sagemathgh-38132: some care for blank lines (pep E30) in groups/ and …

…numerical/

    
fixing pycodestyle warning for blank lines in pyx files in `groups` and
`numerical` folders

### 📝 Checklist

- [x] The title is concise and informative.
- [x] The description explains in detail what this PR is about.
    
URL: sagemath#38132
Reported by: Frédéric Chapoton
Reviewer(s): Matthias Köppe

src/sage/groups/group.pyx

@@ -218,7 +218,6 @@ cdef class Group(Parent):
         """
         return self.order() == 1
 
-
     def is_multiplicative(self):
         r"""
         Return ``True`` if the group operation is given by ``*`` (rather than ``+``).

src/sage/groups/perm_gps/partn_ref/automorphism_group_canonical_label.pyx

@@ -135,6 +135,7 @@ cdef int refine_and_return_invariant_trivial(PartitionStack *PS, void *S, int *c
 cdef int compare_structures_trivial(int *gamma_1, int *gamma_2, void *S1, void *S2, int degree) noexcept:
     return 0
 
+
 def test_get_aut_gp_and_can_lab_trivially(int n=6,
     list partition=[[0,1,2],[3,4],[5]], canonical_label=True, base=False):
     """
@@ -171,6 +172,7 @@ def test_get_aut_gp_and_can_lab_trivially(int n=6,
     PS_dealloc(part)
     deallocate_agcl_output(output)
 
+
 def test_intersect_parabolic_with_alternating(int n=9, list partition=[[0,1,2],[3,4],[5,6,7,8]]):
     """
     A test for nontrivial input group in computing automorphism groups.
@@ -212,6 +214,7 @@ def test_intersect_parabolic_with_alternating(int n=9, list partition=[[0,1,2],[
     SC_dealloc(group)
     deallocate_agcl_output(output)
 
+
 cdef int compare_perms(int *gamma_1, int *gamma_2, void *S1, void *S2, int degree) noexcept:
     cdef list MS1 = <list> S1
     cdef list MS2 = <list> S2
@@ -222,6 +225,7 @@ cdef int compare_perms(int *gamma_1, int *gamma_2, void *S1, void *S2, int degre
             return j
     return 0
 
+
 def coset_rep(list perm=[0,1,2,3,4,5], list gens=[[1,2,3,4,5,0]]):
     """
     Given a group G generated by the given generators, defines a map from the
@@ -293,6 +297,7 @@ def coset_rep(list perm=[0,1,2,3,4,5], list gens=[[1,2,3,4,5,0]]):
     sig_free(c_perm)
     return label
 
+
 cdef aut_gp_and_can_lab *allocate_agcl_output(int n) noexcept:
     r"""
     Allocate an instance of the aut_gp_and_can_lab struct of degree n. This can

src/sage/groups/perm_gps/partn_ref/data_structures.pyx

@@ -75,9 +75,10 @@ cdef OP_string(OrbitPartition *OP):
     for i in range(OP.degree):
         s += " "
         j = OP_find(OP, i)
-        s += "%d -> %d"%(i, j)
+        s += "%d -> %d" % (i, j)
     return s
 
+
 def OP_represent(int n, merges, perm):
     """
     Demonstration and testing.
@@ -397,6 +398,7 @@ cdef list PS_singletons(PartitionStack * part):
 
     return l
 
+
 def PS_represent(partition, splits):
     """
     Demonstration and testing.
@@ -1209,6 +1211,7 @@ cdef bint SC_is_giant(int n, int num_perms, int *perms, float p, bitset_t suppor
     sig_free(perm)
     return False
 
+
 def SC_test_list_perms(list L, int n, int limit, bint gap, bint limit_complain, bint test_contains):
     """
     Test that the permutation group generated by list perms in L of degree n
@@ -1612,6 +1615,7 @@ def SC_test_list_perms(list L, int n, int limit, bint gap, bint limit_complain,
         SC_dealloc(SCCC)
         SC_dealloc(SC_nb)
 
