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def vectors_by_length(self, bound): """ Returns a list of short vectors together with their values. This is a naive algorithm which uses the Cholesky decomposition, but does not use the LLL-reduction algorithm. INPUT: bound -- an integer >= 0 OUTPUT: A list L of length (bound + 1) whose entry L[i] is a list of all v...
def vectors_by_length(self, bound): """ Returns a list of short vectors together with their values. This is a naive algorithm which uses the Cholesky decomposition, but does not use the LLL-reduction algorithm. INPUT: bound -- an integer >= 0 OUTPUT: A list L of length (bound + 1) whose entry L[i] is a list of all v...
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def vectors_by_length(self, bound): """ Returns a list of short vectors together with their values. This is a naive algorithm which uses the Cholesky decomposition, but does not use the LLL-reduction algorithm. INPUT: bound -- an integer >= 0 OUTPUT: A list L of length (bound + 1) whose entry L[i] is a list of all v...
def vectors_by_length(self, bound): """ Returns a list of short vectors together with their values. This is a naive algorithm which uses the Cholesky decomposition, but does not use the LLL-reduction algorithm. INPUT: bound -- an integer >= 0 OUTPUT: A list L of length (bound + 1) whose entry L[i] is a list of all v...
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def vectors_by_length(self, bound): """ Returns a list of short vectors together with their values. This is a naive algorithm which uses the Cholesky decomposition, but does not use the LLL-reduction algorithm. INPUT: bound -- an integer >= 0 OUTPUT: A list L of length (bound + 1) whose entry L[i] is a list of all v...
def vectors_by_length(self, bound): """ Returns a list of short vectors together with their values. This is a naive algorithm which uses the Cholesky decomposition, but does not use the LLL-reduction algorithm. INPUT: bound -- an integer >= 0 OUTPUT: A list L of length (bound + 1) whose entry L[i] is a list of all v...
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def vectors_by_length(self, bound): """ Returns a list of short vectors together with their values. This is a naive algorithm which uses the Cholesky decomposition, but does not use the LLL-reduction algorithm. INPUT: bound -- an integer >= 0 OUTPUT: A list L of length (bound + 1) whose entry L[i] is a list of all v...
def vectors_by_length(self, bound): """ Returns a list of short vectors together with their values. This is a naive algorithm which uses the Cholesky decomposition, but does not use the LLL-reduction algorithm. INPUT: bound -- an integer >= 0 OUTPUT: A list L of length (bound + 1) whose entry L[i] is a list of all v...
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... def __repr__(self):
... def __repr__(self):
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... def __repr__(self):
... def __repr__(self):
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... def __repr__(self):
... def __repr__(self):
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... def __repr__(self):
... def __repr__(self):
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... def __repr__(self):
... def __repr__(self):
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... def __repr__(self):
... def __repr__(self):
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def __mod__(self, args): """ Binds the lazy format with its parameters
def __mod__(self, args): """ Binds the lazy format with its parameters
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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.
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.
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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.
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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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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.
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def evalunitdict(): """ Replace all the string values of the unitdict variable by their evaluated forms, and builds some other tables for ease of use. This function is mainly used internally, for efficiency (and flexibility) purposes, making it easier to describe the units. EXAMPLES:: sage: sage.symbolic.units.evalun...
def evalunitdict(): """ Replace all the string values of the unitdict variable by their evaluated forms, and builds some other tables for ease of use. This function is mainly used internally, for efficiency (and flexibility) purposes, making it easier to describe the units. EXAMPLES:: sage: sage.symbolic.units.evalun...
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def evalunitdict(): """ Replace all the string values of the unitdict variable by their evaluated forms, and builds some other tables for ease of use. This function is mainly used internally, for efficiency (and flexibility) purposes, making it easier to describe the units. EXAMPLES:: sage: sage.symbolic.units.evalun...
def evalunitdict(): """ Replace all the string values of the unitdict variable by their evaluated forms, and builds some other tables for ease of use. This function is mainly used internally, for efficiency (and flexibility) purposes, making it easier to describe the units. EXAMPLES:: sage: sage.symbolic.units.evalun...
