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def points(self): """ Return a list of the generating points in this Mordell-Weil group.
def points(self): """ Return a list of the generating points in this Mordell-Weil group.
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def _latex_(self): """ Return Latex representation of this Maxima object.
def _latex_(self): """ Return Latex representation of this Maxima object.
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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 ModularForms(group = 1, weight = 2, base_ring = None, use_cache = True, prec = defaults.DEFAULT_PRECISION): r""" Create an ambient space of modular forms. INPUT: - ``group`` - A congruence subgroup or a Dirichlet character eps. - ``weight`` - int, the weight, which must be an integer = 1. - ``base_ring`` - ...
def ModularForms(group = 1, weight = 2, base_ring = None, use_cache = True, prec = defaults.DEFAULT_PRECISION): r""" Create an ambient space of modular forms. INPUT: - ``group`` - A congruence subgroup or a Dirichlet character eps. - ``weight`` - int, the weight, which must be an integer = 1. - ``base_ring`` - ...
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points_iterator(self): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
def rational_points(self, algorithm="enum", sort=True): r""" Return the rational points on this curve computed via enumeration.
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def cardinality(self): """ Returns the cardinality of this disjoint union.
def cardinality(self): """ Returns the cardinality of this disjoint union.
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def _allowed_options(self): """ Return the allowed options for the Point class.
def _allowed_options(self): """ Return the allowed options for the Point class.
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def _allowed_options(self): """ Return the allowed options for the Point class.
def _allowed_options(self): """ Return the allowed options for the Point class.
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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 desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False): r""" Solves a 1st or 2nd order linear ODE via maxima. Including IVP and BVP. *Use* ``desolve? <tab>`` *if the output in truncated in notebook.* INPUT: - ``de`` - an expression or equation representing the ODE - ``dvar`` - the dependen...
def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False): r""" Solves a 1st or 2nd order linear ODE via maxima. Including IVP and BVP. *Use* ``desolve? <tab>`` *if the output in truncated in notebook.* INPUT: - ``de`` - an expression or equation representing the ODE - ``dvar`` - the dependen...
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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``.
def associated_primes(self, algorithm='sy'): r""" Return a list of primary ideals (and their associated primes) such that their intersection is `I` = ``self``.
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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``.
def associated_primes(self, algorithm='sy'): r""" Return a list of primary ideals (and their associated primes) such that their intersection is `I` = ``self``.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
@rename_keyword(deprecated='Sage version 4.6.1', l='arg') def iter_morphisms(self, arg=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
def iter_morphisms(self, l=None, codomain=None, min_length=1): r""" Iterate over all morphisms with domain ``self`` and the given codmain.
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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 __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(self): r""" Returns True if self is a square, and False otherwise.
def is_square(self): r""" Returns True if self is a square, and False otherwise.
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def is_square_free(self): r""" Returns True if self does not contain squares, and False otherwise.
def is_square_free(self): r""" Returns True if self does not contain squares, and False otherwise.
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def is_square_free(self): r""" Returns True if self does not contain squares, and False otherwise.
def is_square_free(self): r""" Returns True if self does not contain squares, and False otherwise.
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def is_square_free(self): r""" Returns True if self does not contain squares, and False otherwise.
def is_square_free(self): r""" Returns True if self does not contain squares, and False otherwise.
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def is_cube(self): r""" Returns True if self is a cube, and False otherwise.
def is_cube(self): r""" Returns True if self is a cube, and False otherwise.
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def is_cube_free(self): r""" Returns True if self does not contain cubes, and False otherwise.
def is_cube_free(self): r""" Returns True if self does not contain cubes, and False otherwise.
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def is_cube_free(self): r""" Returns True if self does not contain cubes, and False otherwise.
def is_cube_free(self): r""" Returns True if self does not contain cubes, and False otherwise.
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def is_cube_free(self): r""" Returns True if self does not contain cubes, and False otherwise.
def is_cube_free(self): r""" Returns True if self does not contain cubes, and False otherwise.
