31 KiB
31 KiB
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from sage.structure.parent import Parent from sage.structure.element import Element #from sage.structure.unique_representation import UniqueRepresentation #from sage.categories.rings import Rings from sage.structure.richcmp import rich_to_bool, op_EQ, op_NE, op_LT, op_LE, op_GT, op_GE #from sage.categories.morphism import Morphism #from sage.rings.rational_field import QQ #from sage.algebras.commutative_dga import GradedCommutativeAlgebra #from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing #from sage.functions.other import factorial import itertools import copy import random
Simplicial Complex¶
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def cohomology_gens(S,n): gens = {} basis = S.cohomology_ring().basis() for k in basis.keys(): if k[0] == n: gens[k] = basis[k] return gens
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def lift_cohomology_generator(S, g): cycle = g.to_cycle() cochain = {} for n in range(S.dimension()+1): for simplex in S._n_cells_sorted(n): cochain[simplex.tuple()] = 0 for i in cycle: #print(type(i[0].tuple())) cochain[i[0].tuple()] = i[1] return cochain
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def sublist(lst1, lst2): return set(lst1) <= set(lst2) def sublist_of_lists(list1,list_of_lists): is_sublist = False for list2 in list_of_lists: if sublist(list1,list2): is_sublist = True break return is_sublist def random_simplicial_complex(num_vertices,max_num_facets): vertices = list(range(num_vertices)) simplices = [] partitions = [] for length in range(num_vertices): partitions += [tuple(e) for e in powerset(vertices) if len(e)==length and e] for i in range(max_num_facets+1): r = randint(0,len(partitions)-1) candidate = partitions[r] if not sublist_of_lists(candidate,simplices): simplices.append(candidate) return SimplicialComplex(simplices,is_mutable=False) def random_cochain(S): dim = S.dimension() cochain = {} for n in range(dim+1): for s in S._n_cells_sorted(n): cochain[s.tuple()] = randint(-100,100) return cochain
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S = random_simplicial_complex(5,4) S.facets()
Out[12]:
{(2, 4), (0, 1, 3, 4)}
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Apln¶
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class Apln(UniqueRepresentation): def __init__(self,n): self.n = n self.algebras = [] self.relations = [] for i in range(n+1): [cdga, relations] = self._cdga(i) self.algebras.append(cdga) self.relations.append(relations) def _cdga(self,n): ts = ['t'+str(t) for t in range(n+1)] ys = ['y'+str(t) for t in range(n+1)] degrees = tuple((n+1)*[(0,2)])+tuple((n+1)*[(1,2)]) gca = GradedCommutativeAlgebra(QQ, names=ts+ys, degrees=degrees) t_relation = -1 y_relation = 0 #t_relation = sum(self.gca.gens()[:n])-1 #y_relation = sum(self.gca.gens()[n:]) for i in range(n+1): t_relation += gca.gens()[i] y_relation += gca.gens()[n+1+i] #gca_with_relations = self.gca.quotient(self.gca.ideal(t_relation,y_relation),check=False) dfs = {gca.gens_dict()[ts[i]]:gca.gens_dict()[ys[i]] for i in range(n+1)} return [gca.cdg_algebra(dfs),[t_relation,y_relation]] def __repr__(self): return "Simplicial algebra of dimension %s" % self.n def inject_variables(self, n): self.algebras[n].inject_variables() def generators(self, n): return self.algebras[n].gens() def face(self,i,n): if(self.n==0): return 0 domain = self.algebras[n] codomain = self.algebras[n-1] tImages = [] yImages = [] n = len(domain.gens())//2 for k in range(n): if(k<i): tImages.append(codomain.gens()[k]) yImages.append(codomain.gens()[n-1+k]) elif(k==i): tImages.append(0) yImages.append(0) elif(k>i): tImages.append(codomain.gens()[k-1]) yImages.append(codomain.gens()[n+k-2]) images = tImages+yImages return domain.hom(codomain=codomain,im_gens=images) def reduce_polynomial(self,p): ## Input: # - p: polynomial in variables t_0,..t_n,y_0,...,y_n ## Output: # - The polynomial after applying the corresponding relations to reduce the variables t_0, y_0 apl = p.parent() dim = len(apl.gens())//2-1 return p.reduce(self.relations[dim]) def _convert_integrable_polynomial(self,p): ## Input: # - p: polynomial in variables t_1,..t_n,y_1,...,y_n ## Output: # - A polynomial in the ring Q<t_1,..t_n>. # (After reducing the variables t_{0},y_{0} by the corresponding relations, # the algorithm