, calls the appropriate local_coordinates function. | def local_coord(self, P, prec = 20, name = 't'): """ If P is not infinity, calls the appropriate local_coordinates function. | 459,120 |
'def identity_matrix' | 'def identity_matrix' | 459,121 |
def _pcubicroots(b, c, d): r""" Local function returning the number of roots of `x^3 + b*x^2 + c*x + d` modulo `P`, counting multiplicities """ return sum([rr[1] for rr in PolynomialRing(F, 'x')([d, c, b, 1]).roots()],0) | def _pcubicroots(b, c, d): r""" Local function returning the number of roots of `x^3 + b*x^2 + c*x + d` modulo `P`, counting multiplicities """ return sum([rr[1] for rr in PolynomialRing(F, 'x')([d, c, b, 1]).roots()],0) | 459,122 |
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... | 459,123 |
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... | 459,124 |
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... | 459,125 |
def global_integral_model(self): r""" Return a model of self which is integral at all primes. EXAMPLES:: | def global_integral_model(self): r""" Return a model of self which is integral at all primes. EXAMPLES:: | 459,126 |
def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | 459,127 |
def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | 459,128 |
def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | 459,129 |
def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | 459,130 |
def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | 459,131 |
def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | 459,132 |
def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | 459,133 |
def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | def reduced_rauzy_graph(self, n): r""" Returns the reduced Rauzy graph of order `n` of self. | 459,134 |
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... | 459,135 |
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... | 459,136 |
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 + ... | 459,137 |
def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points. INPUT: | def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points. Used internally by the :meth:`~rank`, :meth:`~rank_bounds` and :meth:`~gens` methods. INPUT: | 459,138 |
def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points. INPUT: | def simon_two_descent(self, verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Computes lower and upper bounds on the rank of the Mordell-Weil group, and a list of independent points. INPUT: | 459,139 |
def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached. | def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached. | 459,140 |
def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached. | def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached. | 459,141 |
def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached. | def rank_bounds(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns the lower and upper bounds using simon_two_descent. The results of simon_two_descent are cached. | 459,142 |
def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | 459,143 |
def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | 459,144 |
def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | 459,145 |
def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | 459,146 |
def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | 459,147 |
def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | def rank(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Return the rank of this elliptic curve, if it can be determined. | 459,148 |
def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators. | def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators. | 459,149 |
def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators. | def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators. | 459,150 |
def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators. | def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators. | 459,151 |
def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators. | def gens(self,verbose=0, lim1=5, lim3=50, limtriv=10, maxprob=20, limbigprime=30): r""" Returns some generators of this elliptic curve. Check rank or rank_bound to verify the number of generators. | 459,152 |
def WeylCharacterRing(ct, base_ring=ZZ, prefix=None, cache=False, style="lattice"): r""" A class for rings of Weyl characters. The Weyl character is a character of a semisimple (or reductive) Lie group or algebra. They form a ring, in which the addition and multiplication correspond to direct sum and tensor product of ... | def WeylCharacterRing(ct, base_ring=ZZ, prefix=None, cache=False, style="lattice"): r""" A class for rings of Weyl characters. The Weyl character is a character of a semisimple (or reductive) Lie group or algebra. They form a ring, in which the addition and multiplication correspond to direct sum and tensor product of ... | 459,153 |
def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Will... | def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Will... | 459,154 |
def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Will... | def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Will... | 459,155 |
