sign_vectors.sign_vectors

Sign vectors

First, we load the functions from the package:

sage: from sign_vectors import *

There are several ways to define sign vectors:

sage: sign_vector([1, 0, -1, 1])
(+0-+)
sage: x = vector([5, 2/5, -1, 0])
sage: sign_vector(x)
(++-0)
sage: sign_vector('++-+-00-')
(++-+-00-)

We construct the zero sign vector of a given length:

sage: zero_sign_vector(6)
(000000)

It is also possible to generate random sign vectors:

sage: random_sign_vector(7) # random
(+-+00+-)

There are different notions of support:

sage: X = sign_vector('+++000--')
sage: X.support()
[0, 1, 2, 6, 7]
sage: X.zero_support()
[3, 4, 5]
sage: X.positive_support()
[0, 1, 2]
sage: X.negative_support()
[6, 7]

Next, we define two sign vectors:

sage: X = sign_vector([-1, 0, 1, -1, 0])
sage: X
(-0+-0)
sage: Y = sign_vector([0, 1, 0, 1, 0])
sage: Y
(0+0+0)

We compose them:

sage: X.compose(Y)
(-++-0)
sage: Y.compose(X)
(-+++0)

Use the operator & as a shorthand for composition:

sage: X & Y
(-++-0)
sage: Y & X
(-+++0)

The conformal relation is a partial order on a set of sign vectors:

sage: X = sign_vector([-1, 1, 0, 0, 1])
sage: Y = sign_vector([-1, 1, 1, 0, 1])
sage: Z = sign_vector([-1, 1, 1, -1, 1])
sage: X
(-+00+)
sage: Y
(-++0+)
sage: Z
(-++-+)
sage: X.conforms(Y)
True
sage: X.conforms(X)
True
sage: Y.conforms(Z)
True
sage: Z.conforms(Y)
False
sage: X.conforms(Z)
True
sage: X < X
False
sage: X <= Y
True
sage: Y > Z
False
sage: Z >= X
True

Similar as for real vectors, we define orthogonality for sign vectors. First, we define some real vectors:

sage: x = vector([0, 0, 1])
sage: y = vector([1, -2, 0])
sage: z = vector([2, 1, 2])

We compute some scalar products to investigate orthogonality of those vectors:

sage: x.dot_product(y)
0
sage: y.dot_product(z)
0
sage: x.dot_product(z)
2

Next, we define the corresponding sign vectors:

sage: X = sign_vector(x)
sage: X
(00+)
sage: Y = sign_vector(y)
sage: Y
(+-0)
sage: Z = sign_vector(z)
sage: Z
(+++)

By definition, if two real vectors are orthogonal, then their corresponding sign vectors are also orthogonal:

sage: X.is_orthogonal_to(Y)
True
sage: Y.is_orthogonal_to(Z)
True
sage: X.is_orthogonal_to(Z)
False

Functions

random_sign_vector(length)	Return a random sign vector of a given length.
sign_symbolic(value)	Return the sign of an expression.
sign_vector(iterable)	Create a sign vector from a list, vector or string.
zero_sign_vector(length)	Return the zero sign vector of a given length.

Classes

SignVector(positive_support, negative_support)

A sign vector.

class sign_vectors.sign_vectors.SignVector(positive_support: sage.data_structures.bitset.FrozenBitset, negative_support: sage.data_structures.bitset.FrozenBitset)

A sign vector.

compose(other: sign_vectors.sign_vectors.SignVector) → sign_vectors.sign_vectors.SignVector

Return the composition of two sign vectors.

INPUT:

• other – a sign vector

OUTPUT:

Composition of this sign vector with other.

Note

Alternatively, the operator & can be used.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('+00'); X
(+00)
sage: Y = sign_vector('--0'); Y
(--0)
sage: X.compose(Y)
(+-0)
sage: Y.compose(X)
(--0)
sage: Y & X
(--0)
sage: X = sign_vector('000+++---')
sage: Y = sign_vector('0+-0+-0+-')
sage: X.compose(Y)
(0+-+++---)

compose_harmonious(other: sign_vectors.sign_vectors.SignVector) → sign_vectors.sign_vectors.SignVector

Return the composition of two harmonious sign vectors.

INPUT:

• other – a sign vector

OUTPUT:

Composition of this sign vector with other.

Note

This method is more efficient than - compose(). However, it does not check whether the sign vectors are harmonious.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('+00')
sage: X
(+00)
sage: Y = sign_vector('0-0')
sage: Y
(0-0)
sage: X.compose_harmonious(Y)
(+-0)
sage: Y.compose_harmonious(X)
(+-0)

conforms(other: sign_vectors.sign_vectors.SignVector) → bool

Conformal relation of two sign vectors.

