simplify empty sets

src/sage/algebras/orlik_solomon.py

@@ -192,7 +192,7 @@ def one_basis(self):
 
             sage: M = matroids.Wheel(3)
             sage: OS = M.orlik_solomon_algebra(QQ)
-            sage: OS.one_basis() == frozenset([])
+            sage: OS.one_basis() == frozenset()
             True
         """
         return frozenset({})
@@ -347,7 +347,7 @@ def subset_image(self, S):
             [[(1, 2), (1, 4), (2, 3), (3, 4)],
              [(3, 5), (3, 6), (5, 6)]]
             sage: OSMG = MG.orlik_solomon_algebra(QQ, ordering=s)
-            sage: OSMG.subset_image(frozenset([]))
+            sage: OSMG.subset_image(frozenset())
             OS{}
             sage: OSMG.subset_image(frozenset([(1,2),(3,4),(1,4),(2,3)]))
             0
@@ -375,7 +375,7 @@ def subset_image(self, S):
              [0, 3, 5], [1, 2, 4], [1, 2, 5], [1, 3, 4],
              [1, 3, 5], [2, 3], [4, 5]]
             sage: OSMG = MG.orlik_solomon_algebra(QQ)
-            sage: OSMG.subset_image(frozenset([]))
+            sage: OSMG.subset_image(frozenset())
             OS{}
             sage: OSMG.subset_image(frozenset([1, 2, 3]))
             0
@@ -394,7 +394,7 @@ def subset_image(self, S):
             sage: sorted([sorted(c) for c in MG.circuits()])
             [[0, 1], [2, 3, 4]]
             sage: OSMG = MG.orlik_solomon_algebra(QQ)
-            sage: OSMG.subset_image(frozenset([]))
+            sage: OSMG.subset_image(frozenset())
             OS{}
             sage: OSMG.subset_image(frozenset([1, 3, 4]))
             -OS{0, 2, 3} + OS{0, 2, 4}

src/sage/algebras/orlik_terao.py

@@ -242,7 +242,7 @@ def one_basis(self):
 
             sage: M = matroids.Wheel(3)
             sage: OT = M.orlik_terao_algebra(QQ)
-            sage: OT.one_basis() == frozenset([])
+            sage: OT.one_basis() == frozenset()
             True
         """
         return frozenset({})
@@ -391,7 +391,7 @@ def subset_image(self, S):
             [[(1, 2), (1, 4), (2, 3), (3, 4)],
              [(3, 5), (3, 6), (5, 6)]]
             sage: OT = M.orlik_terao_algebra(QQ, ordering=s)
-            sage: OT.subset_image(frozenset([]))
+            sage: OT.subset_image(frozenset())
             OT{}
             sage: OT.subset_image(frozenset([(1,2),(3,4),(1,4),(2,3)]))
             0
@@ -419,7 +419,7 @@ def subset_image(self, S):
              [0, 3, 5], [1, 2, 4], [1, 2, 5], [1, 3, 4],
              [1, 3, 5], [2, 3], [4, 5]]
             sage: OT = M.orlik_terao_algebra()
-            sage: OT.subset_image(frozenset([]))
+            sage: OT.subset_image(frozenset())
             OT{}
             sage: OT.subset_image(frozenset([1, 2, 3]))
             0
@@ -438,7 +438,7 @@ def subset_image(self, S):
             sage: sorted([sorted(c) for c in M.circuits()])
             [[0, 1], [2, 3, 4]]
             sage: OT = M.orlik_terao_algebra(QQ)
-            sage: OT.subset_image(frozenset([]))
+            sage: OT.subset_image(frozenset())
             OT{}
             sage: OT.subset_image(frozenset([1, 3, 4]))
             -OT{0, 2, 3} + OT{0, 2, 4}

src/sage/calculus/desolvers.py

@@ -913,7 +913,7 @@ def desolve_system(des, vars, ics=None, ivar=None, algorithm='maxima'):
 
     if len(des) == 1 and algorithm == "maxima":
         return desolve_laplace(des[0], vars[0], ics=ics, ivar=ivar)
-    ivars = set([])
+    ivars = set()
     for i, de in enumerate(des):
         if not (isinstance(de, Expression) and de.is_relational()):
             des[i] = de == 0
@@ -1440,11 +1440,10 @@ def desolve_system_rk4(des, vars, ics=None, ivar=None, end_points=None, step=0.1
 
     - Robert Marik (10-2009)
     """
-
     if ics is None:
         raise ValueError("No initial conditions, specify with ics=[x0,y01,y02,...].")
 
