from sage.misc.lazy_import import lazy_import
lazy_import('sage.manifolds.manifold', 'Manifold')
# interfaces to other interpreters
from __future__ import absolute_import
from sage.misc.lazy_import import lazy_import
from .frobby import frobby
from .four_ti_2 import four_ti_2
from .axiom import Axiom, axiom
from .fricas import FriCAS, fricas
from .gap import gap, gap_reset_workspace, set_gap_memory_pool_size, Gap
from .gap3 import gap3, gap3_version, Gap3
lazy_import('sage.interfaces.genus2reduction',
['genus2reduction', 'Genus2reduction'])
from .gfan import gfan, Gfan
from .giac import giac, Giac
from .gp import gp, gp_version, Gp
from .gnuplot import gnuplot
from .kash import kash, kash_version, Kash
from .lisp import lisp, Lisp
from .magma import magma, magma_version, Magma
from .magma_free import magma_free
from .macaulay2 import macaulay2, Macaulay2
from .maple import maple, Maple
from .maxima import maxima, Maxima
# import problems
#from maxima_lib import maxima_lib
from .mathematica import mathematica, Mathematica
from .matlab import matlab, matlab_version, Matlab
from .mupad import mupad, Mupad # NOT functional yet
from .mwrank import mwrank, Mwrank
from sage.structure.element import Element
from sage.rings.integer_ring import ZZ
from sage.rings.integer import Integer
from sage.arith.functions import LCM_list
from sage.misc.functional import denominator
from .base_mutable import Polyhedron_mutable
from .base_QQ import Polyhedron_QQ
from .base_ZZ import Polyhedron_ZZ
from sage.misc.lazy_import import lazy_import
from sage.features import PythonModule
lazy_import('ppl', [
'C_Polyhedron', 'Generator_System', 'Constraint_System',
'Linear_Expression', 'line', 'ray', 'point'
],
feature=PythonModule("ppl", spkg="pplpy"))
#########################################################################
class Polyhedron_ppl(Polyhedron_mutable):
"""
Polyhedra with ppl
INPUT:
- ``Vrep`` -- a list ``[vertices, rays, lines]`` or ``None``.
- ``Hrep`` -- a list ``[ieqs, eqns]`` or ``None``.
BCHCode, CyclicCode, CyclicCodeFromGeneratingPolynomial,
CyclicCodeFromCheckPolynomial, DuadicCodeEvenPair, DuadicCodeOddPair,
ExtendedBinaryGolayCode, ExtendedQuadraticResidueCode,
ExtendedTernaryGolayCode, HammingCode, LinearCodeFromCheckMatrix,
QuadraticResidueCode, QuadraticResidueCodeEvenPair,
QuadraticResidueCodeOddPair, RandomLinearCode, ReedSolomonCode,
TernaryGolayCode, ToricCode, TrivialCode, WalshCode)
from guava import (BinaryReedMullerCode, QuasiQuadraticResidueCode,
RandomLinearCodeGuava)
from code_bounds import (codesize_upper_bound, dimension_upper_bound,
volume_hamming, gilbert_lower_bound,
plotkin_upper_bound, griesmer_upper_bound,
elias_upper_bound, hamming_upper_bound,
singleton_upper_bound, gv_info_rate, entropy,
gv_bound_asymp, hamming_bound_asymp,
singleton_bound_asymp, plotkin_bound_asymp,
elias_bound_asymp, mrrw1_bound_asymp)
from linear_code import (LinearCode, LinearCodeFromVectorSpace, hamming_weight,
best_known_linear_code, best_known_linear_code_www,
bounds_minimum_distance, self_orthogonal_binary_codes)
from sd_codes import self_dual_codes_binary
lazy_import("sage.coding.delsarte_bounds", [
"Krawtchouk", "delsarte_bound_hamming_space",
"delsarte_bound_additive_hamming_space"
])
from sql_db import SQLQuery, SQLDatabase
from conway import ConwayPolynomials
from cremona import CremonaDatabase
from jones import JonesDatabase
from stein_watkins import SteinWatkinsAllData, SteinWatkinsPrimeData
from install import database_install
from sloane import sloane_sequence, sloane_find, SloaneEncyclopedia
from sage.misc.lazy_import import lazy_import
lazy_import('sage.databases.oeis', 'oeis')
from symbolic_data import SymbolicData
# commented out, since it's broken -- nobody updated the parser
# for the new format; nobody complained it didn't work, so it
# can't be that important.
#from lincodes import linear_code_bound
from odlyzko import zeta_zeros
from db_modular_polynomials import \
ClassicalModularPolynomialDatabase, \
DedekindEtaModularPolynomialDatabase, \
DedekindEtaModularCorrespondenceDatabase, \
AtkinModularPolynomialDatabase, \
from __future__ import absolute_import
#*****************************************************************************
# Copyright (C) 2005 David Kohel <*****@*****.**> and
# William Stein <*****@*****.**>
# 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net>
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.misc.latex import latex
from sage.misc.unknown import Unknown
from .category import JoinCategory, Category, CategoryWithParameters
from sage.misc.lazy_import import lazy_import
lazy_import('sage.categories.objects', 'Objects')
####################################################################
# Different types of categories
####################################################################
#############################################################
# Category of elements of some object
#############################################################
class Elements(Category):
"""
The category of all elements of a given parent.