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def evalunitdict(): """ Replace all the string values of the unitdict variable by their evaluated forms, and builds some other tables for ease of use. This function is mainly used internally, for efficiency (and flexibility) purposes, making it easier to describe the units. EXAMPLES:: sage: sage.symbolic.units.evalun...
def evalunitdict(): """ Replace all the string values of the unitdict variable by their evaluated forms, and builds some other tables for ease of use. This function is mainly used internally, for efficiency (and flexibility) purposes, making it easier to describe the units. EXAMPLES:: sage: sage.symbolic.units.evalun...
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def str_to_unit(name): """ Create the symbolic unit with given name. A symbolic unit is a class that derives from symbolic expression, and has a specialized docstring. INPUT: - ``name`` -- string OUTPUT: - UnitExpression EXAMPLES:: sage: sage.symbolic.units.str_to_unit('acre') acre sage: type(sage.symbolic.units...
def str_to_unit(name): """ Create the symbolic unit with given name. A symbolic unit is a class that derives from symbolic expression, and has a specialized docstring. INPUT: - ``name`` -- string OUTPUT: - UnitExpression EXAMPLES:: sage: sage.symbolic.units.str_to_unit('acre') acre sage: type(sage.symbolic.units...
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def __init__(self, data, name=''): """ EXAMPLES::
def __init__(self, data, name=''): """ EXAMPLES::
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def __getattr__(self, name): """ Return the unit with the given name.
def __getattr__(self, name): """ Return the unit with the given name.
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def __repr__(self): """ Return string representation of this collection of units.
def __repr__(self): """ Return string representation of this collection of units.
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def solve(f, *args, **kwds): r""" Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported. INPUT: - ``f`` - equation or system of equations (given by a list or tuple) - ``*args`` - variables to solve for. - ...
def solve(f, *args, **kwds): r""" Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported. INPUT: - ``f`` - equation or system of equations (given by a list or tuple) - ``*args`` - variables to solve for. - ...
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def solve(f, *args, **kwds): r""" Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported. INPUT: - ``f`` - equation or system of equations (given by a list or tuple) - ``*args`` - variables to solve for. - ...
def solve(f, *args, **kwds): r""" Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported. INPUT: - ``f`` - equation or system of equations (given by a list or tuple) - ``*args`` - variables to solve for. - ...
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def solve(f, *args, **kwds): r""" Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported. INPUT: - ``f`` - equation or system of equations (given by a list or tuple) - ``*args`` - variables to solve for. - ...
def solve(f, *args, **kwds): r""" Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported. INPUT: - ``f`` - equation or system of equations (given by a list or tuple) - ``*args`` - variables to solve for. - ...
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def solve(f, *args, **kwds): r""" Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported. INPUT: - ``f`` - equation or system of equations (given by a list or tuple) - ``*args`` - variables to solve for. - ...
def solve(f, *args, **kwds): r""" Algebraically solve an equation or system of equations (over the complex numbers) for given variables. Inequalities and systems of inequalities are also supported. INPUT: - ``f`` - equation or system of equations (given by a list or tuple) - ``*args`` - variables to solve for. - ...
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def solve_mod(eqns, modulus, solution_dict = False): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. By default the solutions are returned as `n`-tuples, where `n` is the number of variables appearing an...
def solve_mod(eqns, modulus, solution_dict = False): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. By default the solutions are returned as `n`-tuples, where `n` is the number of variables appearing an...
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def solve_mod(eqns, modulus, solution_dict = False): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. By default the solutions are returned as `n`-tuples, where `n` is the number of variables appearing an...
def solve_mod(eqns, modulus, solution_dict = False): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. By default the solutions are returned as `n`-tuples, where `n` is the number of variables appearing an...
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def solve_mod(eqns, modulus, solution_dict = False): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. By default the solutions are returned as `n`-tuples, where `n` is the number of variables appearing an...
def solve_mod(eqns, modulus, solution_dict = False): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. By default the solutions are returned as `n`-tuples, where `n` is the number of variables appearing an...
