معرفی شرکت ها
arithmat-1.1.2
تبلیغات ما
مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.
مشاهده بیشتر
تبلیغات ما
مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.
مشاهده بیشتر
تبلیغات ما
مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.
مشاهده بیشتر
تبلیغات ما
مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.
مشاهده بیشتر
تبلیغات ما
مشتریان به طور فزاینده ای آنلاین هستند. تبلیغات می تواند به آنها کمک کند تا کسب و کار شما را پیدا کنند.
مشاهده بیشترتوضیحات
Sage implementation of arithmetic matroids and toric arrangements
| ویژگی | مقدار |
|---|---|
| سیستم عامل | - |
| نام فایل | arithmat-1.1.2 |
| نام | arithmat |
| نسخه کتابخانه | 1.1.2 |
| نگهدارنده | [] |
| ایمیل نگهدارنده | [] |
| نویسنده | Roberto Pagaria, Giovanni Paolini |
| ایمیل نویسنده | giovanni.paolini@sns.it |
| آدرس صفحه اصلی | https://github.com/giove91/arithmat |
| آدرس اینترنتی | https://pypi.org/project/arithmat/ |
| مجوز | GPLv3 |
# Arithmat
SageMath implementation of arithmetic matroids and toric arrangements.
Authors: Roberto Pagaria and Giovanni Paolini
* [Requirements](#requirements)
* [Installation](#installation)
* [Overview](#overview)
* [Citing Arithmat](#citing-arithmat)
* [Documentation](#documentation)
+ [Import](#import)
+ [Available classes for arithmetic matroids](#available-classes-for-arithmetic-matroids)
+ [Available methods](#available-methods)
* [Bibliography](#bibliography)
* [License](#license)
## Requirements
* [Python](https://www.python.org/) 2.7 or 3.x
* [SageMath](http://www.sagemath.org/) >= 8
## Installation
### From PyPI
The easiest way to start using Arithmat is to install [the latest release from PyPI](https://pypi.org/project/arithmat/):
```bash
sage -pip install --upgrade arithmat
```
### From source
Download source code from the git repository:
```bash
git clone https://github.com/giove91/arithmat.git
```
Change to the root directory and run:
```bash
sage -pip install --upgrade --no-index -v .
```
## Overview
Arithmat is a SageMath package that implements arithmetic matroids and toric arrangements.
At its core there is the class `ArithmeticMatroidMixin`, which is intended to be used in combination with [any existing matroid class of SageMath](http://doc.sagemath.org/html/en/reference/matroids/index.html) (e.g. `RankMatroid`, `BasisMatroid`, `LinearMatroid`) via multiple inheritance.
The most common combinations are already defined: `ArithmeticMatroid` (deriving from `RankMatroid`), `BasisArithmeticMatroid` (deriving from `BasisMatroid`), and `LinearArithmeticMatroid` (deriving from `LinearMatroid`).
An additional class `ToricArithmeticMatroid` is implemented, for arithmetic matroids constructed from a fixed given representation.
## Citing Arithmat
To cite Arithmat, you can use (some variant of) the following BibTeX code:
```latex
@misc{arithmat,
author = {Pagaria, Roberto and Paolini, Giovanni},
title = {{Arithmat}: {SageMath} implementation of arithmetic matroids and toric arrangements},
year = {2019},
publisher = {GitHub},
journal = {GitHub repository},
howpublished = {\url{https://github.com/giove91/arithmat"}},
}
```
## Documentation
### Import
All defined classes and functions can be imported at once:
```sage
from arithmat import *
```
Alternatively, it is possible to import only specific classes and/or functions. For example:
```sage
from arithmat import ArithmeticMatroid, ToricArithmeticMatroid
```
### Available classes for arithmetic matroids
All classes for arithmetic matroids derive from `ArithmeticMatroidMixin` and from some subclass of SageMath's `Matroid`.
The class `ArithmeticMatroidMixin` is not intended to be used by itself, but it is possible to subclass it in order to create new classes for arithmetic matroids.
The classes which are already provided in `arithmat` are the following.