EXAMPLES::
sage: a = IntegerRing()(5)
from bar_chart import bar_chart
from histogram import histogram
from bezier_path import bezier_path
from scatter_plot import scatter_plot
from disk import disk
from point import point, points, point2d
from matrix_plot import matrix_plot
from plot_field import plot_vector_field, plot_slope_field
from text import text
from polygon import polygon, polygon2d
from circle import circle
from ellipse import ellipse
from contour_plot import contour_plot, implicit_plot, region_plot
from density_plot import density_plot
from sage.misc.lazy_import import lazy_import
lazy_import("sage.plot.complex_plot",["complex_plot"])
from arc import arc
from animate import animate
from plot3d.tachyon import Tachyon
from colors import Color, hue, rainbow, colors, colormaps
from step import plot_step_function
from hyperbolic_arc import hyperbolic_arc
from hyperbolic_polygon import hyperbolic_triangle, hyperbolic_polygon
from __future__ import absolute_import
#*****************************************************************************
# Copyright (C) 2005 David Kohel <*****@*****.**> and
# William Stein <*****@*****.**>
# 2008-2009 Nicolas M. Thiery <nthiery at users.sf.net>
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.misc.latex import latex
from sage.misc.unknown import Unknown
from .category import JoinCategory, Category, CategoryWithParameters
from sage.misc.lazy_import import lazy_import
lazy_import('sage.categories.objects', 'Objects')
####################################################################
# Different types of categories
####################################################################
#############################################################
# Category of elements of some object
#############################################################
class Elements(Category):
"""
The category of all elements of a given parent.
EXAMPLES::
sage: a = IntegerRing()(5)
# interfaces to other interpreters
from sage.misc.lazy_import import lazy_import
from frobby import frobby
from four_ti_2 import four_ti_2
from axiom import Axiom, axiom, axiom_console
from fricas import FriCAS, fricas, fricas_console
from gap import gap, gap_reset_workspace, gap_console, set_gap_memory_pool_size, Gap
from gap3 import gap3, gap3_console, gap3_version, Gap3
lazy_import('sage.interfaces.genus2reduction', ['genus2reduction', 'Genus2reduction'])
from gfan import gfan, Gfan
from giac import giac, giac_console, Giac
from gp import gp, gp_console, gp_version, Gp
from gnuplot import gnuplot, gnuplot_console
from kash import kash, kash_console, kash_version, Kash
from lisp import lisp, lisp_console, Lisp
from magma import magma, magma_console, magma_version, Magma
from magma_free import magma_free
from macaulay2 import macaulay2, macaulay2_console, Macaulay2
from maple import maple, maple_console, Maple
from maxima_abstract import maxima_console
from maxima import maxima, Maxima
# import problems
#from maxima_lib import maxima_lib
from mathematica import mathematica, mathematica_console, Mathematica
from matlab import matlab, matlab_console, matlab_version, Matlab
from mupad import mupad, mupad_console, Mupad # NOT functional yet
from mwrank import mwrank, Mwrank, mwrank_console
from octave import octave, octave_console, octave_version, Octave
from qepcad import qepcad, qepcad_console, qepcad_version, qepcad_formula
from sage.misc.lazy_import import lazy_import
lazy_import("sage.coding.code_constructions", ["permutation_action",
"walsh_matrix"])
from sage.misc.superseded import deprecated_callable_import
deprecated_callable_import(19315,
"sage.coding.code_bounds",
globals(),
locals(),
["codesize_upper_bound",
"dimension_upper_bound",
"volume_hamming",
"gilbert_lower_bound",
"plotkin_upper_bound",
"griesmer_upper_bound",
"elias_upper_bound",
"hamming_upper_bound",
"singleton_upper_bound",
"gv_info_rate",
"entropy",
"gv_bound_asymp",
"hamming_bound_asymp",
"singleton_bound_asymp",
"plotkin_bound_asymp",
"elias_bound_asymp",
"mrrw1_bound_asymp"],
("This method soon will not be available in that way."
"Please call codes.bounds.%(name)s instead"))
from code_bounds import (codesize_upper_bound,
dimension_upper_bound,
volume_hamming,
gilbert_lower_bound,
plotkin_upper_bound,
griesmer_upper_bound,
elias_upper_bound,
hamming_upper_bound,
singleton_upper_bound,
gv_info_rate,
entropy,
gv_bound_asymp,
hamming_bound_asymp,
singleton_bound_asymp,
plotkin_bound_asymp,
elias_bound_asymp,
mrrw1_bound_asymp)
from linear_code import (LinearCode, LinearCodeFromVectorSpace,
hamming_weight,
best_known_linear_code,
best_known_linear_code_www,
bounds_minimum_distance,
self_orthogonal_binary_codes)
from sd_codes import self_dual_codes_binary
lazy_import("sage.coding.delsarte_bounds",
["Krawtchouk", "delsarte_bound_hamming_space", "delsarte_bound_additive_hamming_space"])
from pari_group import PariGroup
from matrix_gps.all import (GL, SL, Sp, SU, GU, SO, GO,
MatrixGroup, is_MatrixGroup,
