Asymptotic Expansions — Miscellaneous#

AUTHORS:

  • Daniel Krenn (2015)

ACKNOWLEDGEMENT:

  • Benjamin Hackl, Clemens Heuberger and Daniel Krenn are supported by the Austrian Science Fund (FWF): P 24644-N26.

  • Benjamin Hackl is supported by the Google Summer of Code 2015.

Functions, Classes and Methods#

class sage.rings.asymptotic.misc.Locals#

Bases: dict

A frozen dictionary-like class for storing locals of an AsymptoticRing.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import Locals
sage: locals = Locals({'a': 42})
sage: locals['a']
42

The object contains default values (see default_locals()) for some keys:

sage: locals['log']
<function log at 0x...>
default_locals()#

Return the default locals used in the AsymptoticRing.

OUTPUT:

A dictionary.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import Locals
sage: locals = Locals({'a': 2, 'b': 1})
sage: locals
{'a': 2, 'b': 1}
sage: locals.default_locals()
{'log': <function log at 0x...>}
sage: locals['log']
<function log at 0x...>
exception sage.rings.asymptotic.misc.NotImplementedBZero(asymptotic_ring=None, var=None, exact_part=0)#

Bases: NotImplementedError

A special NotImplementedError which is raised when the result is B(0) which means 0 for sufficiently large values of the variable.

exception sage.rings.asymptotic.misc.NotImplementedOZero(asymptotic_ring=None, var=None, exact_part=0)#

Bases: NotImplementedError

A special NotImplementedError which is raised when the result is O(0) which means 0 for sufficiently large values of the variable.

class sage.rings.asymptotic.misc.WithLocals#

Bases: SageObject

A class extensions for handling local values; see also Locals.

This is used in the AsymptoticRing.

EXAMPLES:

sage: A.<n> = AsymptoticRing('n^ZZ', QQ, locals={'a': 42})
sage: A.locals()
{'a': 42}
locals(locals=None)#

Return the actual Locals object to be used.

INPUT:

  • locals – an object

    If locals is not None, then a Locals object is created and returned. If locals is None, then a stored Locals object, if any, is returned. Otherwise, an empty (i.e. no values except the default values) Locals object is created and returned.

OUTPUT:

A Locals object.

sage.rings.asymptotic.misc.bidirectional_merge_overlapping(A, B, key=None)#

Merge the two overlapping tuples/lists.

INPUT:

  • A – a list or tuple (type has to coincide with type of B).

  • B – a list or tuple (type has to coincide with type of A).

  • key – (default: None) a function. If None, then the identity is used. This key-function applied on an element of the list/tuple is used for comparison. Thus elements with the same key are considered as equal.

OUTPUT:

A pair of lists or tuples (depending on the type of A and B).

Note

Suppose we can decompose the list \(A=ac\) and \(B=cb\) with lists \(a\), \(b\), \(c\), where \(c\) is nonempty. Then bidirectional_merge_overlapping() returns the pair \((acb, acb)\).

Suppose a key-function is specified and \(A=ac_A\) and \(B=c_Bb\), where the list of keys of the elements of \(c_A\) equals the list of keys of the elements of \(c_B\). Then bidirectional_merge_overlapping() returns the pair \((ac_Ab, ac_Bb)\).

After unsuccessfully merging \(A=ac\) and \(B=cb\), a merge of \(A=ca\) and \(B=bc\) is tried.

sage.rings.asymptotic.misc.bidirectional_merge_sorted(A, B, key=None)#

Merge the two tuples/lists, keeping the orders provided by them.

INPUT:

  • A – a list or tuple (type has to coincide with type of B).

  • B – a list or tuple (type has to coincide with type of A).

  • key – (default: None) a function. If None, then the identity is used. This key-function applied on an element of the list/tuple is used for comparison. Thus elements with the same key are considered as equal.

Note

The two tuples/list need to overlap, i.e. need at least one key in common.

OUTPUT:

A pair of lists containing all elements totally ordered. (The first component uses A as a merge base, the second component B.)