* `ArithmeticMatroid(groundset, rank_function, multiplicity_function)`
This is the simplest arithmetic matroid class, and derives from `ArithmeticMatroidMixin` and `RankMatroid`.
```sage
E = [1,2,3,4,5]
def rk(X):
return min(2, len(X))
def m(X):
if len(X) == 2 and all(x in [3,4,5] for x in X):
return 2
else:
return 1
M = ArithmeticMatroid(E, rk, m)
print(M)
# Arithmetic matroid of rank 2 on 5 elements
print(M.arithmetic_tutte_polynomial())
# y^3 + x^2 + 2*y^2 + 3*x + 3*y + 3
print(M.representation())
# None (this arithmetic matroid is not representable)
```
* `ToricArithmeticMatroid(arrangement_matrix, torus_matrix=None, ordered_groundset=None)`
Arithmetic matroid associated with a given toric arrangement. This class derives from `ArithmeticMatroidMixin` and `Matroid`.
The constructor requires an integer matrix `arrangement_matrix` representing the toric arrangement. Optionally it accepts another integer matrix `torus_matrix` (whose cokernel describes the ambient torus, and defaults to `matrix(ZZ, arrangement_matrix.nrows(), 0)`) and/or an ordered copy `ordered_groundset` of the groundset (defaults to `range(matrix.ncols())`). The number of rows of `arrangement_matrix` must be equal to the numer of rows of `torus_matrix`.
The two matrices are not guaranteed to remain unchanged: internally,`torus_matrix` is kept in Smith normal form (this also affects `arrangement_matrix`).
```sage
A = matrix(ZZ, [[-1, 1, 0, 2], [3, 1, -1, -2]])
M = ToricArithmeticMatroid(A)
print(M)
# Toric arithmetic matroid of rank 2 on 4 elements
print(M.arrangement_matrix())
# [-1 1 0 2]
# [ 3 1 -1 -2]
print(M.torus_matrix())
# []
P = M.poset_of_layers()
print(P)
# Finite poset containing 11 elements
P.show(label_elements=False)
```
<img src="https://github.com/giove91/arithmat/raw/master/figures/poset.png" height="240">
```sage
A = matrix(ZZ, [[-1, 1, 0, 2], [3, 1, -1, -2]])
Q = matrix(ZZ, [[5], [1]])
N = ToricArithmeticMatroid(A, Q)
print(N)
# Toric arithmetic matroid of rank 1 on 4 elements
print(N.arrangement_matrix())
# [-16 -4 5 12]
print(N.torus_matrix())
# []
```
The classes `BasisArithmeticMatroid` and `LinearArithmeticMatroid` derive from the SageMath classes `BasisMatroid` and `LinearMatroid`, respectively.
An instance of `XxxArithmeticMatroid` can be constructed with `XxxArithmeticMatroid(..., multiplicity_function=m)`, where `...` should be replaced by arguments to construct an instance of `XxxMatroid`, and `m` is the multiplicity function.
The multiplicity function needs to be passed as a keyword argument (and not as a positional argument).
* `BasisArithmeticMatroid(M=None, groundset=None, bases=None, ..., multiplicity_function)`
```sage
def m(X):
if len(X) == 2 and all(x in ['b','c','d'] for x in X):
return 2
else:
return 1
M = BasisArithmeticMatroid(groundset='abcd', bases=['ab', 'ac', 'ad', 'bc', 'bd', 'cd'], multiplicity_function=m)
print(M)
# Basis arithmetic matroid of rank 2 on 4 elements
print(M.is_valid())
# True
```
It is possible to cast any arithmetic matroid to a `BasisArithmeticMatroid`:
```sage
M1 = ToricArithmeticMatroid(matrix(ZZ, [[-1, 1, 0, 7], [6, 1, -1, -2]]))
M2 = BasisArithmeticMatroid(M1)
print(M1)
# Toric arithmetic matroid of rank 2 on 4 elements
print(M2)
# Basis arithmetic matroid of rank 2 on 4 elements
print(M2.full_multiplicity() == M1.full_multiplicity())
# True
```
* `LinearArithmeticMatroid(matrix=None, groundset=None, ..., multiplicity_function)`
```sage
A = matrix(GF(2), [[1, 0, 0, 1, 1], [0, 1, 0, 1, 0], [0, 0, 1, 0, 1]])
def m(X):
return 1
M = LinearArithmeticMatroid(A, multiplicity_function=m)
print(M)
# Linear arithmetic matroid of rank 3 on 5 elements
print(M.is_valid())
# True
```
Finally, `MinorArithmeticMatroid` and `DualArithmeticMatroid` are the analogs of `MinorMatroid` and `DualMatroid`.