is_MatrixGroupElement)
from abelian_gps.all import *
from perm_gps.all import *
from generic import *
from class_function import ClassFunction
from additive_abelian.all import *
from conjugacy_classes import *
from sage.misc.lazy_import import lazy_import
lazy_import('sage.groups.free_group', 'FreeGroup')
lazy_import('sage.groups.braid', 'BraidGroup')
import groups_catalog as groups
from tuple import Tuples, UnorderedTuples
#Alternating sign matrices
from alternating_sign_matrix import AlternatingSignMatrix, AlternatingSignMatrices, MonotoneTriangles, ContreTableaux, TruncatedStaircases
# Parking Functions
from non_decreasing_parking_function import NonDecreasingParkingFunctions, NonDecreasingParkingFunction
from parking_functions import ParkingFunctions, ParkingFunction
# Trees and Tamari interval posets
from sage.misc.lazy_import import lazy_import
from ordered_tree import (OrderedTree, OrderedTrees,
LabelledOrderedTree, LabelledOrderedTrees)
from binary_tree import (BinaryTree, BinaryTrees,
LabelledBinaryTree, LabelledBinaryTrees)
lazy_import('sage.combinat.interval_posets', ['TamariIntervalPoset', 'TamariIntervalPosets'])
from combination import Combinations
from cartesian_product import CartesianProduct
from set_partition import SetPartition, SetPartitions
from set_partition_ordered import OrderedSetPartition, OrderedSetPartitions
from subset import Subsets
#from subsets_pairwise import PairwiseCompatibleSubsets
from necklace import Necklaces
from lyndon_word import LyndonWord, LyndonWords, StandardBracketedLyndonWords
from dyck_word import DyckWords, DyckWord
from sloane_functions import sloane
from root_system.all import *
from sf.all import *
from sage.misc.lazy_import import lazy_import
lazy_import('sage.manifolds.manifold', 'Manifold')
lazy_import('sage.manifolds.differentiable.real_line', 'OpenInterval')
lazy_import('sage.manifolds.differentiable.real_line', 'RealLine')
from functional import (diff, derivative,
expand,
taylor, simplify)
from functions import (wronskian,jacobian)
from desolvers import (desolve, desolve_laplace, desolve_system,
eulers_method, eulers_method_2x2,
eulers_method_2x2_plot, desolve_rk4, desolve_system_rk4,
desolve_odeint)
from var import (var, function, clear_vars)
# We lazy_import the following modules since they import numpy which slows down sage startup
from sage.misc.lazy_import import lazy_import
lazy_import("sage.calculus.riemann",["Riemann_Map"])
lazy_import("sage.calculus.interpolators",["polygon_spline","complex_cubic_spline"])
from sage.modules.all import vector
def symbolic_expression(x):
"""
Create a symbolic expression or vector of symbolic expressions from x.
INPUT:
- ``x`` - an object
OUTPUT:
- a symbolic expression.
from sage.structure.unique_representation import UniqueRepresentation
from sage.structure.element import FieldElement, parent
from sage.structure.coerce import py_scalar_to_element
from sage.rings.ring import Field
from sage.rings.integer import Integer
from sage.rings.rational import Rational
from sage.rings.integer_ring import ZZ
from sage.rings.rational_field import QQ
from sage.rings.infinity import Infinity
# Deprecations from the old universal cyclotomic field
from sage.misc.lazy_import import lazy_import
lazy_import("sage.rings", "universal_cyclotomic_field", deprecation=18152)
libgap = GapElement_Integer = GapElement_Rational = GapElement_Cyclotomic = None
gap = gap3 = None
def late_import():
r"""
This function avoids importing libgap on startup. It is called once through
the constrcturo of :class:`UniversalCyclotomicField`.
EXAMPLES::
sage: import sage.rings.universal_cyclotomic_field as ucf
sage: _ = UniversalCyclotomicField() # indirect doctest
sage: ucf.libgap is None # indirect doctest
False
# code exports
from fano_variety import CPRFanoToricVariety
from ideal import ToricIdeal
from library import toric_varieties
from variety import AffineToricVariety, ToricVariety
from sage.misc.lazy_import import lazy_import
lazy_import("sage.schemes.toric.weierstrass", "WeierstrassForm")
# Copyright (C) 2006 David Joyner
# Copyright (C) 2019 Vincent Delecroix <*****@*****.**>
#
# This program is free software: you can redistribute it and/or modify
# it under the terms of the GNU General Public License as published by
# the Free Software Foundation, either version 2 of the License, or
# (at your option) any later version.
# http://www.gnu.org/licenses/
#*****************************************************************************
from __future__ import absolute_import
from . import permgroup_element
from sage.misc.sage_eval import sage_eval
from sage.misc.lazy_import import lazy_import
from sage.interfaces.gap import GapElement
lazy_import('sage.combinat.permutation', ['Permutation', 'from_cycles'])
from sage.libs.pari.all import pari_gen
from sage.libs.gap.element import GapElement_Permutation
def PermutationGroupElement(g, parent=None, check=True):
r"""
Builds a permutation from ``g``.