removes the variables y_{1},..,y_{n}). apl = p.parent() n = len(apl.gens())//2 pol = self.reduce_polynomial(p) relations = [] for i in range(n): relations.append(apl.gens()[n+i]-1) return pol.reduce(relations) def integral(self,p): ## Input: # - p: polynomial in variables t_0,..t_n,y_0,...,y_n ## Output: # - the integral of the polynomial after reducing the variables t_0, y_0 by the corresponding relations # apl = p.parent() n = len(apl.gens())//2-1 pol = self._convert_integrable_polynomial(p) result = 0 for k in pol.dict().keys(): # k is a monomial numerator = 1 denominator = n for e in k: # e is an exponents numerator *= factorial(e) denominator += e coef = pol.dict()[k] result += coef*numerator/factorial(denominator) return ((-1)**(n*(n-1)/2))*result def lift(self, value, n): apl = self.algebras[n] gens = apl.gens() return value*((-1)**(n*(n-1)/2))*factorial(n)*prod(gens[n+2:]) def homogeneous_part(self,pol,degree): parts = [pol.homogeneous_parts()[k] for k in pol.homogeneous_parts() if list(k)[0]==degree] return sum(parts) def propagate(self, face_list, n): """ - face_list: indexes of the elements that form the face of the simplex we want to propagate - EXAMPLE: p = a.propagate([0,3,4],5) t4*y0*y3 - t3*y0*y4 + t0*y3*y4 """ simplex_dim = len(face_list)-1 indexes = list(range(simplex_dim+1)) apl = self.algebras[n] y_gens = [apl.gens()[n+1+i] for i in face_list] t_gens = [apl.gens()[i] for i in face_list] result = apl(0) for i in reversed(indexes): t = t_gens[i] y_prod = copy.deepcopy(y_gens) del y_prod[i] y_prod = prod(y_prod) result += ((-1)**i)*t*y_prod#((-1)**(simplex_dim+i))*t*y_prod return result
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Apl(K)¶
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class AplK(UniqueRepresentation, Parent): def __init__(self,K, category=None): self.K = K self.n = K.dimension() self.apln = Apln(K.dimension()) Parent.__init__(self,category=category ) f = MorphismFromQQToAplK(self) self.register_coercion(f) def _repr_(self): return "Algebra AplK over %s" % self.K def _element_constructor_(self,values): return self.element_class(self,values) def dimension(self): return self.K.dimension() def simplicialComplex(self): return self.K def _n_cells_sorted(self,n): return self.K._n_cells_sorted(n) def lift(self, cochain): # first we find polynomials compatible with the integration value values = {} max_n = 0 for k in cochain.keys(): n = len(k)-1 if n > max_n: max_n = n apl = self.apln.algebras[n] if n>0: values[k] = self.apln.lift(cochain[k],n) else: values[k]= apl(cochain[k]) # second, we propagate the simplicial information simplicial_structure = {} for n in reversed(range(max_n+1)): keys = [k for k in values.keys() if len(k) == n+1] # keys of the n-simplices for k in keys: simplicial_structure[k] = values[k] + self._propagate(cochain, k) #print(simplicial_structure) return AplKElement(self,simplicial_structure) def _faces_of_length(self, simplex, length): return [tuple(e) for e in powerset(simplex) if len(e)==length] def _propagate(self, cochain, simplex): dim = len(simplex)-1 apl = self.apln.algebras[dim] propagation = apl(0) for n in reversed(range(dim+1)): faces = self._faces_of_length(simplex, n) for f in faces: if f in cochain.keys(): dim_face = len(f)-1 value = cochain[f] face_indexes = [simplex.index(i) for i in f] propagation += value*((-1)**(dim_face*(dim_face-1)/2))*factorial(dim_face)*self.apln.propagate(face_indexes, dim) return propagation
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Apl(K) Element¶
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class AplKElement(Element): def __init__(self, parent, values): Element.__init__(self,parent) self.apln = Apln(parent.dimension()) self.elements = self.reduce(values) self.check() def reduce(self, values): reduced_values = {} for k in values.keys(): n = len(k)-1 apl = self.apln.algebras[n] reduced_values[k] = self.apln.reduce_polynomial(apl(values[k])) return reduced_values def _repr_(self): return str(self.elements) def dimension(self): return self.parent().dimension() def nullElement(self): C = self.__class__ values = {} for k in self.elements.keys(): values[k]=0 return C(self.parent(),values) def _add_(self,other): C = self.__class__ if(self.elements.keys()!