-    ivars = set([])
+    ivars = set()
 
     for de in des:
         ivars = ivars.union(set(de.variables()))

src/sage/combinat/root_system/pieri_factors.py

@@ -519,7 +519,7 @@ class PieriFactors_type_A_affine(PieriFactors_affine_type):
 
     @staticmethod
     def __classcall__(cls, W, min_length=0, max_length=infinity,
-                      min_support=frozenset([]), max_support=None):
+                      min_support=frozenset(), max_support=None):
         r"""
         TESTS::
 

src/sage/doctest/util.py

@@ -606,7 +606,7 @@ def make_recording_dict(D, st, gt):
     EXAMPLES::
 
         sage: from sage.doctest.util import make_recording_dict
-        sage: D = make_recording_dict({'a':4,'d':42},set([]),set(['not_here']))
+        sage: D = make_recording_dict({'a':4,'d':42},set(),set(['not_here']))
         sage: sorted(D.items())
         [('a', 4), ('d', 42)]
         sage: D.got

src/sage/geometry/fan.py

@@ -636,7 +636,7 @@ def result():
                         (len(cones), len(generating_cones)))
         elif discard_faces:
             cones = _discard_faces(cones)
-        ray_set = set([])
+        ray_set = set()
         for cone in cones:
             ray_set.update(cone.rays())
         if rays:    # Preserve the initial order of rays, if they were given

src/sage/geometry/hasse_diagram.py

@@ -146,7 +146,7 @@ def default_face_constructor(atoms, coatoms, **kwds):
                 atoms = atoms.intersection(coatom_to_atoms[coatom])
             H[atom] = (atoms, coatoms)
         # 8: compute the set G of minimal sets in H
-        minimals = set([])
+        minimals = set()
         while candidates:
             candidate = candidates.pop()
             atoms = H[candidate][0]

src/sage/geometry/hyperplane_arrangement/arrangement.py

@@ -2199,7 +2199,7 @@ def poset_of_regions(self, B=None, numbered_labels=True):
         while R:
             # Transfer the "next step" to the "current step"
             curTest = list(nextTest)
-            nextTest = set([])
+            nextTest = set()
             # we want to test each region that we haven't hit yet
             for r in R:
                 # Since it's graded, it suffices to look at the regions of the previous rank

src/sage/geometry/lattice_polytope.py

@@ -3947,11 +3947,10 @@ def skeleton_points(self, k=1):
         """
         if k >= self.dim():
             return list(range(self.npoints()))
-        skeleton = set([])
+        skeleton = set()
         for face in self.faces(dim=k):
             skeleton.update(face.ambient_point_indices())
-        skeleton = sorted(skeleton)
-        return skeleton
+        return sorted(skeleton)
 
     def skeleton_show(self, normal=None):
         r"""Show the graph of one-skeleton of this polytope.

src/sage/geometry/polyhedral_complex.py

@@ -370,7 +370,7 @@ def cells(self, subcomplex=None) -> dict:
         for k in range(self._dim, -1, -1):
             if k in maximal_cells:
                 if k not in cells:
-                    cells[k] = set([])
+                    cells[k] = set()
                 cells[k].update(maximal_cells[k])
             if k in cells:
                 for cell in cells[k]:
@@ -382,7 +382,7 @@ def cells(self, subcomplex=None) -> dict:
                             covers[p] = []
                         covers[p].append(cell)
                         if (k-1) not in cells:
-                            cells[k-1] = set([])
+                            cells[k-1] = set()
                         cells[k-1].add(p)
         self._face_poset = Poset(covers)
         self._cells = cells
@@ -1055,7 +1055,7 @@ def is_subcomplex(self, other) -> bool:
         """
         other_cells = other.cells()
         for (d, stratum) in self.maximal_cells().items():
-            if not stratum.issubset(other_cells.get(d, set([]))):
+            if not stratum.issubset(other_cells.get(d, set())):
                 return False
         return True
 