INPUT:
- ``g`` -- either
- a list of images
- a tuple describing a single cycle
from sage.misc.lazy_import import lazy_import
lazy_import('sage.groups.matrix_gps.linear', 'GL')
lazy_import('sage.groups.matrix_gps.linear', 'SL')
lazy_import('sage.groups.matrix_gps.symplectic', 'Sp')
lazy_import('sage.groups.matrix_gps.unitary', 'SU')
lazy_import('sage.groups.matrix_gps.unitary', 'GU')
lazy_import('sage.groups.matrix_gps.orthogonal', 'GO')
lazy_import('sage.groups.matrix_gps.orthogonal', 'SO')
lazy_import('sage.groups.matrix_gps.finitely_generated', 'MatrixGroup')
lazy_import('sage.groups.matrix_gps.finitely_generated', 'QuaternionMatrixGroupGF3')
import sage.groups.matrix_gps.pickling_overrides
# Free modules and friends
from .free_module import CombinatorialFreeModule
from .combinatorial_algebra import CombinatorialAlgebra
from .debruijn_sequence import DeBruijnSequences
from .schubert_polynomial import SchubertPolynomialRing
from .symmetric_group_algebra import SymmetricGroupAlgebra, HeckeAlgebraSymmetricGroupT
from .symmetric_group_representations import SymmetricGroupRepresentation, SymmetricGroupRepresentations
from .yang_baxter_graph import YangBaxterGraph
#from hall_littlewood import HallLittlewood_qp, HallLittlewood_q, HallLittlewood_p
#Permutations
from .permutation import Permutation, Permutations, Arrangements, CyclicPermutations, CyclicPermutationsOfPartition
from .affine_permutation import AffinePermutationGroup
lazy_import('sage.combinat.colored_permutations',
['ColoredPermutations', 'SignedPermutations'])
from .derangements import Derangements
lazy_import('sage.combinat.baxter_permutations', ['BaxterPermutations'])
#RSK
from .rsk import RSK, RSK_inverse, robinson_schensted_knuth, robinson_schensted_knuth_inverse, InsertionRules
#HillmanGrassl
lazy_import("sage.combinat.hillman_grassl",
["WeakReversePlanePartition", "WeakReversePlanePartitions"])
#PerfectMatchings
from .perfect_matching import PerfectMatching, PerfectMatchings
# Integer lists
from .integer_lists import IntegerListsLex
r"""
Symmetric function features that are imported by default in the interpreter namespace
"""
from __future__ import absolute_import
from sage.misc.lazy_import import lazy_import
# In the long run, this will be the single entry point
# Nothing else will be exported
from .sf import SymmetricFunctions
# Advanced stuff:
lazy_import('sage.combinat.sf.kfpoly', 'KostkaFoulkesPolynomial')
from .ns_macdonald import NonattackingFillings, AugmentedLatticeDiagramFilling, LatticeDiagram
from graded_algebras import GradedAlgebras
from graded_coalgebras import GradedCoalgebras
from graded_bialgebras import GradedBialgebras
from graded_hopf_algebras import GradedHopfAlgebras
# graded * with basis
from graded_modules_with_basis import GradedModulesWithBasis
from graded_algebras_with_basis import GradedAlgebrasWithBasis
from graded_coalgebras_with_basis import GradedCoalgebrasWithBasis
from graded_bialgebras_with_basis import GradedBialgebrasWithBasis
from graded_hopf_algebras_with_basis import GradedHopfAlgebrasWithBasis
# coxeter groups
from coxeter_groups import CoxeterGroups
from finite_coxeter_groups import FiniteCoxeterGroups
from weyl_groups import WeylGroups
from finite_weyl_groups import FiniteWeylGroups
from affine_weyl_groups import AffineWeylGroups
# crystal bases
from crystals import Crystals
from highest_weight_crystals import HighestWeightCrystals
from regular_crystals import RegularCrystals
from finite_crystals import FiniteCrystals
from classical_crystals import ClassicalCrystals
# polyhedra
from sage.misc.lazy_import import lazy_import
lazy_import('sage.categories.polyhedra', 'PolyhedralSets')
"""
Crystal features that are imported by default in the interpreter namespace
"""
from __future__ import absolute_import
from sage.misc.lazy_import import lazy_import
lazy_import('sage.combinat.crystals', 'catalog', 'crystals')
from sage.misc.lazy_import import lazy_import
lazy_import("sage.dynamics.interval_exchanges", "constructors", "iet")
# Free modules and friends
from .free_module import CombinatorialFreeModule
from .combinatorial_algebra import CombinatorialAlgebra
from .debruijn_sequence import DeBruijnSequences
from .schubert_polynomial import SchubertPolynomialRing
from .symmetric_group_algebra import SymmetricGroupAlgebra, HeckeAlgebraSymmetricGroupT
from .symmetric_group_representations import SymmetricGroupRepresentation, SymmetricGroupRepresentations
from .yang_baxter_graph import YangBaxterGraph
#from hall_littlewood import HallLittlewood_qp, HallLittlewood_q, HallLittlewood_p
#Permutations
from .permutation import Permutation, Permutations, Arrangements, CyclicPermutations, CyclicPermutationsOfPartition
from .affine_permutation import AffinePermutationGroup
lazy_import('sage.combinat.colored_permutations', ['ColoredPermutations',
'SignedPermutations'])
from .derangements import Derangements
lazy_import('sage.combinat.baxter_permutations', ['BaxterPermutations'])
#RSK
from .rsk import RSK, RSK_inverse, robinson_schensted_knuth, robinson_schensted_knuth_inverse
#HillmanGrassl
lazy_import("sage.combinat.hillman_grassl", ["WeakReversePlanePartition", "WeakReversePlanePartitions"])