They are used when calling `contract(X)`, `delete(X)`, `dual()` (see below).
### Available methods
All classes for arithmetic matroids must also derive from some subclass of SageMath's `Matroid`.
In particular, `Matroid` methods are still available. For example:
```sage
M = ToricArithmeticMatroid(matrix(ZZ, [[1,2,3], [0,1, 1]]))
print(list(M.bases()))
# [frozenset([0, 1]), frozenset([0, 2]), frozenset([1, 2])]
```
All subclasses of `ArithmeticMatroidMixin` also (re-)implement the following methods.
* `multiplicity(X)`
Return the multiplicity of the set `X`.
* `full_multiplicity()`
Return the multiplicity of the groundset.
* `is_valid()`
Check if the arithmetic matroid axioms are satisfied.
This method overwrites `Matroid.is_valid`.
* `is_torsion_free()`
Check if the matroid is torsion-free, i.e. the multiplicity of the empty set is equal to 1.
* `is_surjective()`
Check if the matroid is surjective, i.e. the multiplicity of the groundset is equal to 1.
* `is_gcd()`
Check if the matroid satisfies the gcd property, defined in [DM13, Section 3].
* `is_strong_gcd()`
Check if the matroid satisfies the strong gcd property, defined in [PP19, Section 3].
* `is_isomorphic(other, morphism)`
Check if the two arithmetic matroids are isomorphic.
This method overwrites `Matroid.is_isomorphic`.
* `is_isomorphism(other, morphism)`
Check if the given morphism of groundsets is an isomorphism of arithmetic matroids.
It works also when comparing instances of different subclasses of `ArithmeticMatroid`.
This method overwrites `Matroid.is_isomorphism`.
* `contract(X)`
Contract elements.
This method overwrites `Matroid.contract`.
* `delete(X)`
Delete elements.
This method overwrites `Matroid.delete`.
* `minor(contractions=None, deletions=None)`
Contract some elements and delete some other elements.
This method overwrites `Matroid.minor`.
* `dual()`
Return the dual of the matroid.
This method overwrites `Matroid.dual`.
* `arithmetic_tutte_polynomial(x=None, y=None)`
Return the arithmetic Tutte polynomial of the matroid.
* `reduction()`
Return the reduction of the matroid, defined in [PP19, Section 4].
* `check_representation(A, ordered_groundset=None)`
Check if the given integer matrix `A` is a representation of the matroid.
The optional parameter `ordered_groundset` specifies the bijection between the columns of the matrix and the groundset.
* `all_representations(ordered_groundset=None, shnf=True)`
Generator of all non-equivalent essential representations of the matroid, computed using the algorithm of [PP19, Section 5].
If `shnf=True`, the output matrices are guaranteed to be in signed Hermite normal form [PP19, Section 6].
* `num_representations()`
Return the number of non-equivalent essential representations of the matroid.
This calls `all_representations()`.
* `representation(ordered_groundset=None, shnf=True)`
Return any essential representation of the matroid, or `None` if the matroid is not representable.
* `is_representable()`
Check if the matroid is representable.
This calls `representation()`.
* `is_orientable()`
Check if the matroid is orientable as an arithmetic matroid, as defined in [Pag18].
In addition, `ToricArithmeticMatroid` has the following methods.
* `ordered_groundset()`
Return the groundset as a list, in the same order as the columns of the matrix of the defining toric arrangement.