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def old_cremona_letter_code(n): r""" Returns the *old* Cremona letter code corresponding to an integer. integer. For example, :: 1 --> A 26 --> Z 27 --> AA 52 --> ZZ 53 --> AAA etc. INPUT: - ``n`` - int OUTPUT: str EXAMPLES:: sage: old_cremona_letter_code(1) 'A' sage: old_cremona_letter_code(26) 'Z' sage: old_...
def old_cremona_letter_code(n): r""" Returns the *old* Cremona letter code corresponding to an integer. integer. For example:: 1 --> A 26 --> Z 27 --> AA 52 --> ZZ 53 --> AAA etc. INPUT: - ``n`` - int OUTPUT: str EXAMPLES:: sage: old_cremona_letter_code(1) 'A' sage: old_cremona_letter_code(26) 'Z' sage: old_cr...
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def __iter__(self): """ Returns an iterator through all EllipticCurve objects in the Cremona database.
def __iter__(self): """ Returns an iterator through all EllipticCurve objects in the Cremona database.
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def iter(self, conductors): """ Returns an iterator through all curves with conductor between Nmin and Nmax-1, inclusive, in the database.
def iter(self, conductors): """ Returns an iterator through all curves with conductor between Nmin and Nmax-1, inclusive, in the database.
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def iter_optimal(self, conductors): """ Returns an iterator through all optimal curves with conductor between Nmin and Nmax-1 in the database.
def iter_optimal(self, conductors): """ Returns an iterator through all optimal curves with conductor between Nmin and Nmax-1 in the database.
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def list(self, conductors): """ Returns a list of all curves with conductor between Nmin and Nmax-1, inclusive, in the database.
def list(self, conductors): """ Returns a list of all curves with conductor between Nmin and Nmax-1, inclusive, in the database.
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def list_optimal(self, conductors): """ Returns a list of all optimal curves with conductor between Nmin and Nmax-1, inclusive, in the database.
def list_optimal(self, conductors): """ Returns a list of all optimal curves with conductor between Nmin and Nmax-1, inclusive, in the database.
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def smallest_conductor(self): """ The smallest conductor for which the database is complete. (Always 1.)
def smallest_conductor(self): """ The smallest conductor for which the database is complete. (Always 1.)
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def conductor_range(self): """ Return the range of conductors that are covered by the database.
def conductor_range(self): """ Return the range of conductors that are covered by the database.
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def _init_allgens(self, ftpdata, largest_conductor=0): """ Initialize the allgens table by reading the corresponding ftpdata files and importing them into the database. """ if self.read_only: raise RuntimeError, "The database must not be read_only." files = os.listdir(ftpdata) files.sort() name = "allgens" c = _map[nam...
def _init_allgens(self, ftpdata, largest_conductor=0): """ Initialize the allgens table by reading the corresponding ftpdata files and importing them into the database. """ if self.read_only: raise RuntimeError, "The database must not be read_only." files = os.listdir(ftpdata) files.sort() name = "allgens" c = _map[nam...
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def face_lattice(self): """ Computes the face-lattice poset. Elements are tuples of (vertices, facets) - i.e. this keeps track of both the vertices in each face, and all the facets containing them.
def face_lattice(self): """ Computes the face-lattice poset. Elements are tuples of (vertices, facets) - i.e. this keeps track of both the vertices in each face, and all the facets containing them.
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def library_interact(f): """ This is a decorator for using interacts in the Sage library. EXAMPLES:: sage: @interacts.decorator.library_interact ... def f(n=5): print n ... sage: f() # an interact appears <html>...</html> """ @sage_wraps(f) def library_wrapper(): # Maybe program around bug (?) in the notebook: html(...
def library_interact(f): """ This is a decorator for using interacts in the Sage library. EXAMPLES:: sage: @interacts.library.library_interact ... def f(n=5): print n ... sage: f() # an interact appears <html>...</html> """ @sage_wraps(f) def library_wrapper(): # Maybe program around bug (?) in the notebook: html("<...
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def demo(n=tuple(range(10)), m=tuple(range(10))): """ This is a demo interact that sums two numbers. INPUT: - `n` -- integer slider - `m` -- integer slider EXAMPLES:: sage: interacts.decorator.demo() <html>...</html> """ print n+m
def demo(n=tuple(range(10)), m=tuple(range(10))): """ This is a demo interact that sums two numbers. INPUT: - `n` -- integer slider - `m` -- integer slider EXAMPLES:: sage: interacts.library.demo() <html>...</html> """ print n+m
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... def variance(self, bias = False):
... def variance(self, bias = False):
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def homchain(complex=None, **kwds): r""" Compute the homology of a chain complex using the CHomP program ``homchain``. :param complex: a chain complex :param generators: if True, also return list of generators :type generators: boolean; optional, default False :param verbose: if True, print helpful messages as the com...
def homchain(complex=None, **kwds): r""" Compute the homology of a chain complex using the CHomP program ``homchain``. :param complex: a chain complex :param generators: if True, also return list of generators :type generators: boolean; optional, default False :param verbose: if True, print helpful messages as the com...