Note

Alternatively, the operator <= can be used. Use >=, < and > for the other relations.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-+00+'); X
(-+00+)
sage: Y = sign_vector('-++0+'); Y
(-++0+)
sage: Z = sign_vector('-++-+'); Z
(-++-+)
sage: X.conforms(Y)
True
sage: X.conforms(X)
True
sage: Y.conforms(Z)
True
sage: Z.conforms(Y)
False
sage: X.conforms(Z)
True

delete_components(indices: list[int]) → sign_vectors.sign_vectors.SignVector

Delete the given components from the sign vector.

INPUT:

• indices – list of indices to delete

OUTPUT:

Returns a new sign vector with the specified components removed.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('+0-')
sage: X
(+0-)
sage: X.delete_components([1])
(+-)

sage: X = sign_vector('-+0+0')
sage: X
(-+0+0)
sage: X.delete_components([0, 2])
(++0)
sage: X.delete_components([1])
(-0+0)

disjoint_support(other: sign_vectors.sign_vectors.SignVector) → bool

Return whether these two sign vectors have disjoint support.

INPUT:

• other – a sign vector or real vector

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('++00')
sage: Y = sign_vector('0+0-')
sage: Z = sign_vector('00--')
sage: X.disjoint_support(Y)
False
sage: Y.disjoint_support(X)
False
sage: X.disjoint_support(Z)
True
sage: Z.disjoint_support(X)
True
sage: Y.disjoint_support(Z)
False

flip_signs(indices: list[int]) → sign_vectors.sign_vectors.SignVector

Flips entries of given indices.

INPUT:

• indices – list of indices

OUTPUT: Returns a new sign vector. Components of indices are multiplied by -1.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-++0+')
sage: X
(-++0+)
sage: X.flip_signs([0, 2, 3])
(++-0+)

sage: X = sign_vector('+-0+0')
sage: X
(+-0+0)
sage: X.flip_signs([0, 3, 4])
(--0-0)

classmethod from_iterable(iterable) → sign_vectors.sign_vectors.SignVector

Create a sign vector from an iterable.

EXAMPLES:

sage: from sign_vectors import *
sage: SignVector.from_iterable([5, 0, -1, -2])
(+0--)
sage: v = vector([5, 0, -1, 0, 8])
sage: SignVector.from_iterable(v)
(+0-0+)

classmethod from_string(s: str) → sign_vectors.sign_vectors.SignVector

Create a sign vector from a string.

EXAMPLES:

sage: from sign_vectors import *
sage: SignVector.from_string("+-0+0")
(+-0+0)

classmethod from_support(positive_support: list[int], negative_support: list[int], length: int) → sign_vectors.sign_vectors.SignVector

Return a sign vector that is given by lists representing positive support and negative support.

INPUT:

• positive_support – a list of integers.

• negative_support – a list of integers.

• length – a nonnegative integer

OUTPUT: a sign vector

Note

The list positive_support and negative_support should be disjoint. For efficiency, this is not checked.

EXAMPLES:

sage: from sign_vectors import *
sage: SignVector.from_support([1, 4], [2], 6)
(0+-0+0)

is_harmonious_to(other: sign_vectors.sign_vectors.SignVector) → bool

Check whether these two sign vectors are harmonious.

INPUT:

• other – a sign vector or real vector

OUTPUT: Returns true if there are no separating elements. Otherwise, false is returned.

Note

Two sign vectors are harmonious if there is no component where one sign vector has + and the other has -.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('++00-')
sage: X
(++00-)
sage: Y = sign_vector('+-+++')
sage: Y
(+-+++)
sage: X.is_harmonious_to(Y)
False
sage: sign_vector('0+00').is_harmonious_to(sign_vector('-+0+'))
True
sage: v = vector([1, 2/3, 0, -1, -1])
sage: X.is_harmonious_to(v)
True
sage: Y.is_harmonious_to(v)
False

is_orthogonal_to(other: sign_vectors.sign_vectors.SignVector) → bool

Return whether two sign vectors are orthogonal.

INPUT:

• other – a sign vector.

OUTPUT:

• Returns True if the sign vectors are orthogonal and False otherwise.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-+00+'); X
(-+00+)
sage: X.is_orthogonal_to(X)
False
sage: X.is_orthogonal_to(sign_vector('++000'))
True
sage: X.is_orthogonal_to(sign_vector('++00+'))
True
sage: X.is_orthogonal_to(sign_vector('00++0'))
True

is_vector() → bool

Return False since sign vectors are not vectors.

length() → int

Return the length of the sign vector.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('0+-'); X
(0+-)
sage: X.length()
3

list_from_positions(positions: list[int]) → list[int]

Return a list of components that are in the list of indices positions.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-+00+'); X
(-+00+)
sage: X.list_from_positions([0, 1, 4])
[-1, 1, 1]

negative_support() → list[int]

Return a list of indices where the sign vector is negative.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-0+-0'); X
(-0+-0)
sage: X.negative_support()
[0, 3]

positive_support() → list[int]

Return a list of indices where the sign vector is positive.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-0+-0'); X
(-0+-0)
sage: X.positive_support()
[2]

separating_elements(other: sign_vectors.sign_vectors.SignVector) → list[int]

Compute the list of separating elements of two sign vectors.