+
 # Functions
 
 cdef int sort_by_function(PartitionStack *PS, int start, int *degrees) noexcept:

src/sage/groups/perm_gps/partn_ref/double_coset.pyx

@@ -117,6 +117,7 @@ cdef int compare_perms(int *gamma_1, int *gamma_2, void *S1, void *S2, int degre
             return j
     return 0
 
+
 def coset_eq(list perm1=[0,1,2,3,4,5], list perm2=[1,2,3,4,5,0], list gens=[[1,2,3,4,5,0]]):
     """
     Given a group G generated by the given generators, tests whether the given
@@ -199,6 +200,7 @@ def coset_eq(list perm1=[0,1,2,3,4,5], list perm2=[1,2,3,4,5,0], list gens=[[1,2
     sig_free(isomorphism)
     return x
 
+
 cdef dc_work_space *allocate_dc_work_space(int n) noexcept:
     r"""
     Allocates work space for the double_coset function. It can be

src/sage/groups/perm_gps/partn_ref/refinement_graphs.pyx

@@ -506,6 +506,7 @@ def search_tree(G_in, partition, lab=True, dig=False, dict_rep=False, certificat
     else:
         return tuple(return_tuple)
 
+
 cdef int refine_by_degree(PartitionStack *PS, void *S, int *cells_to_refine_by, int ctrb_len) noexcept:
     r"""
     Refine the input partition by checking degrees of vertices to the given
@@ -746,6 +747,7 @@ cdef inline int degree(PartitionStack *PS, CGraph G, int entry, int cell_index,
                 break
     return num_arcs
 
+
 def all_labeled_graphs(n):
     """
     Return all labeled graphs on n vertices {0,1,...,n-1}.
@@ -948,6 +950,7 @@ def orbit_partition(gamma, list_perm=False):
                     i[j] = 0
         return l
 
+
 def coarsest_equitable_refinement(CGraph G, list partition, bint directed):
     """
     Return the coarsest equitable refinement of ``partition`` for ``G``.
@@ -1012,6 +1015,7 @@ def coarsest_equitable_refinement(CGraph G, list partition, bint directed):
 
     return eq_part
 
+
 def get_orbits(list gens, int n):
     """
     Compute orbits given a list of generators of a permutation group, in list
@@ -1054,8 +1058,6 @@ def get_orbits(list gens, int n):
     return list(orbit_dict.itervalues())
 
 
-
-
 # Canonical augmentation
 from cpython.ref cimport *
 
@@ -1386,7 +1388,6 @@ def generate_dense_graphs_edge_addition(int n, bint loops, G=None, depth=None,
         return number
 
 
-
 # Dense graphs: adding vertices
 
 # This implements an augmentation scheme as follows:

src/sage/groups/perm_gps/partn_ref/refinement_lists.pyx

@@ -62,6 +62,7 @@ def is_isomorphic(self, other):
     sig_free(output)
     return output_py
 
+
 cdef bint all_list_children_are_equivalent(PartitionStack *PS, void *S) noexcept:
     return 0
 

src/sage/groups/perm_gps/partn_ref/refinement_matrices.pyx

@@ -170,7 +170,6 @@ cdef class MatrixStruct:
 
         PS_dealloc(part)
 
-
     def automorphism_group(self):
         """
         Return a list of generators of the automorphism group, along with its
@@ -296,6 +295,7 @@ cdef int compare_matrices(int *gamma_1, int *gamma_2, void *S1, void *S2, int de
 cdef bint all_matrix_children_are_equivalent(PartitionStack *PS, void *S) noexcept:
     return 0
 
+
 def random_tests(n=10, nrows_max=50, ncols_max=50, nsymbols_max=10, perms_per_matrix=5, density_range=(.1,.9)):
     """
     Test to make sure that ``C(gamma(M)) == C(M)`` for random permutations ``gamma``

src/sage/groups/perm_gps/partn_ref/refinement_python.pyx

@@ -333,6 +333,7 @@ cdef class PythonPartitionStack:
                     cell = []
         return partition
 