def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Will... | def branch_weyl_character(chi, R, S, rule="default"): r""" A Branching rule describes the restriction of representations from a Lie group or algebra G to a smaller one. See for example, R. C. King, Branching rules for classical Lie groups using tensor and spinor methods. J. Phys. A 8 (1975), 429-449, Howe, Tan and Will... | 459,156 |
def rule(x) : x[len(x)-1] = -x[len(x)-1]; return x | def rule(x) : x[len(x)-1] = -x[len(x)-1]; return x | 459,157 |
def __call__(self, *args): """ Coerces the element into the ring. INPUT: - ``x`` - a ring element EXAMPLES:: sage: a2 = WeightRing(WeylCharacterRing(['A',2])) sage: a2(-1) -a2(0,0,0) """ if len(args) == 1: x = args[0] else: x = args if x == 0: return WeightRingElement(self, {}) if x in ZZ: mdict = {self._origin:... | def __call__(self, *args): """ Coerces the element into the ring. INPUT: - ``x`` - a ring element EXAMPLES:: sage: a2 = WeightRing(WeylCharacterRing(['A',2])) sage: a2(-1) -a2(0,0,0) """ if len(args) == 1: x = args[0] else: x = args if x == 0 and not x in self._space: return WeightRingElement(self, {}) if x in ... | 459,158 |
def steiner_tree(self,vertices, weighted = False): r""" Returns a tree of minimum weight connecting the given set of vertices. | def steiner_tree(self,vertices, weighted = False): r""" Returns a tree of minimum weight connecting the given set of vertices. | 459,159 |
def steiner_tree(self,vertices, weighted = False): r""" Returns a tree of minimum weight connecting the given set of vertices. | def steiner_tree(self,vertices, weighted = False): r""" Returns a tree of minimum weight connecting the given set of vertices. | 459,160 |
def steiner_tree(self,vertices, weighted = False): r""" Returns a tree of minimum weight connecting the given set of vertices. | def steiner_tree(self,vertices, weighted = False): r""" Returns a tree of minimum weight connecting the given set of vertices. | 459,161 |
sage: def naive_height(P): | sage: def naive_height(P): | 459,162 |
def minpoly(ex, var='x', algorithm=None, bits=None, degree=None, epsilon=0): r""" Return the minimal polynomial of self, if possible. INPUT: - ``var`` - polynomial variable name (default 'x') - ``algorithm`` - 'algebraic' or 'numerical' (default both, but with numerical first) - ``bits`` - the number of bits to ... | def minpoly(ex, var='x', algorithm=None, bits=None, degree=None, epsilon=0): r""" Return the minimal polynomial of self, if possible. INPUT: - ``var`` - polynomial variable name (default 'x') - ``algorithm`` - 'algebraic' or 'numerical' (default both, but with numerical first) - ``bits`` - the number of bits to ... | 459,163 |
def _limit_latex_(self, f, x, a): r""" Return latex expression for limit of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _limit_latex_ sage: var('x,a') (x, a) sage: f = function('f',x) sage: _limit_latex_(0, f, x, a) '\\lim_{x \\to a}\\, f\\left(x\\right)' sage: latex(limit(f, x=oo)) \lim_... | def _limit_latex_(self, f, x, a): r""" Return latex expression for limit of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _limit_latex_ sage: var('x,a') (x, a) sage: f = function('f',x) sage: _limit_latex_(0, f, x, a) '\\lim_{x \\to a}\\, f\\left(x\\right)' sage: latex(limit(f, x=oo)) \lim_... | 459,164 |
def _laplace_latex_(self, *args): r""" Return LaTeX expression for Laplace transform of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _laplace_latex_ sage: var('s,t') (s, t) sage: f = function('f',t) sage: _laplace_latex_(0,f,t,s) '\\mathcal{L}\\left(f\\left(t\\right), t, s\\right)' sage: l... | def _laplace_latex_(self, *args): r""" Return LaTeX expression for Laplace transform of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _laplace_latex_ sage: var('s,t') (s, t) sage: f = function('f',t) sage: _laplace_latex_(0,f,t,s) '\\mathcal{L}\\left(f\\left(t\\right), t, s\\right)' sage: l... | 459,165 |
def _inverse_laplace_latex_(self, *args): r""" Return LaTeX expression for inverse Laplace transform of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _inverse_laplace_latex_ sage: var('s,t') (s, t) sage: F = function('F',s) sage: _inverse_laplace_latex_(0,F,s,t) '\\mathcal{L}^{-1}\\left(F\\... | def _inverse_laplace_latex_(self, *args): r""" Return LaTeX expression for inverse Laplace transform of a symbolic function. EXAMPLES:: sage: from sage.calculus.calculus import _inverse_laplace_latex_ sage: var('s,t') (s, t) sage: F = function('F',s) sage: _inverse_laplace_latex_(0,F,s,t) '\\mathcal{L}^{-1}\\left(F\\... | 459,166 |
def metaclass(name, bases): """ Creates a new class in this metaclass INPUT:: - name: a string - bases: a tuple of classes EXAMPLES:: sage: from sage.misc.test_class_pickling import metaclass, bar sage: c = metaclass("foo2", (object, bar,)) constructing class sage: c <class 'sage.misc.test_class_pickling.foo2'> sage... | def metaclass(name, bases): """ Creates a new class in this metaclass INPUT:: - name: a string - bases: a tuple of classes EXAMPLES:: sage: from sage.misc.test_class_pickling import metaclass, bar sage: c = metaclass("foo2", (object, bar,)) constructing class sage: c <class 'sage.misc.test_class_pickling.foo2'> sage... | 459,167 |
def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT: | def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT: | 459,168 |