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def solve_mod(eqns, modulus, solution_dict = False): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. By default the solutions are returned as `n`-tuples, where `n` is the number of variables appearing an...
def solve_mod(eqns, modulus, solution_dict = False): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. By default the solutions are returned as `n`-tuples, where `n` is the number of variables appearing an...
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def solve_mod_enumerate(eqns, modulus): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. The solutions are returned as `n`-tuples, where `n` is the number of variables appearing anywhere in the given equa...
def solve_mod_enumerate(eqns, modulus): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. The solutions are returned as `n`-tuples, where `n` is the number of variables appearing anywhere in the given equa...
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def solve_mod_enumerate(eqns, modulus): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. The solutions are returned as `n`-tuples, where `n` is the number of variables appearing anywhere in the given equa...
def solve_mod_enumerate(eqns, modulus): r""" Return all solutions to an equation or list of equations modulo the given integer modulus. Each equation must involve only polynomials in 1 or many variables. The solutions are returned as `n`-tuples, where `n` is the number of variables appearing anywhere in the given equa...
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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...
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def _sympy_(self): """ Converts pi to sympy pi.
def _sympy_(self): """ Converts pi to sympy pi.
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def is_integral_domain(self, proof = True): r""" If this function returns ``True`` then self is definitely an integral domain. If it returns ``False``, then either self is definitely not an integral domain or this function was unable to determine whether or not self is an integral domain.
def is_integral_domain(self, proof = True): r""" If this function returns ``True`` then self is definitely an integral domain. If it returns ``False``, then either self is definitely not an integral domain or this function was unable to determine whether or not self is an integral domain.
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def is_integral_domain(self, proof = True): r""" If this function returns ``True`` then self is definitely an integral domain. If it returns ``False``, then either self is definitely not an integral domain or this function was unable to determine whether or not self is an integral domain.
def is_integral_domain(self, proof = True): r""" If this function returns ``True`` then self is definitely an integral domain. If it returns ``False``, then either self is definitely not an integral domain or this function was unable to determine whether or not self is an integral domain.
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def is_integral_domain(self, proof = True): r""" If this function returns ``True`` then self is definitely an integral domain. If it returns ``False``, then either self is definitely not an integral domain or this function was unable to determine whether or not self is an integral domain.
def is_integral_domain(self, proof = True): r""" If this function returns ``True`` then self is definitely an integral domain. If it returns ``False``, then either self is definitely not an integral domain or this function was unable to determine whether or not self is an integral domain.
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def global_integral_model(self): r""" Return a model of self which is integral at all primes.
def global_integral_model(self): r""" Return a model of self which is integral at all primes.
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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: - ...
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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: - ``pack...
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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...
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def _ambient_space_point(self, data): r""" Try to convert ``data`` to a point of the ambient space of ``self``.
def _ambient_space_point(self, data): r""" Try to convert ``data`` to a point of the ambient space of ``self``.
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def contains(self, *args): r""" Check if a given point is contained in ``self``.
def contains(self, *args): r""" Check if a given point is contained in ``self``.
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def contains(self, *args): r""" Check if a given point is contained in ``self``.
def contains(self, *args): r""" Check if a given point is contained in ``self``.
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def dual(self): r""" Return the dual cone of ``self``.
def dual(self): r""" Return the dual cone of ``self``.
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def facet_normals(self): r""" Return normals to facets of ``self``.
def facet_normals(self): r""" Return normals to facets of ``self``.
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def _split_ambient_lattice(self): r""" Compute a decomposition of the ``N``-lattice into `N_\sigma` and its complement `N(\sigma)`.
def _split_ambient_lattice(self): r""" Compute a decomposition of the ``N``-lattice into `N_\sigma` and its complement `N(\sigma)`.
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def _split_ambient_lattice(self): r""" Compute a decomposition of the ``N``-lattice into `N_\sigma` and its complement `N(\sigma)`.
def _split_ambient_lattice(self): r""" Compute a decomposition of the ``N``-lattice into `N_\sigma` and its complement `N(\sigma)`.