+
 class PythonObjectWrapper:
     """
     Instances of this class wrap a Python object and the refinement functions.
@@ -368,6 +369,7 @@ class PythonObjectWrapper:
         self.rari_fn = rari_fn
         self.cs_fn = cs_fn
 
+
 cdef bint all_children_are_equivalent_python(PartitionStack *PS, void *S) noexcept:
     """
     Python conversion of all_children_are_equivalent function.
@@ -398,6 +400,7 @@ cdef int compare_structures_python(int *gamma_1, int *gamma_2, void *S1, void *S
     cdef list gamma_2_py = [gamma_2[i] for i from 0 <= i < degree]
     return S1_obj.cs_fn(gamma_1_py, gamma_2_py, S1_obj.obj, S2_obj.obj, degree)
 
+
 def aut_gp_and_can_lab_python(S, partition, n,
     all_children_are_equivalent,
     refine_and_return_invariant,

src/sage/groups/perm_gps/partn_ref/refinement_sets.pyx

@@ -181,6 +181,7 @@ def set_stab_py(generators, sett, relab=False):
         return stab_gens, relabeling
     return stab_gens
 
+
 cdef aut_gp_and_can_lab *set_stab(StabilizerChain *supergroup, subset *sett, bint relab) noexcept:
     r"""
     Compute the set stabilizer of ``sett`` within ``supergroup``. (Note that
@@ -200,6 +201,7 @@ cdef aut_gp_and_can_lab *set_stab(StabilizerChain *supergroup, subset *sett, bin
         return NULL
     return output
 
+
 def sets_isom_py(generators, set1, set2):
     r"""
     Compute whether ``set1`` and ``set2`` are isomorphic under the action of
@@ -424,6 +426,7 @@ def sets_isom_py(generators, set1, set2):
     sig_free(isom)
     return output_py
 
+
 cdef int sets_isom(StabilizerChain *supergroup, subset *set1, subset *set2, int *isom) except -1:
     r"""
     Underlying C function for testing two sets for isomorphism.

src/sage/groups/perm_gps/partn_ref2/refinement_generic.pyx

@@ -857,7 +857,6 @@ cdef class PartitionRefinement_generic:
         """
         return self._is_candidate_initialized and self._known_automorphisms.has_empty_intersection(self._part)
 
-
     ###########################################################################
     # These functions are used to produce some latex output:
     # it writes the actual node

src/sage/groups/perm_gps/permgroup_element.pyx

@@ -152,6 +152,7 @@ cdef extern from *:
 cdef int etuple_index_cmp(const void * a, const void * b) noexcept nogil:
     return ((<int *> a)[0] > (<int *> b)[0]) - ((<int *> a)[0] < (<int *> b)[0])
 
+
 def make_permgroup_element(G, x):
     r"""
     Return a :class:`PermutationGroupElement` given the permutation group
@@ -172,6 +173,7 @@ def make_permgroup_element(G, x):
     domain = FiniteEnumeratedSet(range(1, len(x)+1))
     return make_permgroup_element_v2(G, x, domain)
 
+
 def make_permgroup_element_v2(G, x, domain):
     r"""
     Return a :class:`PermutationGroupElement` given the permutation group
@@ -199,6 +201,7 @@ def make_permgroup_element_v2(G, x, domain):
     G._domain_from_gap = {i+1: key for i, key in enumerate(domain)}
     return G.element_class(x, G, check=False)
 
+
 def is_PermutationGroupElement(x):
     r"""
     Return ``True`` if ``x`` is a :class:`PermutationGroupElement`.
@@ -212,6 +215,7 @@ def is_PermutationGroupElement(x):
     """
     return isinstance(x, PermutationGroupElement)
 
+
 cdef class PermutationGroupElement(MultiplicativeGroupElement):
     r"""
     An element of a permutation group.
@@ -791,7 +795,6 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
 
             i = j + 1
 
-
     def __reduce__(self):
         r"""
         Return a function and its arguments needed to create this
@@ -1252,7 +1255,6 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
             if isinstance(x, Matrix):
                 return x.with_permuted_rows(self)
 