If merging fails, then a RuntimeError is raised.

sage.rings.asymptotic.misc.combine_exceptions(e, *f)#

Helper function which combines the messages of the given exceptions.

INPUT:

  • e – an exception.

  • *f – exceptions.

OUTPUT:

An exception.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import combine_exceptions
sage: raise combine_exceptions(ValueError('Outer.'), TypeError('Inner.'))
Traceback (most recent call last):
...
ValueError: Outer.
> *previous* TypeError: Inner.
sage: raise combine_exceptions(ValueError('Outer.'),
....:                          TypeError('Inner1.'), TypeError('Inner2.'))
Traceback (most recent call last):
...
ValueError: Outer.
> *previous* TypeError: Inner1.
> *and* TypeError: Inner2.
sage: raise combine_exceptions(ValueError('Outer.'),
....:                          combine_exceptions(TypeError('Middle.'),
....:                                             TypeError('Inner.')))
Traceback (most recent call last):
...
ValueError: Outer.
> *previous* TypeError: Middle.
>> *previous* TypeError: Inner.
sage.rings.asymptotic.misc.log_string(element, base=None)#

Return a representation of the log of the given element to the given base.

INPUT:

  • element – an object.

  • base – an object or None.

OUTPUT:

A string.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import log_string
sage: log_string(3)
'log(3)'
sage: log_string(3, base=42)
'log(3, base=42)'
sage.rings.asymptotic.misc.parent_to_repr_short(P)#

Helper method which generates a short(er) representation string out of a parent.

INPUT:

  • P – a parent.

OUTPUT:

A string.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import parent_to_repr_short
sage: parent_to_repr_short(ZZ)
'ZZ'
sage: parent_to_repr_short(QQ)
'QQ'
sage: parent_to_repr_short(SR)
'SR'
sage: parent_to_repr_short(RR)
'RR'
sage: parent_to_repr_short(CC)
'CC'
sage: parent_to_repr_short(ZZ['x'])
'ZZ[x]'
sage: parent_to_repr_short(QQ['d, k'])
'QQ[d, k]'
sage: parent_to_repr_short(QQ['e'])
'QQ[e]'
sage: parent_to_repr_short(SR[['a, r']])
'SR[[a, r]]'
sage: parent_to_repr_short(Zmod(3))
'Ring of integers modulo 3'
sage: parent_to_repr_short(Zmod(3)['g'])
'Univariate Polynomial Ring in g over Ring of integers modulo 3'
sage.rings.asymptotic.misc.repr_op(left, op, right=None, latex=False)#

Create a string left op right with taking care of parentheses in its operands.

INPUT:

  • left – an element.

  • op – a string.

  • right – an element.

  • latex – (default: False) a boolean. If set, then LaTeX-output is returned.

OUTPUT:

A string.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import repr_op
sage: repr_op('a^b', '^', 'c')
'(a^b)^c'
sage.rings.asymptotic.misc.repr_short_to_parent(s)#

Helper method for the growth group factory, which converts a short representation string to a parent.

INPUT:

  • s – a string, short representation of a parent.

OUTPUT:

A parent.

The possible short representations are shown in the examples below.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import repr_short_to_parent
sage: repr_short_to_parent('ZZ')
Integer Ring
sage: repr_short_to_parent('QQ')
Rational Field
sage: repr_short_to_parent('SR')
Symbolic Ring
sage: repr_short_to_parent('NN')
Non negative integer semiring
sage: repr_short_to_parent('UU')
Group of Roots of Unity
sage.rings.asymptotic.misc.split_str_by_op(string, op, strip_parentheses=True)#

Split the given string into a tuple of substrings arising by splitting by op and taking care of parentheses.

INPUT:

  • string – a string.

  • op – a string. This is used by str.split. Thus, if this is None, then any whitespace string is a separator and empty strings are removed from the result.

  • strip_parentheses – (default: True) a boolean.