INPUT:

• other – sign vector

OUTPUT: List of elements e such that self[e] == -other[e] != 0.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('++00-')
sage: X
(++00-)
sage: Y = sign_vector('+-+++')
sage: Y
(+-+++)
sage: X.separating_elements(Y)
[1, 4]

set_to_zero(indices: list[int]) → sign_vectors.sign_vectors.SignVector

Set given entries to zero.

INPUT:

• indices – list of indices

OUTPUT: Returns a new sign vector of same length. Components with indices in indices are multiplied by -1.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-++0+')
sage: X
(-++0+)
sage: X.set_to_zero([0, 2, 3])
(0+00+)

sage: X = sign_vector('+-+0+0')
sage: X
(+-+0+0)
sage: X.set_to_zero([0, 1, 4])
(00+000)

support() → list[int]

Return a list of indices where the sign vector is nonzero.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-0+-0'); X
(-0+-0)
sage: X.support()
[0, 2, 3]

to_string() → str

Return a string representation of this sign vector (without parentheses).

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector("0+-")
sage: X
(0+-)
sage: X.to_string()
'0+-'

Note that that str and to_string are different:

sage: str(X)
'(0+-)'

classmethod zero(length: int) → sign_vectors.sign_vectors.SignVector

Create a zero sign vector of a given length.

EXAMPLES:

sage: from sign_vectors import *
sage: SignVector.zero(5)
(00000)

zero_support() → list[int]

Return a list of indices where the sign vector is zero.

EXAMPLES:

sage: from sign_vectors import *
sage: X = sign_vector('-0+-0'); X
(-0+-0)
sage: X.zero_support()
[1, 4]

sign_vectors.sign_vectors.random_sign_vector(length: int) → sign_vectors.sign_vectors.SignVector

Return a random sign vector of a given length.

INPUT:

• length – length

EXAMPLES:

sage: from sign_vectors import *
sage: random_sign_vector(5) # random
(++-0-)

TEST:

sage: len(random_sign_vector(5))
5

sign_vectors.sign_vectors.sign_symbolic(value) → int

Return the sign of an expression. Supports symbolic expressions.

OUTPUT: If the sign cannot be determined, a warning is shown and 0 is returned.

EXAMPLES:

For real numbers, the sign is determined:

sage: from sign_vectors.sign_vectors import sign_symbolic
sage: sign_symbolic(1)
1
sage: sign_symbolic(-1/2)
-1
sage: sign_symbolic(0)
0

For symbolic expressions, the sign is determined using assumptions:

sage: var('a')
a
sage: sign_symbolic(a) # not tested
...
UserWarning: Cannot determine sign of symbolic expression, using ``0`` instead.
0
sage: assume(a > 0)
sage: sign_symbolic(a)
1
sage: forget()

sage: var('a, b, c, d')
(a, b, c, d)
sage: assume(a > 0, b > 0, c > 0, d > 0)
sage: sign_symbolic((a + b)*(c + d) - b*c)
1

sign_vectors.sign_vectors.sign_vector(iterable: list[int] | str | sign_vectors.sign_vectors.SignVector) → sign_vectors.sign_vectors.SignVector

Create a sign vector from a list, vector or string.

INPUT:

• iterable – different inputs are accepted:

  • an iterable (e.g. a list or vector) of real values. Variables can also occur.

  • a string consisting of "-", "+", "0". Other characters are treated as "0".

OUTPUT:

Returns a sign vector.

EXAMPLES:

sage: from sign_vectors import *
sage: sign_vector([5, 0, -1, -2])
(+0--)
sage: v = vector([5, 0, -1, -2])
sage: sign_vector(v)
(+0--)

We can also use a string to construct a sign vector:

sage: sign_vector("00--")
(00--)
sage: sign_vector('++-+-00-')
(++-+-00-)

Variables are supported to some extent:

sage: var('a')
a
sage: v = vector([1, a, -1])
sage: sign_vector(v)
...
UserWarning: Cannot determine sign of symbolic expression, using ``0`` instead.
(+0-)
sage: assume(a > 0)
sage: sign_vector(v)
(++-)
sage: forget()

TESTS:

sage: sign_vector(sign_vector("+-+"))
(+-+)

sign_vectors.sign_vectors.zero_sign_vector(length: int) → sign_vectors.sign_vectors.SignVector

Return the zero sign vector of a given length.

INPUT:

• length – length

EXAMPLES:

sage: from sign_vectors import *
sage: zero_sign_vector(4)
(0000)