#PerfectMatchings
from .perfect_matching import PerfectMatching, PerfectMatchings
# Integer lists
from .integer_lists import IntegerListsLex
from __future__ import absolute_import
from .chain_complex import ChainComplex
from .chain_complex_morphism import ChainComplexMorphism
from .simplicial_complex import SimplicialComplex, Simplex
from .simplicial_complex_morphism import SimplicialComplexMorphism
from . import simplicial_complexes_catalog as simplicial_complexes
from .delta_complex import DeltaComplex, delta_complexes
from .cubical_complex import CubicalComplex, cubical_complexes
from sage.misc.lazy_import import lazy_import
lazy_import('sage.homology.koszul_complex', 'KoszulComplex')
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#******************************************************************************
from __future__ import print_function
import itertools
from sage.categories.category_with_axiom import CategoryWithAxiom
from sage.categories.enumerated_sets import EnumeratedSets
from sage.categories.sets_cat import Sets
from sage.categories.cartesian_product import CartesianProductsCategory
from sage.categories.isomorphic_objects import IsomorphicObjectsCategory
from sage.misc.cachefunc import cached_method
from sage.misc.lazy_import import lazy_import
lazy_import("sage.rings.integer", "Integer")
class FiniteEnumeratedSets(CategoryWithAxiom):
"""
The category of finite enumerated sets
EXAMPLES::
sage: FiniteEnumeratedSets()
Category of finite enumerated sets
sage: FiniteEnumeratedSets().super_categories()
[Category of enumerated sets, Category of finite sets]
sage: FiniteEnumeratedSets().all_super_categories()
[Category of finite enumerated sets,
Category of enumerated sets,
Category of finite sets,
import sage.dynamics.cellular_automata.catalog as cellular_automata
from sage.misc.lazy_import import lazy_import
lazy_import("sage.dynamics.cellular_automata.solitons",
["SolitonCellularAutomata", "PeriodicSolitonCellularAutomata"])
from sage.misc.lazy_import import lazy_import
lazy_import("sage.graphs.graph_generators", "graphs")
lazy_import("sage.graphs.digraph_generators", "digraphs")
lazy_import("sage.graphs.hypergraph_generators", "hypergraphs")
from graph_database import GraphDatabase, GenericGraphQuery, GraphQuery
from graph import Graph
from digraph import DiGraph
from bipartite_graph import BipartiteGraph
from graph_bundle import GraphBundle
import weakly_chordal
import graph_list as graphs_list
import sage.graphs.generic_graph_pyx
import sage.graphs.generic_graph
import sage.graphs.graph_decompositions
import sage.graphs.graph
import sage.graphs.digraph
lazy_import("sage.graphs", "graph_coloring")
import sage.graphs.graph_decompositions
import sage.graphs.modular_decomposition.modular_decomposition
import sage.graphs.comparability
from sage.graphs.cliquer import *
from graph_database import graph_db_info
lazy_import("sage.graphs.graph_editor", "graph_editor")
import sage.graphs.isgci
from sage.graphs.isgci import graph_classes
import sage.graphs.distances_all_pairs
import sage.graphs.trees
import sage.graphs.graph_latex
from sage.misc.lazy_import import lazy_import
lazy_import('sage.arith.all', '*', deprecation=19879)
from graded_modules import GradedModules
from graded_algebras import GradedAlgebras
from graded_coalgebras import GradedCoalgebras
from graded_bialgebras import GradedBialgebras
from graded_hopf_algebras import GradedHopfAlgebras
# graded * with basis
from graded_modules_with_basis import GradedModulesWithBasis
from graded_algebras_with_basis import GradedAlgebrasWithBasis
from graded_coalgebras_with_basis import GradedCoalgebrasWithBasis
from graded_bialgebras_with_basis import GradedBialgebrasWithBasis
from graded_hopf_algebras_with_basis import GradedHopfAlgebrasWithBasis
# Coxeter groups
from coxeter_groups import CoxeterGroups
from finite_coxeter_groups import FiniteCoxeterGroups
from weyl_groups import WeylGroups
from finite_weyl_groups import FiniteWeylGroups
from affine_weyl_groups import AffineWeylGroups
# crystal bases
from crystals import Crystals
from highest_weight_crystals import HighestWeightCrystals
from regular_crystals import RegularCrystals
from finite_crystals import FiniteCrystals
from classical_crystals import ClassicalCrystals
# polyhedra
from sage.misc.lazy_import import lazy_import
lazy_import('sage.categories.polyhedra', 'PolyhedralSets')
and much more. For more information, see
https://www.labri.fr/perso/vdelecro/surface-dynamics/latest/
You can install the ``surface_dynamics`` package via:
sage -pip install surface_dynamics
If you do not have write access to the Sage installation you can
alternatively do:
sage -pip install surface_dynamics --user
"""
# install the docstring of this module to the containing package
from sage.misc.namespace_package import install_doc
install_doc(__package__, __doc__)
from sage.misc.lazy_import import lazy_import
from sage.dynamics.arithmetic_dynamics.all import *
from sage.dynamics.complex_dynamics.all import *
from sage.dynamics.cellular_automata.all import *
# Discrete dynamical systems
lazy_import('sage.dynamics.finite_dynamical_system',
['DiscreteDynamicalSystem'])