def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT: | def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT: | 459,169 |
def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT: | def __call__(self, w, order=1, datatype='iter'): r""" Returns the image of ``w`` under self to the given order. INPUT: | 459,170 |
def is_prolongable(self, letter): r""" Returns ``True`` if ``self`` is prolongable on ``letter``. A morphism `\varphi` is prolongable on a letter `a` if `a` is a prefix of `\varphi(a)`. INPUT: | def is_prolongable(self, letter): r""" Returns ``True`` if ``self`` is prolongable on ``letter``. A morphism `\varphi` is prolongable on a letter `a` if `a` is a prefix of `\varphi(a)`. INPUT: | 459,171 |
def is_prolongable(self, letter): r""" Returns ``True`` if ``self`` is prolongable on ``letter``. A morphism `\varphi` is prolongable on a letter `a` if `a` is a prefix of `\varphi(a)`. INPUT: | defis_prolongable(self,letter):r"""Returns``True``if``self``isprolongableon``letter``.Amorphism`\varphi`isprolongableonaletter`a`if`a`isaprefixof`\varphi(a)`.INPUT: | 459,172 |
def is_prolongable(self, letter): r""" Returns ``True`` if ``self`` is prolongable on ``letter``. A morphism `\varphi` is prolongable on a letter `a` if `a` is a prefix of `\varphi(a)`. INPUT: | def is_prolongable(self, letter): r""" Returns ``True`` if ``self`` is prolongable on ``letter``. A morphism `\varphi` is prolongable on a letter `a` if `a` is a prefix of `\varphi(a)`. INPUT: | 459,173 |
def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | def _fixed_point_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | 459,174 |
def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | 459,175 |
def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | 459,176 |
def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | 459,177 |
def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | def letter_iterator(self, letter): r""" Returns an iterator of the letters of the fixed point of ``self`` starting with ``letter``. | 459,178 |
def fixed_point(self, letter): r""" Returns the fixed point of ``self`` beginning by the given ``letter``. | def fixed_point(self, letter): r""" Returns the fixed point of ``self`` beginning by the given ``letter``. | 459,179 |
def fixed_point(self, letter): r""" Returns the fixed point of ``self`` beginning by the given ``letter``. | def fixed_point(self, letter): r""" Returns the fixed point of ``self`` beginning by the given ``letter``. | 459,180 |
def install_package(package=None, force=False): """ Install a package or return a list of all packages that have been installed into this Sage install. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. It is not needed to provide the version number. INPUT: - ... | def install_package(package=None, force=False): """ Install a package or return a list of all packages that have been installed into this Sage install. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. It is not needed to provide the version number. INPUT: - ... | 459,181 |
def install_package(package=None, force=False): """ Install a package or return a list of all packages that have been installed into this Sage install. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. It is not needed to provide the version number. INPUT: - ... | def install_package(package=None, force=False): """ Install a package or a list of all packages that have been installed into this Sage install. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. It is not needed to provide the version number. INPUT: - ``pac... | 459,182 |
def upgrade(): """ Download and build the latest version of Sage. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. This upgrades to the latest version of core packages (optional packages are not automatically upgraded). This will not work on systems that don't... | def upgrade(): """ Download and build the latest version of Sage. You must have an internet connection. Also, you will have to restart Sage for the changes to take affect. This upgrades to the latest version of core packages (optional packages are not automatically upgraded). This will not work on systems that don't... | 459,183 |
def __mul__(self, other): r""" Calculate the product self * other. | def __mul__(self, other): r""" Calculate the product self * other. | 459,184 |
def _compute_faces(self): r""" Compute and cache faces of this polytope. If this polytope is reflexive and the polar polytope was already computed, computes faces of both in order to save time and preserve the one-to-one correspondence between the faces of this polytope of dimension d and the faces of the polar polyto... | def _compute_faces(self): r""" Compute and cache faces of this polytope. If this polytope is reflexive and the polar polytope was already computed, computes faces of both in order to save time and preserve the one-to-one correspondence between the faces of this polytope of dimension d and the faces of the polar polyto... | 459,185 |