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def __cmp__(self, right): r""" Compare ``self`` and ``right``.
def __cmp__(self, right): r""" Compare ``self`` and ``right``.
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def bound_kato(self): r""" Returns a list `p` of primes such that the theorems of Kato's [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if `p` divides the order of Sha(E) then `p` is in the list.
def bound_kato(self): r""" Returns a list `p` of primes such that the theorems of Kato's [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if `p` divides the order of Sha(E) then `p` is in the list.
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def bound_kato(self): r""" Returns a list `p` of primes such that the theorems of Kato's [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if `p` divides the order of Sha(E) then `p` is in the list.
def bound_kato(self): r""" Returns a list `p` of primes such that the theorems of Kato's [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if `p` divides the order of Sha(E) then `p` is in the list.
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def bound_kato(self): r""" Returns a list `p` of primes such that the theorems of Kato's [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if `p` divides the order of Sha(E) then `p` is in the list.
def bound_kato(self): r""" Returns a list `p` of primes such that the theorems of Kato's [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if `p` divides the order of Sha(E) then `p` is in the list.
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def bound_kato(self): r""" Returns a list `p` of primes such that the theorems of Kato's [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if `p` divides the order of Sha(E) then `p` is in the list.
def bound_kato(self): r""" Returns a list `p` of primes such that the theorems of Kato's [Ka] and others (e.g., as explained in a paper/thesis of Grigor Grigorov [Gri]) imply that if `p` divides the order of Sha(E) then `p` is in the list.
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def sage_getvariablename(obj, omit_underscore_names=True): """ Attempt to get the name of a Sage object. INPUT: - ``obj`` - an object - ``omit_underscore_names`` (optional, default True) If the user has assigned an object ``obj`` to a variable name, then return that variable name. If several variables point to ``ob...
def sage_getvariablename(obj, omit_underscore_names=True): """ Attempt to get the name of a Sage object. INPUT: - ``obj`` - an object - ``omit_underscore_names`` (optional, default True) If the user has assigned an object ``obj`` to a variable name, then return that variable name. If several variables point to ``ob...
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def sage_getvariablename(obj, omit_underscore_names=True): """ Attempt to get the name of a Sage object. INPUT: - ``obj`` - an object - ``omit_underscore_names`` (optional, default True) If the user has assigned an object ``obj`` to a variable name, then return that variable name. If several variables point to ``ob...
def sage_getvariablename(obj, omit_underscore_names=True): """ Attempt to get the name of a Sage object. INPUT: - ``obj`` - an object - ``omit_underscore_names`` (optional, default True) If the user has assigned an object ``obj`` to a variable name, then return that variable name. If several variables point to ``ob...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
def algdep(z, degree, known_bits=None, use_bits=None, known_digits=None, use_digits=None, height_bound=None, proof=False): """ Returns a polynomial of degree at most `degree` which is approximately satisfied by the number `z`. Note that the returned polynomial need not be irreducible, and indeed usually won't be if `z`...
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def is_pseudoprime_small_power(n, bound=1024, get_data=False): r""" Return True if `n` is a small power of a pseudoprime, and False otherwise. The result is *NOT* proven correct - *this IS a pseudo-primality test!*. If `get_data` is set to true and `n = p^d`, for a pseudoprime `p` and power `d`, return [(p, d)]. INP...
def is_pseudoprime_small_power(n, bound=1024, get_data=False): r""" Return True if `n` is a small power of a pseudoprime, and False otherwise. The result is *NOT* proven correct - *this IS a pseudo-primality test!*. If `get_data` is set to true and `n = p^d`, for a pseudoprime `p` and power `d`, return [(p, d)]. INP...
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def factor(n, proof=None, int_=False, algorithm='pari', verbose=0, **kwds): """ Returns the factorization of n. The result depends on the type of n. If n is an integer, factor returns the factorization of the integer n as an object of type Factorization. If n is not an integer, ``n.factor(proof=proof, **kwds)`` gets ...
def factor(n, proof=None, int_=False, algorithm='pari', verbose=0, **kwds): """ Returns the factorization of n. The result depends on the type of n. If n is an integer, factor returns the factorization of the integer n as an object of type Factorization. If n is not an integer, ``n.factor(proof=proof, **kwds)`` gets ...