lazy_import('sage.dynamics', 'finite_dynamical_system_catalog',
'finite_dynamical_systems')
- :class:`algebras.Tensor <sage.algebras.tensor_algebra.TensorAlgebra>`
"""
from sage.algebras.free_algebra import FreeAlgebra as Free
from sage.algebras.iwahori_hecke_algebra import IwahoriHeckeAlgebra as IwahoriHecke
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra as Quaternion
from sage.algebras.steenrod.steenrod_algebra import SteenrodAlgebra as Steenrod
from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra as FiniteDimensional
from sage.algebras.group_algebra import GroupAlgebra as Group
from sage.algebras.clifford_algebra import CliffordAlgebra as Clifford
from sage.algebras.clifford_algebra import ExteriorAlgebra as Exterior
from sage.algebras.weyl_algebra import DifferentialWeylAlgebra as DifferentialWeyl
from sage.algebras.lie_algebras.lie_algebra import LieAlgebra as Lie
from sage.misc.lazy_import import lazy_import
lazy_import('sage.algebras.nil_coxeter_algebra', 'NilCoxeterAlgebra',
'NilCoxeter')
lazy_import('sage.algebras.free_zinbiel_algebra', 'FreeZinbielAlgebra',
'FreeZinbiel')
lazy_import('sage.algebras.hall_algebra', 'HallAlgebra', 'Hall')
lazy_import('sage.algebras.jordan_algebra', 'JordanAlgebra', 'Jordan')
lazy_import('sage.algebras.orlik_solomon', 'OrlikSolomonAlgebra',
'OrlikSolomon')
lazy_import('sage.algebras.shuffle_algebra', 'ShuffleAlgebra', 'Shuffle')
lazy_import('sage.algebras.schur_algebra', 'SchurAlgebra', 'Schur')
lazy_import('sage.algebras.commutative_dga', 'GradedCommutativeAlgebra',
'GradedCommutative')
lazy_import('sage.algebras.rational_cherednik_algebra',
'RationalCherednikAlgebra', 'RationalCherednik')
lazy_import('sage.algebras.yokonuma_hecke_algebra', 'YokonumaHeckeAlgebra',
'YokonumaHecke')
lazy_import('sage.combinat.posets.incidence_algebras', 'IncidenceAlgebra',
from sage.misc.lazy_import import lazy_import
lazy_import('sage.rings.semirings.non_negative_integer_semiring',
['NonNegativeIntegerSemiring', 'NN'])
#
# http://www.gnu.org/licenses/
#*****************************************************************************
from sage.misc.lazy_import import lazy_import
from quatalg.all import *
# Algebra base classes
from algebra import Algebra, is_Algebra
# Ring element base classes
from algebra_element import AlgebraElement, is_AlgebraElement
from free_algebra import FreeAlgebra, is_FreeAlgebra
from free_algebra_quotient import FreeAlgebraQuotient
from steenrod.all import *
from finite_dimensional_algebras.all import FiniteDimensionalAlgebra
from group_algebra_new import GroupAlgebra
from iwahori_hecke_algebra import IwahoriHeckeAlgebra, IwahoriHeckeAlgebraT
from affine_nil_temperley_lieb import AffineNilTemperleyLiebTypeA
lazy_import('sage.algebras.nil_coxeter_algebra', 'NilCoxeterAlgebra')
lazy_import('sage.algebras.hall_algebra', 'HallAlgebra')
lazy_import('sage.algebras.shuffle_algebra', 'ShuffleAlgebra')
from sage.misc.lazy_import import lazy_import
lazy_import('sage.crypto.sbox',
['SBox', 'feistel_construction', 'misty_construction'],
deprecation=22986)
from __future__ import absolute_import
from sage.misc.lazy_import import lazy_import
lazy_import('sage.functions.piecewise_old', 'Piecewise') # deprecated
lazy_import('sage.functions.piecewise', 'piecewise')
lazy_import('sage.functions.error', ['erf', 'erfc', 'erfi', 'erfinv'])
from .trig import (sin, cos, sec, csc, cot, tan, asin, acos, atan, acot, acsc,
asec, arcsin, arccos, arctan, arccot, arccsc, arcsec,
arctan2, atan2)
from .hyperbolic import (tanh, sinh, cosh, coth, sech, csch, asinh, acosh,
atanh, acoth, asech, acsch, arcsinh, arccosh, arctanh,
arccoth, arcsech, arccsch)
reciprocal_trig_functions = {
'sec': cos,
'csc': sin,
'cot': tan,
'sech': cosh,
'csch': sinh,
'coth': tanh
}
from .other import (ceil, floor, gamma, psi, factorial, beta, binomial,
abs_symbolic, sqrt, log_gamma, gamma_inc, incomplete_gamma,
gamma_inc_lower, arg, real_part, real, frac, imag_part,
imag, imaginary, conjugate)
from .log import (exp, exp_polar, log, ln, polylog, dilog, lambert_w,
from sage.misc.lazy_import import lazy_import
lazy_import('sage.functions.piecewise_old', 'Piecewise') # deprecated
lazy_import('sage.functions.piecewise', 'piecewise')
from trig import ( sin, cos, sec, csc, cot, tan,
asin, acos, atan,
acot, acsc, asec,
arcsin, arccos, arctan,
arccot, arccsc, arcsec,
arctan2, atan2)
from hyperbolic import ( tanh, sinh, cosh, coth, sech, csch,
asinh, acosh, atanh, acoth, asech, acsch,
arcsinh, arccosh, arctanh, arccoth, arcsech, arccsch )
reciprocal_trig_functions = {'sec': cos, 'csc': sin, 'cot': tan, 'sech': cosh, 'csch': sinh, 'coth': tanh}
from other import ( ceil, floor, gamma, psi, factorial, beta, binomial,
abs_symbolic, erf, sqrt, log_gamma,
gamma_inc, incomplete_gamma,
arg, real_part, real,
imag_part, imag, imaginary, conjugate)
from log import (exp, exp_polar, log, ln, polylog, dilog, lambert_w)
from transcendental import (zeta, zetaderiv, zeta_symmetric, hurwitz_zeta,
dickman_rho, stieltjes)
from sage.misc.lazy_import import lazy_import
from cone import Cone
from fan import Fan, FaceFan, NormalFan, Fan2d
from fan_morphism import FanMorphism
from polytope import polymake
from polyhedron.all import *