@@ -1594,9 +1594,9 @@ def is_convex(self) -> bool:
         # After making sure that the affine hulls of the cells are the same,
         # it does not matter that is not full dimensional.
         boundaries = self.relative_boundary_cells()
-        vertices = set([])
-        rays = set([])
-        lines = set([])
+        vertices = set()
+        rays = set()
+        lines = set()
         for cell in boundaries:
             # it suffices to consider only vertices on the boundaries
             # Note that a line (as polyhedron) has vertex too
@@ -2055,7 +2055,7 @@ def add_cell(self, cell):
                 for facet in c.facets():
                     p = facet.as_polyhedron()
                     if d not in cells:
-                        cells[d] = set([])
+                        cells[d] = set()
                     if p not in cells[d]:
                         cells[d].add(p)
                         covers[p] = [c]
@@ -2358,7 +2358,7 @@ def subdivide(self, make_simplicial=False,
             if new_rays:
                 raise ValueError("rays/lines cannot be used for subdivision")
             # bounded version of `fan.subdivide`; not require rational.
-            vertices = set([])
+            vertices = set()
             if make_simplicial and not self.is_simplicial_complex():
                 for p in self.maximal_cell_iterator():
                     for v in p.vertices_list():
@@ -2390,7 +2390,7 @@ def subdivide(self, make_simplicial=False,
                 raise ValueError("new vertices cannot be used for subdivision")
             # mimic :meth:`~sage.geometry.fan <RationalPolyhedralFan>.subdivide`
             # but here we allow for non-pointed cones, and we subdivide them.
-            rays_normalized = set([])
+            rays_normalized = set()
             self_rays = []
             cones = []
             for p in self.maximal_cell_iterator():

src/sage/graphs/generators/families.py

@@ -3135,7 +3135,7 @@ def YDeltaTrans(G, v):
     # for as long as we generate new graphs.
     P = PetersenGraph()
 
-    l = set([])
+    l = set()
     l_new = [P.canonical_label().graph6_string()]
 
     while l_new:

src/sage/graphs/generators/smallgraphs.py

@@ -5642,7 +5642,7 @@ def pi_vec(x):
     # Associate a matrix to every entry of W
     int_to_matrix = {0: matrix.zero(45)}
     for i in range(15):
-        vec = [frozenset([]), L[0, 0], L[1, 0], L[2, 0], L[3, 0]]
+        vec = [frozenset(), L[0, 0], L[1, 0], L[2, 0], L[3, 0]]
         vec = f_pow(pi_vec, i % 3, vec)
         vec = f_pow(sigma2, i % 5, vec)
         int_to_matrix[i + 1] = N(vec)

src/sage/matrix/matrix_modn_sparse.pyx

@@ -361,7 +361,7 @@ cdef class Matrix_modn_sparse(Matrix_sparse):
         cdef c_vector_modint* v
 
         # Build a table that gives the nonzero positions in each column of right
-        nonzero_positions_in_columns = [set([]) for _ in range(right._ncols)]
+        nonzero_positions_in_columns = [set() for _ in range(right._ncols)]
         cdef Py_ssize_t i, j, k
         for i from 0 <= i < right._nrows:
             v = &(right.rows[i])

src/sage/matrix/matrix_rational_sparse.pyx

@@ -236,7 +236,7 @@ cdef class Matrix_rational_sparse(Matrix_sparse):
         cdef mpq_vector* v
 