-
     def __mul__(left, right):
         r"""
         TESTS::
@@ -1696,7 +1698,6 @@ cdef class PermutationGroupElement(MultiplicativeGroupElement):
         sig_free(seen)
         return 1 - 2*(cycle_len_sum % 2) # == (-1)^cycle_len
 
-
     def orbit(self, n, bint sorted=True):
         r"""
         Return the orbit of the integer `n` under this group

src/sage/groups/semimonomial_transformations/semimonomial_transformation.pyx

@@ -90,6 +90,7 @@ def _inverse(f, R):
         g *= f
     return g
 
+
 cdef class SemimonomialTransformation(MultiplicativeGroupElement):
     r"""
     An element in the semimonomial group over a ring `R`. See

src/sage/numerical/backends/cvxopt_backend.pyx

@@ -497,10 +497,10 @@ cdef class CVXOPTBackend(GenericBackend):
         from cvxopt import matrix, solvers
         h = []
 
-        #for the equation bounds
+        # for the equation bounds
         for eq_index in range(self.nrows()):
             h.append(self.row_upper_bound[eq_index])
-            #upper bound is already in G
+            # upper bound is already in G
             if self.row_lower_bound[eq_index] is not None:
                 h.append(-1 * self.row_lower_bound[eq_index])
                 for cindex in range(len(self.G_matrix)):
@@ -509,9 +509,7 @@ cdef class CVXOPTBackend(GenericBackend):
                     else:
                         self.G_matrix[cindex].append(0)
 
-
-
-        #for the upper bounds (if there are any)
+        # for the upper bounds (if there are any)
         for i in range(len(self.col_upper_bound)):
             if self.col_upper_bound[i] is not None:
                 h.append(self.col_upper_bound[i])
@@ -563,7 +561,6 @@ cdef class CVXOPTBackend(GenericBackend):
             raise MIPSolverException("CVXOPT: Terminated early due to numerical difficulties or because the maximum number of iterations was reached.")
         return 0
 
-
     cpdef get_objective_value(self):
         """
         Return the value of the objective function.
@@ -659,7 +656,6 @@ cdef class CVXOPTBackend(GenericBackend):
         """
         return len(self.row_upper_bound)
 
-
     cpdef bint is_maximization(self) noexcept:
         """
         Test whether the problem is a maximization
@@ -702,7 +698,6 @@ cdef class CVXOPTBackend(GenericBackend):
             return self.prob_name
         self.prob_name = name
 
-
     cpdef row(self, int i):
         """
         Return a row
@@ -740,7 +735,6 @@ cdef class CVXOPTBackend(GenericBackend):
             index += 1
         return (idx, coeff)
 
-
     cpdef row_bounds(self, int index):
         """
         Return the bounds of a specific constraint.

src/sage/numerical/backends/cvxopt_sdp_backend.pyx

@@ -56,7 +56,6 @@ cdef class CVXOPTSDPBackend(MatrixSDPBackend):
                       "refinement":1 }
         self.answer = {}
 
-
     cpdef int solve(self) except -1:
         """
         Solve the problem.
@@ -161,7 +160,6 @@ cdef class CVXOPTSDPBackend(MatrixSDPBackend):
             raise SDPSolverException("CVXOPT: Terminated early due to numerical difficulties or because the maximum number of iterations was reached.")
         return 0
 
-
     cpdef get_objective_value(self):
         """
         Return the value of the objective function.
@@ -358,7 +356,6 @@ cdef class CVXOPTSDPBackend(MatrixSDPBackend):
         assert(n == self.answer['ss'][i].size[1]) # must be square matrix
         return Matrix(n, n, list(self.answer['ss'][i]), sparse=sparse)
 
-
     cpdef solver_parameter(self, name, value=None):
         """
         Return or define a solver parameter