from lattice_polytope import (LatticePolytope, NefPartition, ReflexivePolytope,
ReflexivePolytopes)
import lattice_polytope
from toric_lattice import ToricLattice
import toric_plotter
from hyperbolic_space.all import *
lazy_import('sage.geometry.hyperplane_arrangement.arrangement',
'HyperplaneArrangements')
lazy_import('sage.geometry.hyperplane_arrangement.library',
'hyperplane_arrangements')
from sage.misc.lazy_import import lazy_import
from sage.features.databases import DatabaseKnotInfo
lazy_import('sage.knots.knot', ['Knot', 'Knots'])
lazy_import('sage.knots.link', 'Link')
if DatabaseKnotInfo().is_present():
lazy_import('sage.knots.knotinfo', ['KnotInfo', 'KnotInfoSeries'])
from sage.misc.lazy_import import lazy_import
lazy_import(
'sage.symbolic.expression',
['normalize_index_for_doctests', 'OperandsWrapper', 'restore_op_wrapper'],
deprecation=32386)
from .r import ttest
from .basic_stats import (mean, mode, std, variance, median, moving_average)
from .hmm import all as hmm
# We lazy_import the following modules since they import numpy which
# slows down sage startup
from sage.misc.lazy_import import lazy_import
lazy_import("sage.stats.time_series", ["TimeSeries", "autoregressive_fit"])
lazy_import("sage.stats.intlist", ["IntList"])
from sage.misc.lazy_import import lazy_import
lazy_import('sage.rings.asymptotic.asymptotic_ring', 'AsymptoticRing')
lazy_import('sage.rings.asymptotic.asymptotic_expansion_generators', 'asymptotic_expansions')
from __future__ import absolute_import
from .chain_complex import ChainComplex
from .chain_complex_morphism import ChainComplexMorphism
from .simplicial_complex import SimplicialComplex, Simplex
from .simplicial_complex_morphism import SimplicialComplexMorphism
from .delta_complex import DeltaComplex, delta_complexes
from .cubical_complex import CubicalComplex, cubical_complexes
from sage.misc.lazy_import import lazy_import
lazy_import('sage.homology.koszul_complex', 'KoszulComplex')
lazy_import('sage.homology', 'simplicial_complexes_catalog', 'simplicial_complexes')
lazy_import('sage.homology', 'simplicial_set_catalog', 'simplicial_sets')
# For taking care of old pickles
from sage.structure.sage_object import register_unpickle_override
register_unpickle_override('sage.homology.examples', 'SimplicialSurface', SimplicialComplex)
Moebius, Hirzebruch_Jung_continued_fraction_list, dedekind_sum, \
prime_factors
from fast_arith import prime_range
from bernoulli_mod_p import bernoulli_mod_p, bernoulli_mod_p_single
from monomials import monomials
#from fast_polynomial.compiled_polynomial import compiled_polynomial
CC = ComplexField()
CIF = ComplexIntervalField()
# i = I = QuadraticField(-1, 'I').gen()
I = CC.gen()
from residue_field import ResidueField
from misc import composite_field
import tests
# Universal Cyclotomic Field
from sage.rings.universal_cyclotomic_field.all import *
from sage.misc.lazy_import import lazy_import
lazy_import('sage.rings.invariant_theory', 'invariant_theory')
from .misc import (
algdep, bernoulli, is_prime, is_prime_power, is_pseudoprime,
is_pseudoprime_power, prime_powers, primes_first_n, eratosthenes, primes,
next_prime_power, next_probable_prime, next_prime, previous_prime,
previous_prime_power, random_prime, divisors, sigma, gcd, GCD, xlcm, xgcd,
xkcd, inverse_mod, get_gcd, get_inverse_mod, power_mod,
rational_reconstruction, mqrr_rational_reconstruction, trial_division,
factor, prime_divisors, odd_part, prime_to_m_part, is_square,
is_squarefree, euler_phi, crt, CRT, CRT_list, CRT_basis, CRT_vectors,
multinomial, multinomial_coefficients, binomial, factorial,
kronecker_symbol, kronecker, legendre_symbol, primitive_root, nth_prime,
quadratic_residues, moebius, continuant, number_of_divisors,
hilbert_symbol, hilbert_conductor, hilbert_conductor_inverse,
falling_factorial, rising_factorial, integer_ceil, integer_floor,
two_squares, three_squares, four_squares, sum_of_k_squares, subfactorial,
is_power_of_two, differences, sort_complex_numbers_for_display,
fundamental_discriminant, squarefree_divisors, radical,
binomial_coefficients, jacobi_symbol, dedekind_sum, prime_factors,
prime_range, valuation)
lazy_import('sage.arith.misc', ('Sigma', 'Moebius', 'Euler_Phi'),
deprecation=30322)
from .functions import lcm
LCM = lcm
from .srange import xsrange, srange, ellipsis_iter, ellipsis_range
sxrange = xsrange
σ = sigma
import sage.crypto.sbox
from sage.misc.persist import register_unpickle_override
register_unpickle_override('sage.crypto.mq.sbox', 'SBox',
sage.crypto.sbox.SBox)
from sage.misc.lazy_import import lazy_import
lazy_import('sage.crypto.classical', [
'AffineCryptosystem',
'HillCryptosystem',
'SubstitutionCryptosystem',
'ShiftCryptosystem',
'TranspositionCryptosystem',
'VigenereCryptosystem',
])
lazy_import('sage.crypto.stream', [
'LFSRCryptosystem',
'ShrinkingGeneratorCryptosystem',
])
lazy_import('sage.crypto.lfsr', [
'lfsr_sequence',
'lfsr_autocorrelation',
'lfsr_connection_polynomial',
])
<sage.algebras.yokonuma_hecke_algebra.YokonumaHeckeAlgebra>`
- :class:`algebras.Tensor <sage.algebras.tensor_algebra.TensorAlgebra>`
"""
from sage.algebras.free_algebra import FreeAlgebra as Free