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def orthogonal_sublattice(self, *args, **kwds): r""" The sublattice (in the dual lattice) orthogonal to the sublattice spanned by the cone.
def orthogonal_sublattice(self, *args, **kwds): r""" The sublattice (in the dual lattice) orthogonal to the sublattice spanned by the cone.
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def spherical_bessel_J(n, var, algorithm="maxima"): r""" Returns the spherical Bessel function of the first kind for integers n -1. Reference: AS 10.1.8 page 437 and AS 10.1.15 page 439. EXAMPLES:: sage: spherical_bessel_J(2,x) ((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x """ if algorithm=="scipy": import scipy.special ans =...
def spherical_bessel_J(n, var, algorithm="maxima"): r""" Returns the spherical Bessel function of the first kind for integers n >= 1. Reference: AS 10.1.8 page 437 and AS 10.1.15 page 439. EXAMPLES:: sage: spherical_bessel_J(2,x) ((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x """ if algorithm=="scipy": import scipy.special ans...
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def spherical_bessel_J(n, var, algorithm="maxima"): r""" Returns the spherical Bessel function of the first kind for integers n -1. Reference: AS 10.1.8 page 437 and AS 10.1.15 page 439. EXAMPLES:: sage: spherical_bessel_J(2,x) ((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x """ if algorithm=="scipy": import scipy.special ans =...
def spherical_bessel_J(n, var, algorithm="maxima"): r""" Returns the spherical Bessel function of the first kind for integers n -1. Reference: AS 10.1.8 page 437 and AS 10.1.15 page 439. EXAMPLES:: sage: spherical_bessel_J(2,x) ((3/x^2 - 1)*sin(x) - 3*cos(x)/x)/x """ if algorithm=="scipy": from scipy.special.specfun...
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def __cmp__(self, other): r""" Define comparison for finite posets.
def __hash__(self): """ TESTS:: sage: P = Poset([[1,2],[3],[3]]) sage: P.__hash__() 6557284140853143473 584755121 sage: P = Poset([[1],[3],[3]]) sage: P.__hash__() 5699294501102840900 278031428 """ if self._hash is None: self._hash = tuple(map(tuple, self.cover_relations())).__hash__() return self._hash def __eq__(se...
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def __cmp__(self, other): r""" Define comparison for finite posets.
def __cmp__(self, other): r""" Define comparison for finite posets.
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def __cmp__(self, other): r""" Define comparison for finite posets.
def __cmp__(self, other): r""" Define comparison for finite posets.
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def __cmp__(self, other): r""" Define comparison for finite posets.
def __cmp__(self, other): r""" Define comparison for finite posets.
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def __cmp__(self, other): r""" Define comparison for finite posets.
def __cmp__(self, other): r""" Define comparison for finite posets.
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def __cmp__(self, other): r""" Define comparison for finite posets.
def __cmp__(self, other): r""" Define comparison for finite posets.
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def squarefree_part(x): """ Returns the square free part of `x`, i.e., a divisor `z` such that `x = z y^2`, for a perfect square `y^2`. EXAMPLES:: sage: squarefree_part(100) 1 sage: squarefree_part(12) 3 sage: squarefree_part(10) 10 :: sage: x = QQ['x'].0 sage: S = squarefree_part(-9*x*(x-6)^7*(x-3)^2); S -9*x^2 + ...
def squarefree_part(x): """ Returns the square free part of `x`, i.e., a divisor `z` such that `x = z y^2`, for a perfect square `y^2`. EXAMPLES:: sage: squarefree_part(100) 1 sage: squarefree_part(12) 3 sage: squarefree_part(10) 10 :: sage: x = QQ['x'].0 sage: S = squarefree_part(-9*x*(x-6)^7*(x-3)^2); S -9*x^2 + ...