def faces(self, dim=None, codim=None): r""" Return the sequence of faces of this polytope. If ``dim`` or ``codim`` are specified, returns a sequence of faces of the corresponding dimension or codimension. Otherwise returns the sequence of such sequences for all dimensions. EXAMPLES: All faces of the 3-dimensional oct... | def faces(self, dim=None, codim=None): r""" Return the sequence of proper faces of this polytope. If ``dim`` or ``codim`` are specified, returns a sequence of faces of the corresponding dimension or codimension. Otherwise returns the sequence of such sequences for all dimensions. EXAMPLES: All faces of the 3-dimensio... | 459,186 |
def points(self): r""" Return all lattice points of this polytope as columns of a matrix. EXAMPLES: The lattice points of the 3-dimensional octahedron and its polar cube:: sage: o = lattice_polytope.octahedron(3) sage: o.points() [ 1 0 0 -1 0 0 0] [ 0 1 0 0 -1 0 0] [ 0 0 1 0 0 -1 0] sage: cube = o.pola... | def points(self): r""" Return all lattice points of this polytope as columns of a matrix. EXAMPLES: The lattice points of the 3-dimensional octahedron and its polar cube:: sage: o = lattice_polytope.octahedron(3) sage: o.points() [ 1 0 0 -1 0 0 0] [ 0 1 0 0 -1 0 0] [ 0 0 1 0 0 -1 0] sage: cube = o.pola... | 459,187 |
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. | 459,188 |
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. | 459,189 |
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. | 459,190 |
def __cmp__(self, other): P = self.parent() if P.eval("%s %s %s"%(self.name(), P._equality_symbol(), other.name())) == P._true_symbol(): return 0 elif P.eval("%s %s %s"%(self.name(), P._lessthan_symbol(), other.name())) == P._true_symbol(): return -1 elif P.eval("%s %s %s"%(self.name(), P._greaterthan_symbol(), other.n... | def __cmp__(self, other): P = self.parent() if P.eval("%s %s %s"%(self.name(), P._equality_symbol(), other.name())) == P._true_symbol(): return 0 elif P.eval("%s %s %s"%(self.name(), P._lessthan_symbol(), other.name())) == P._true_symbol(): return -1 elif P.eval("%s %s %s"%(self.name(), P._greaterthan_symbol(), other.n... | 459,191 |
def NumberField(polynomial, name=None, check=True, names=None, cache=True, embedding=None, latex_name=None): r""" Return *the* number field defined by the given irreducible polynomial and with variable with the given name. If check is True (the default), also verify that the defining polynomial is irreducible and over ... | def NumberField(polynomial, name=None, check=True, names=None, cache=True, embedding=None, latex_name=None): r""" Return *the* number field defined by the given irreducible polynomial and with variable with the given name. If check is True (the default), also verify that the defining polynomial is irreducible and over ... | 459,192 |
def NumberField(polynomial, name=None, check=True, names=None, cache=True, embedding=None, latex_name=None): r""" Return *the* number field defined by the given irreducible polynomial and with variable with the given name. If check is True (the default), also verify that the defining polynomial is irreducible and over ... | def NumberField(polynomial, name=None, check=True, names=None, cache=True, embedding=None, latex_name=None): r""" Return *the* number field defined by the given irreducible polynomial and with variable with the given name. If check is True (the default), also verify that the defining polynomial is irreducible and over ... | 459,193 |
def __init__(self, polynomial, name=None, check=True, embedding=None, latex_name=None): """ Create a quadratic number field. EXAMPLES:: sage: k.<a> = QuadraticField(5, check=False); k Number Field in a with defining polynomial x^2 - 5 Don't do this:: sage: k.<a> = QuadraticField(4, check=False); k Number Field in a... | def __init__(self, polynomial, name=None, latex_name=None, check=True, embedding=None): """ Create a quadratic number field. EXAMPLES:: sage: k.<a> = QuadraticField(5, check=False); k Number Field in a with defining polynomial x^2 - 5 Don't do this:: sage: k.<a> = QuadraticField(4, check=False); k Number Field in a... | 459,194 |
def is_S_integral(self,S): r""" Return True if this fractional ideal is integral with respect to the list of primes ``S``. | def is_S_integral(self,S): r""" Return True if this fractional ideal is integral with respect to the list of primes ``S``. | 459,195 |
def __call__(self, im_gens, check=True): """ Return the homomorphism defined by images of generators. | def __call__(self, im_gens, check=True): """ Return the homomorphism defined by images of generators. | 459,196 |
def _call_element_(self, _the_element, *args, **kwds): """ Calling a callable symbolic expression returns a symbolic expression with the appropriate arguments substituted. EXAMPLES:: sage: var('a, x, y, z') (a, x, y, z) sage: f(x,y) = a + 2*x + 3*y + z sage: f (x, y) |--> a + 2*x + 3*y + z sage: f(1,2) a + z + 8 sage... | def _call_element_(self, _the_element, *args, **kwds): """ Calling a callable symbolic expression returns a symbolic expression with the appropriate arguments substituted. EXAMPLES:: sage: var('a, x, y, z') (a, x, y, z) sage: f(x,y) = a + 2*x + 3*y + z sage: f (x, y) |--> a + 2*x + 3*y + z sage: f(1,2) a + z + 8 sage... | 459,197 |
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. ... | 459,198 |
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. ... | 459,199 |
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