from sage.algebras.iwahori_hecke_algebra import IwahoriHeckeAlgebra as IwahoriHecke
from sage.algebras.quatalg.quaternion_algebra import QuaternionAlgebra as Quaternion
from sage.algebras.steenrod.steenrod_algebra import SteenrodAlgebra as Steenrod
from sage.algebras.finite_dimensional_algebras.finite_dimensional_algebra import FiniteDimensionalAlgebra as FiniteDimensional
from sage.algebras.group_algebra import GroupAlgebra as Group
from sage.algebras.clifford_algebra import CliffordAlgebra as Clifford
from sage.algebras.clifford_algebra import ExteriorAlgebra as Exterior
from sage.algebras.weyl_algebra import DifferentialWeylAlgebra as DifferentialWeyl
from sage.misc.lazy_import import lazy_import
lazy_import('sage.algebras.nil_coxeter_algebra', 'NilCoxeterAlgebra', 'NilCoxeter')
lazy_import('sage.algebras.free_zinbiel_algebra', 'FreeZinbielAlgebra', 'FreeZinbiel')
lazy_import('sage.algebras.hall_algebra', 'HallAlgebra', 'Hall')
lazy_import('sage.algebras.jordan_algebra', 'JordanAlgebra', 'Jordan')
lazy_import('sage.algebras.orlik_solomon', 'OrlikSolomonAlgebra', 'OrlikSolomon')
lazy_import('sage.algebras.shuffle_algebra', 'ShuffleAlgebra', 'Shuffle')
lazy_import('sage.algebras.schur_algebra', 'SchurAlgebra', 'Schur')
lazy_import('sage.algebras.commutative_dga', 'GradedCommutativeAlgebra', 'GradedCommutative')
lazy_import('sage.algebras.yokonuma_hecke_algebra', 'YokonumaHeckeAlgebra', 'YokonumaHecke')
lazy_import('sage.combinat.posets.incidence_algebras', 'IncidenceAlgebra', 'Incidence')
lazy_import('sage.combinat.descent_algebra', 'DescentAlgebra', 'Descent')
lazy_import('sage.combinat.diagram_algebras', 'BrauerAlgebra', 'Brauer')
lazy_import('sage.combinat.diagram_algebras', 'PartitionAlgebra', 'Partition')
lazy_import('sage.combinat.diagram_algebras', 'PlanarAlgebra', 'PlanarPartition')
lazy_import('sage.combinat.diagram_algebras', 'TemperleyLiebAlgebra', 'TemperleyLieb')
lazy_import('sage.combinat.posets.moebius_algebra', 'MoebiusAlgebra', 'Moebius')
import sage.libs.ntl.all as ntl
from sage.libs.pari.all import pari, pari_gen, allocatemem, PariError
from sage.libs.mwrank.all import (mwrank_EllipticCurve, mwrank_MordellWeil,
mwrank_initprimes,
get_precision as mwrank_get_precision,
set_precision as mwrank_set_precision)
import symmetrica.all as symmetrica
from cremona.all import CremonaModularSymbols
from sage.misc.lazy_import import lazy_import
lazy_import('sage.libs.gap.libgap', 'libgap')
#
# Distributed under the terms of the GNU General Public License (GPL)
# http://www.gnu.org/licenses/
#******************************************************************************
from __future__ import print_function
import itertools
from sage.categories.category_with_axiom import CategoryWithAxiom
from sage.categories.enumerated_sets import EnumeratedSets
from sage.categories.sets_cat import Sets
from sage.categories.cartesian_product import CartesianProductsCategory
from sage.categories.isomorphic_objects import IsomorphicObjectsCategory
from sage.misc.cachefunc import cached_method
from sage.misc.lazy_import import lazy_import
lazy_import("sage.rings.integer", "Integer")
class FiniteEnumeratedSets(CategoryWithAxiom):
"""
The category of finite enumerated sets
EXAMPLES::
sage: FiniteEnumeratedSets()
Category of finite enumerated sets
sage: FiniteEnumeratedSets().super_categories()
[Category of enumerated sets, Category of finite sets]
sage: FiniteEnumeratedSets().all_super_categories()
[Category of finite enumerated sets,
Category of enumerated sets,
from __future__ import absolute_import
from sage.misc.lazy_import import lazy_import
lazy_import(
"sage.dynamics.complex_dynamics.mandel_julia",
["mandelbrot_plot", "external_ray", "kneading_sequence", "julia_plot"])
r"""
Combinatorics on matrix features that are imported by default in the interpreter namespace
"""
from sage.misc.lazy_import import lazy_import
lazy_import('sage.combinat.matrices.latin',
['LatinSquare', 'LatinSquare_generator'])
lazy_import('sage.combinat.matrices.dlxcpp', 'DLXCPP')
lazy_import('sage.combinat.matrices.hadamard_matrix',
['hadamard_matrix', 'hadamard_matrix_www'])
from .simplicial_complex import SimplicialComplex, Simplex
from .simplicial_complex_morphism import SimplicialComplexMorphism
from .delta_complex import DeltaComplex, delta_complexes
from .cubical_complex import CubicalComplex, cubical_complexes
from sage.misc.lazy_import import lazy_import
lazy_import('sage.topology.filtered_simplicial_complex',
'FilteredSimplicialComplex')
lazy_import('sage.topology', 'simplicial_complex_catalog',
'simplicial_complexes')
lazy_import('sage.topology', 'simplicial_set_catalog', 'simplicial_sets')
# # For taking care of old pickles
# from sage.misc.persist import register_unpickle_override
# register_unpickle_override('sage.topology.simplicial_complex_examples', 'SimplicialSurface', SimplicialComplex)
"""
Combinatorial Hopf Algebras
"""
from sage.misc.lazy_import import lazy_import
lazy_import('sage.combinat.chas.fsym', ['FreeSymmetricFunctions'])
lazy_import('sage.combinat.chas.wqsym', ['WordQuasiSymmetricFunctions'])
from sage.misc.lazy_import import lazy_import
lazy_import('sage.geometry.hyperbolic_space.hyperbolic_interface', 'HyperbolicPlane')