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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 edges(self, labels=True, sort=True, key=None): r""" Return a list of the edges of the graph as triples (u,v,l) where u and v are vertices and l is a label.
def edges(self, labels=True, sort=True, key=None): r""" Return a list of the edges of the graph as triples (u,v,l) where u and v are vertices and l is a label.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True, sort=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
def edge_boundary(self, vertices1, vertices2=None, labels=True): """ Returns a list of edges `(u,v,l)` with `u` in ``vertices1`` and `v` in ``vertices2``. If ``vertices2`` is ``None``, then it is set to the complement of ``vertices1``.
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def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
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def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
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def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
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def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
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def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
def edge_iterator(self, vertices=None, labels=True, ignore_direction=False): """ Returns an iterator over the edges incident with any vertex given. If the graph is directed, iterates over edges going out only. If vertices is None, then returns an iterator over all edges. If self is directed, returns outgoing edges only...
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def edges_incident(self, vertices=None, labels=True): """ Returns a list of edges incident with any vertex given. If vertices is None, returns a list of all edges in graph. For digraphs, only lists outward edges.
def edges_incident(self, vertices=None, labels=True): """ Returns a list of edges incident with any vertex given. If vertices is None, returns a list of all edges in graph. For digraphs, only lists outward edges.
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def edges_incident(self, vertices=None, labels=True): """ Returns a list of edges incident with any vertex given. If vertices is None, returns a list of all edges in graph. For digraphs, only lists outward edges.
def edges_incident(self, vertices=None, labels=True): """ Returns a list of edges incident with any vertex given. If vertices is None, returns a list of all edges in graph. For digraphs, only lists outward edges.
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def hom(self, im_gens, codomain=None, check=True): """ Homomorphism defined by giving the images of ``self.gens()`` in some fixed fg R-module.
def hom(self, im_gens, codomain=None, check=True): """ Homomorphism defined by giving the images of ``self.gens()`` in some fixed fg R-module.
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def hom(self, im_gens, codomain=None, check=True): """ Homomorphism defined by giving the images of ``self.gens()`` in some fixed fg R-module.
def hom(self, im_gens, codomain=None, check=True): """ Homomorphism defined by giving the images of ``self.gens()`` in some fixed fg R-module.
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def canonical_label(self, partition=None, certify=False, verbosity=0, edge_labels=False): """ Returns the unique graph on \{0,1,...,n-1\} ( n = self.order() ) which - is isomorphic to self, - is invariant in the isomorphism class.
def canonical_label(self, partition=None, certify=False, verbosity=0, edge_labels=False): """ Returns the unique graph on \{0,1,...,n-1\} ( n = self.order() ) which - is isomorphic to self, - is invariant in the isomorphism class.
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def canonical_label(self, partition=None, certify=False, verbosity=0, edge_labels=False): """ Returns the unique graph on \{0,1,...,n-1\} ( n = self.order() ) which - is isomorphic to self, - is invariant in the isomorphism class.
def canonical_label(self, partition=None, certify=False, verbosity=0, edge_labels=False): """ Returns the unique graph on \{0,1,...,n-1\} ( n = self.order() ) which - is isomorphic to self, - is invariant in the isomorphism class.
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def has_good_reduction(self, P=None): r""" Returns True iff this point has good reduction modulo a prime.
def has_good_reduction(self, P=None): r""" Returns True iff this point has good reduction modulo a prime.
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def has_good_reduction(self, P=None): r""" Returns True iff this point has good reduction modulo a prime.
def has_good_reduction(self, P=None): r""" Returns True iff this point has good reduction modulo a prime.
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def longest_path(self, s=None, t=None, weighted=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
def longest_path(self, s=None, t=None, weighted=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
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def longest_path(self, s=None, t=None, weighted=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
def longest_path(self, s=None, t=None, weighted=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
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def longest_path(self, s=None, t=None, weighted=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
def longest_path(self, s=None, t=None, weighted=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
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def longest_path(self, s=None, t=None, weighted=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
def longest_path(self, s=None, t=None, weighted=False, algorithm="MILP", solver=None, verbose=0): r""" Returns a longest path of ``self``.
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