         # Build a table that gives the nonzero positions in each column of right
-        nonzero_positions_in_columns = [set([]) for _ in range(right._ncols)]
+        nonzero_positions_in_columns = [set() for _ in range(right._ncols)]
         cdef Py_ssize_t i, j, k
         for i in range(right._nrows):
             v = &(right._matrix[i])

src/sage/matroids/dual_matroid.py

@@ -332,8 +332,8 @@ def _minor(self, contractions=None, deletions=None):
         EXAMPLES::
 
             sage: M = matroids.catalog.Vamos().dual()
-            sage: N = M._minor(contractions=set(['a']), deletions=set([]))
-            sage: N._minor(contractions=set([]), deletions=set(['b', 'c']))
+            sage: N = M._minor(contractions=set(['a']), deletions=set())
+            sage: N._minor(contractions=set(), deletions=set(['b', 'c']))
             Dual of 'M / {'b', 'c'} \ {'a'}, where M is Vamos:
             Matroid of rank 4 on 8 elements with circuit-closures
             {3: {{'a', 'b', 'c', 'd'}, {'a', 'b', 'e', 'f'},

src/sage/matroids/linear_matroid.pyx

@@ -1332,8 +1332,8 @@ cdef class LinearMatroid(BasisExchangeMatroid):
             sage: M = Matroid(groundset='abcdefgh', ring=GF(5),
             ....: reduced_matrix=[[2, 1, 1, 0],
             ....:                 [1, 1, 0, 1], [1, 0, 1, 1], [0, 1, 1, 2]])
-            sage: N = M._minor(contractions=set(['a']), deletions=set([]))
-            sage: N._minor(contractions=set([]), deletions=set(['b', 'c']))
+            sage: N = M._minor(contractions=set(['a']), deletions=set())
+            sage: N._minor(contractions=set(), deletions=set(['b', 'c']))
             Linear matroid of rank 3 on 5 elements represented over the Finite
             Field of size 5
         """
@@ -2847,7 +2847,7 @@ cdef class LinearMatroid(BasisExchangeMatroid):
             if sol:
                 if certificate:
                     (certX, certY) = cert_pair
-                    cert = set([])
+                    cert = set()
                     for x in certX:
                         cert.add(dX[x])
                     for y in certY:
@@ -3783,8 +3783,8 @@ cdef class BinaryMatroid(LinearMatroid):
         EXAMPLES::
 
             sage: M = matroids.catalog.Fano()
-            sage: N = M._minor(contractions=set(['a']), deletions=set([]))
-            sage: N._minor(contractions=set([]), deletions=set(['b', 'c']))
+            sage: N = M._minor(contractions=set(['a']), deletions=set())
+            sage: N._minor(contractions=set(), deletions=set(['b', 'c']))
             Binary matroid of rank 2 on 4 elements, type (0, 6)
         """
         self._move_current_basis(contractions, deletions)
@@ -4720,8 +4720,8 @@ cdef class TernaryMatroid(LinearMatroid):
         EXAMPLES::
 
             sage: M = matroids.catalog.P8()
-            sage: N = M._minor(contractions=set(['a']), deletions=set([]))
-            sage: N._minor(contractions=set([]), deletions=set(['b', 'c']))
+            sage: N = M._minor(contractions=set(['a']), deletions=set())
+            sage: N._minor(contractions=set(), deletions=set(['b', 'c']))
             Ternary matroid of rank 3 on 5 elements, type 0-
         """
         self._move_current_basis(contractions, deletions)
@@ -5488,8 +5488,8 @@ cdef class QuaternaryMatroid(LinearMatroid):
         EXAMPLES::
 
             sage: M = matroids.catalog.Q10()                                            # needs sage.rings.finite_rings
-            sage: N = M._minor(contractions=set(['a']), deletions=set([]))              # needs sage.rings.finite_rings
-            sage: N._minor(contractions=set([]), deletions=set(['b', 'c']))             # needs sage.rings.finite_rings
+            sage: N = M._minor(contractions=set(['a']), deletions=set())              # needs sage.rings.finite_rings
+            sage: N._minor(contractions=set(), deletions=set(['b', 'c']))             # needs sage.rings.finite_rings
             Quaternary matroid of rank 4 on 7 elements
         """
         self._move_current_basis(contractions, deletions)