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def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False): """ Solves a 1st or 2nd order linear ODE via maxima. Including IVP and BVP. *Use* ``desolve? <tab>`` *if the output in truncated in notebook.* INPUT: - ``de`` - an expression or equation representing the ODE - ``dvar`` - the dependent...
def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False): """ Solves a 1st or 2nd order linear ODE via maxima. Including IVP and BVP. *Use* ``desolve? <tab>`` *if the output in truncated in notebook.* INPUT: - ``de`` - an expression or equation representing the ODE - ``dvar`` - the dependent...
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def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False): """ Solves a 1st or 2nd order linear ODE via maxima. Including IVP and BVP. *Use* ``desolve? <tab>`` *if the output in truncated in notebook.* INPUT: - ``de`` - an expression or equation representing the ODE - ``dvar`` - the dependent...
def desolve(de, dvar, ics=None, ivar=None, show_method=False, contrib_ode=False): """ Solves a 1st or 2nd order linear ODE via maxima. Including IVP and BVP. *Use* ``desolve? <tab>`` *if the output in truncated in notebook.* INPUT: - ``de`` - an expression or equation representing the ODE - ``dvar`` - the dependent...
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def desolve_system_strings(des,vars,ics=None): """ Solves any size system of 1st order ODE's. Initials conditions are optional. INPUT: de -- a list of strings representing the ODEs in maxima notation (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)") vars -- a list of strings representing the variables (eg, vars = ["...
def desolve_system_strings(des,vars,ics=None): r""" Solves any size system of 1st order ODE's. Initials conditions are optional. INPUT: de -- a list of strings representing the ODEs in maxima notation (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)") vars -- a list of strings representing the variables (eg, vars = [...
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def desolve_system_strings(des,vars,ics=None): """ Solves any size system of 1st order ODE's. Initials conditions are optional. INPUT: de -- a list of strings representing the ODEs in maxima notation (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)") vars -- a list of strings representing the variables (eg, vars = ["...
def desolve_system_strings(des,vars,ics=None): """ Solves any size system of 1st order ODE's. Initials conditions are optional. INPUT: - ``de`` - a list of strings representing the ODEs in maxima notation (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)") - ``vars`` - a list of strings representing the variables (eg, v...
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def desolve_system_strings(des,vars,ics=None): """ Solves any size system of 1st order ODE's. Initials conditions are optional. INPUT: de -- a list of strings representing the ODEs in maxima notation (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)") vars -- a list of strings representing the variables (eg, vars = ["...
def desolve_system_strings(des,vars,ics=None): """ Solves any size system of 1st order ODE's. Initials conditions are optional. INPUT: de -- a list of strings representing the ODEs in maxima notation (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)") vars -- a list of strings representing the variables (eg, vars = ["...
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def desolve_system_strings(des,vars,ics=None): """ Solves any size system of 1st order ODE's. Initials conditions are optional. INPUT: de -- a list of strings representing the ODEs in maxima notation (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)") vars -- a list of strings representing the variables (eg, vars = ["...
def desolve_system_strings(des,vars,ics=None): """ Solves any size system of 1st order ODE's. Initials conditions are optional. INPUT: de -- a list of strings representing the ODEs in maxima notation (eg, de = "diff(f(x),x,2)=diff(f(x),x)+sin(x)") vars -- a list of strings representing the variables (eg, vars = ["...
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def eulers_method(f,x0,y0,h,x1,method="table"): """ This implements Euler's method for finding numerically the solution of the 1st order ODE ``y' = f(x,y)``, ``y(a)=c``. The "x" column of the table increments from ``x0`` to ``x1`` by ``h`` (so ``(x1-x0)/h`` must be an integer). In the "y" column, the new y-value equals...
def eulers_method(f,x0,y0,h,x1,method="table"): r""" This implements Euler's method for finding numerically the solution of the 1st order ODE ``y' = f(x,y)``, ``y(a)=c``. The "x" column of the table increments from ``x0`` to ``x1`` by ``h`` (so ``(x1-x0)/h`` must be an integer). In the "y" column, the new y-value equal...