OUTPUT:

A tuple of strings.

sage.rings.asymptotic.misc.strip_symbolic(expression)#

Return, if possible, the underlying (numeric) object of the symbolic expression.

If expression is not symbolic, then expression is returned.

INPUT:

  • expression – an object

OUTPUT:

An object.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import strip_symbolic
sage: strip_symbolic(SR(2)); _.parent()
2
Integer Ring
sage: strip_symbolic(SR(2/3)); _.parent()
2/3
Rational Field
sage: strip_symbolic(SR('x')); _.parent()
x
Symbolic Ring
sage: strip_symbolic(pi); _.parent()
pi
Symbolic Ring
sage.rings.asymptotic.misc.substitute_raise_exception(element, e)#

Raise an error describing what went wrong with the substitution.

INPUT:

  • element – an element.

  • e – an exception which is included in the raised error message.

OUTPUT:

Raise an exception of the same type as e.

sage.rings.asymptotic.misc.transform_category(category, subcategory_mapping, axiom_mapping, initial_category=None)#

Transform category to a new category according to the given mappings.

INPUT:

  • category – a category.

  • subcategory_mapping – a list (or other iterable) of triples (from, to, mandatory), where

    • from and to are categories and

    • mandatory is a boolean.

  • axiom_mapping – a list (or other iterable) of triples (from, to, mandatory), where

    • from and to are strings describing axioms and

    • mandatory is a boolean.

  • initial_category – (default: None) a category. When transforming the given category, this initial_category is used as a starting point of the result. This means the resulting category will be a subcategory of initial_category. If initial_category is None, then the category of objects is used.

OUTPUT:

A category.

Note

Consider a subcategory mapping (from, to, mandatory). If category is a subcategory of from, then the returned category will be a subcategory of to. Otherwise and if mandatory is set, then an error is raised.

Consider an axiom mapping (from, to, mandatory). If category is has axiom from, then the returned category will have axiom to. Otherwise and if mandatory is set, then an error is raised.

EXAMPLES:

sage: from sage.rings.asymptotic.misc import transform_category
sage: from sage.categories.additive_semigroups import AdditiveSemigroups
sage: from sage.categories.additive_monoids import AdditiveMonoids
sage: from sage.categories.additive_groups import AdditiveGroups
sage: S = [
....:     (Sets(), Sets(), True),
....:     (Posets(), Posets(), False),
....:     (AdditiveMagmas(), Magmas(), False)]
sage: A = [
....:     ('AdditiveAssociative', 'Associative', False),
....:     ('AdditiveUnital', 'Unital', False),
....:     ('AdditiveInverse', 'Inverse', False),
....:     ('AdditiveCommutative', 'Commutative', False)]
sage: transform_category(Objects(), S, A)
Traceback (most recent call last):
...
ValueError: Category of objects is not
a subcategory of Category of sets.
sage: transform_category(Sets(), S, A)
Category of sets
sage: transform_category(Posets(), S, A)
Category of posets
sage: transform_category(AdditiveSemigroups(), S, A)
Category of semigroups
sage: transform_category(AdditiveMonoids(), S, A)
Category of monoids
sage: transform_category(AdditiveGroups(), S, A)
Category of groups
sage: transform_category(AdditiveGroups().AdditiveCommutative(), S, A)
Category of commutative groups

sage: transform_category(AdditiveGroups().AdditiveCommutative(), S, A,
....:     initial_category=Posets())
Join of Category of commutative groups
    and Category of posets

sage: transform_category(ZZ.category(), S, A)
Category of commutative groups
sage: transform_category(QQ.category(), S, A)
Category of commutative groups
sage: transform_category(SR.category(), S, A)
Category of commutative groups
sage: transform_category(Fields(), S, A)
Category of commutative groups
sage: transform_category(ZZ['t'].category(), S, A)
Category of commutative groups

sage: A[-1] = ('Commutative', 'AdditiveCommutative', True)
sage: transform_category(Groups(), S, A)
Traceback (most recent call last):
...
ValueError: Category of groups does not have
axiom Commutative.