* `arrangement_matrix()`
Return the matrix of the defining toric arrangement.
* `torus_matrix()`
Return the matrix describing the ambient torus.
* `poset_of_layers()`
Return the poset of layers of the toric arrangement, computed using Lenz's algorithm [Len17a].
* `arithmetic_independence_poset()`
Return the arithmetic independence poset of the toric arrangement, defined in [Len17b, Definition 5], [Mar18, Definitions 2.1 and 2.2], [DD18, Section 7].
Notice that it is not the same as the independence poset of the underlying matroid.
* `decomposition()`
Return the partition of the groundset corresponding to the decomposition of the matroid as a direct sum of indecomposable matroids.
Uses the algorithm of [PP19, Section 7].
* `is_decomposable()`
Check if the matroid is decomposable.
* `is_indecomposable()`
Check if the matroid is not decomposable.
* `is_equivalent(other, morphism=None)`
Check if the two ToricArithmeticMatroids are equivalent (i.e. the defining representations are equivalent; see [PP19, Section 2]).
If the morphism is not given, the two ordered groundsets are assumed to be the same.
The following function is available outside of arithmetic matroid classes.
* `signed_hermite_normal_form(A)`
Return the signed Hermite normal form of the integer matrix `A`, defined in [PP19, Section 6].
```sage
from arithmat import signed_hermite_normal_form
print(signed_hermite_normal_form(matrix([[3, 2, 1], [-1, 1, 3]])))
# [ 1 1 -3]
# [ 0 5 -10]
```
More examples can be found in the [test file](https://github.com/giove91/arithmat/blob/master/arithmat/test.sage).
## Bibliography
[BM14] P. Brändén and L. Moci, *The multivariate arithmetic Tutte polynomial*, Transactions of the American Mathematical Society **366** (2014), no. 10, 5523-5540.
[DM13] M. D'Adderio and L. Moci, *Arithmetic matroids, the Tutte polynomial and toric arrangements*, Advances in Mathematics **232** (2013), 335-367.
[DD18] A. D'Alì and E. Delucchi, *Stanley-Reisner rings for symmetric simplicial complexes, G-semimatroids and Abelian arrangements*, ArXiv preprint 1804.07366 (2018).
[DGP17] E. Delucchi, N. Girard, and G. Paolini, *Shellability of posets of labeled partitions and arrangements defined by root systems*, ArXiv preprint 1706.06360 (2017).
[Len17a] M. Lenz, *Computing the poset of layers of a toric arrangement*, ArXiv preprint 1708.06646 (2017).
[Len17b] M. Lenz, *Stanley-Reisner rings for quasi-arithmetic matroids*, ArXiv preprint 1709.03834 (2017).
[Len19] M. Lenz, *Representations of weakly multiplicative arithmetic matroids are unique*, Annals of Combinatorics **23** (2019), no. 2, 335-346.
[Mar18] I. Martino, *Face module for realizable Z-matroids*, Contributions to Discrete Mathematics **13** (2018).
[Pag17] R. Pagaria, *Combinatorics of Toric Arrangements*, ArXiv preprint 1710.00409 (2017).
[Pag18] R. Pagaria, *Orientable arithmetic matroids*, ArXiv preprint 1805.11888 (2018).
[Pag19] R. Pagaria, *Two Examples of Toric Arrangements*, Journal of Combinatorial Theory, Series A **167** (2019), 389-402.
[PP19] R. Pagaria and G. Paolini, *Representations of torsion-free arithmetic matroids*, ArXiv preprint 1908.04137 (2019).
## License
This project is licensed under the [GNU General Public License v3.0](LICENSE.md).
نیازمندی
| مقدار | نام |
|---|---|
| - | sage-package |
| ~=2.2 | networkx |
زبان مورد نیاز
| مقدار | نام |
|---|---|
| >=2.7 | Python |
نحوه نصب
نصب پکیج whl arithmat-1.1.2:
pip install arithmat-1.1.2.whl
نصب پکیج tar.gz arithmat-1.1.2:
pip install arithmat-1.1.2.tar.gz