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def eulers_method_2x2(f,g, t0, x0, y0, h, t1,method="table"): """ This implements Euler's method for finding numerically the solution of the 1st order system of two ODEs ``x' = f(t, x, y), x(t0)=x0.`` ``y' = g(t, x, y), y(t0)=y0.`` The "t" column of the table increments from `t_0` to `t_1` by `h` (so `\\frac{t_1-t_0...
def eulers_method_2x2(f,g, t0, x0, y0, h, t1,method="table"): r""" This implements Euler's method for finding numerically the solution of the 1st order system of two ODEs ``x' = f(t, x, y), x(t0)=x0.`` ``y' = g(t, x, y), y(t0)=y0.`` The "t" column of the table increments from `t_0` to `t_1` by `h` (so `\\frac{t_1-t_...
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def eulers_method_2x2_plot(f,g, t0, x0, y0, h, t1): """ This plots the soln in the rectangle ``(xrange[0],xrange[1]) x (yrange[0],yrange[1])`` and plots using Euler's method the numerical solution of the 1st order ODEs `x' = f(t,x,y)`, `x(a)=x_0`, `y' = g(t,x,y)`, `y(a) = y_0`. *For pedagogical purposes only.* EXAMPL...
def eulers_method_2x2_plot(f,g, t0, x0, y0, h, t1): r""" Plots solution of ODE This plots the soln in the rectangle ``(xrange[0],xrange[1]) x (yrange[0],yrange[1])`` and plots using Euler's method the numerical solution of the 1st order ODEs `x' = f(t,x,y)`, `x(a)=x_0`, `y' = g(t,x,y)`, `y(a) = y_0`. *For pedagogical...
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def eulers_method_2x2_plot(f,g, t0, x0, y0, h, t1): """ This plots the soln in the rectangle ``(xrange[0],xrange[1]) x (yrange[0],yrange[1])`` and plots using Euler's method the numerical solution of the 1st order ODEs `x' = f(t,x,y)`, `x(a)=x_0`, `y' = g(t,x,y)`, `y(a) = y_0`. *For pedagogical purposes only.* EXAMPL...
def eulers_method_2x2_plot(f,g, t0, x0, y0, h, t1): """ This plots the soln in the rectangle ``(xrange[0],xrange[1]) x (yrange[0],yrange[1])`` and plots using Euler's method the numerical solution of the 1st order ODEs `x' = f(t,x,y)`, `x(a)=x_0`, `y' = g(t,x,y)`, `y(a) = y_0`. *For pedagogical purposes only.* EXAMPL...
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def eulers_method_2x2_plot(f,g, t0, x0, y0, h, t1): """ This plots the soln in the rectangle ``(xrange[0],xrange[1]) x (yrange[0],yrange[1])`` and plots using Euler's method the numerical solution of the 1st order ODEs `x' = f(t,x,y)`, `x(a)=x_0`, `y' = g(t,x,y)`, `y(a) = y_0`. *For pedagogical purposes only.* EXAMPL...
def eulers_method_2x2_plot(f,g, t0, x0, y0, h, t1): """ This plots the soln in the rectangle ``(xrange[0],xrange[1]) x (yrange[0],yrange[1])`` and plots using Euler's method the numerical solution of the 1st order ODEs `x' = f(t,x,y)`, `x(a)=x_0`, `y' = g(t,x,y)`, `y(a) = y_0`. *For pedagogical purposes only.* EXAMPL...
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def _compute_faces(self): r""" Compute and cache faces of this polytope.
def _compute_faces(self): r""" Compute and cache faces of this polytope.
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def faces(self, dim=None, codim=None): r""" Return the sequence of faces of this polytope.
def faces(self, dim=None, codim=None): r""" Return the sequence of faces of this polytope.
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def points(self): r""" Return all lattice points of this polytope as columns of a matrix.
def points(self): r""" Return all lattice points of this polytope as columns of a matrix.
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def integral(self, x=None, a=None, b=None, definite=False): r""" By default, returns the indefinite integral of the function. If definite=True is given, returns the definite integral.
def integral(self, x=None, a=None, b=None, definite=False): r""" By default, returns the indefinite integral of the function. If definite=True is given, returns the definite integral.
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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)`
def change_weierstrass_model(self, *urst): r""" Return a new Weierstrass model of self under the standard transformation `(u,r,s,,t)`
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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)`
def change_weierstrass_model(self, *urst): r""" Return a new Weierstrass model of self under the standard transformation `(u,r,s,,t)`
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