=other.elements.keys()): raise ValueError("error: keys do not match") newValues = {} for k in self.elements.keys(): newValues[k] = self.elements[k]+other.elements[k] return C(self.parent(),newValues) def _sub_(self,other): C = self.__class__ if(self.elements.keys()!=other.elements.keys()): raise ValueError("error: keys do not match") newValues = {} for k in self.elements.keys(): newValues[k] = self.elements[k]-other.elements[k] return C(self.parent(),newValues) def _mul_(self,other): C = self.__class__ if(self.elements.keys()!=other.elements.keys()): raise ValueError("error: keys do not match") newValues = {} for k in self.elements.keys(): newValues[k] = self.elements[k]*other.elements[k] return C(self.parent(),newValues) def _richcmp_(self, other, op): if(op==op_EQ): if(self.elements.keys()!=other.elements.keys()): return False for k in self.elements.keys(): if(self.elements[k]!=other.elements[k]): return False return True elif(op==op_NE): return not self._richcmp_(other,op_EQ) else: raise NotImplementedError("operation not implemented for this element class") def differential(self): C = self.__class__ newValues = {} for k in self.elements.keys(): newValues[k] = self.elements[k].differential() return C(self.parent(),newValues) def check(self): S = self.parent().simplicialComplex() dim = S.dimension() for n in range(1,dim+1): #print("Checking dimension " + str(n)) for simplex in S._n_cells_sorted(n): for f in range(n+1): sFace = tuple(S.face(simplex,f)) aFace = self.apln.face(f,n)(self.elements[simplex.tuple()]) aFace = self.apln.reduce_polynomial(aFace) #print("face: " + str(f),"simplex: " + str(simplex),"sFace: " + str(self.elements[sFace]), "aFace: " + str(aFace)) if(self.elements[sFace] != aFace): raise ValueError("the assignation is not valid") def homogeneous_part(self,degree): C = self.__class__ part = {} for k in self.elements.keys(): part[k] = self.apln.homogeneous_part(self.elements[k],degree) return C(self.parent(),part) def homogeneous_parts(self): parts = {} for n in range(self.dimension()+1): parts[n] = self.homogeneous_part(n) return parts def is_homogeneous(self): isHomogeneous = True for k in self.elements.keys(): isHomogeneous = isHomogeneous and self.elements[k].is_homogeneous() return isHomogeneous def is_closed(self): diff = self.differential() if(diff == self.nullElement()): return True else: return False def _integrable_component(self): C = self.__class__ component = {} for k in self.elements.keys(): degree = len(k)-1 component[k] = self.apln.homogeneous_part(self.elements[k],degree) #component[k] = self.homogeneous_component(dim).elements[k] return component def integral(self): component = self._integrable_component() cochain = {} #integration of each polynomial for k in component.keys(): n = len(k)-1 apl = self.apln.algebras[n] cochain[k] = self.apln.integral(apl(component[k])) return cochain ##################################################################### class MorphismFromQQToAplK(Morphism): def __init__(self, codomain): self.simplicialComplex= codomain.K Morphism.__init__(self, QQ, codomain) def _call_(self, x): S = self.simplicialComplex params1 = {} for d in range(S.dimension()+1): for c in S._n_cells_sorted(d): params1[tuple(c)]=x A = AplK(S) return AplKElement(A, params1)
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Test lift == integral¶
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def test(max_vertices, verbose=False): success = True data = [] for i in range(10): v = randint(2,max_vertices) f = randint(1,v) S = random_simplicial_complex(num_vertices=v,max_num_facets=f) A = AplK(S) cochain = random_cochain(S) #print(cochain) data = [S, cochain] if verbose: print("iteration: " + str(i), "num_vertices: " + str(v), "max_num_facets: " + str(f), "facets: " + str(S.facets())) p = A.lift(cochain) if verbose: print(p) if not p.integral()==cochain: success = False print("Error") break if success: print("success: " + str(success))
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test(max_vertices=6, verbose=True)
iteration: 0 num_vertices: 6 max_num_facets: 4 facets: {(1, 2, 5), (0, 3, 4, 5), (0, 1, 3), (1, 3, 4)}
{(0, 3, 4, 5): 288*y1*y2*y3 + 84*t3*y1*y2 - 84*t2*y1*y3 + 84*t1*y2*y3 - 174*y1*y2 - 112*y1*y3 + 124*y2*y3 + 43*t2*y1 + 225*t3*y1 - 43*t1*y2 + 3*t3*y2 - 225*t1*y3 - 3*t2*y3 - 63*y1 + 37*y2 + 81*y3 - 86*t1 - 69*t2 - 114*t3 + 97, (0, 1, 3): 200*y1*y2 + 13*t2*y1 - 13*t1*y2 + 18*y1 - 63*y2 - 178*t1 - 86*t2 + 97, (0, 3, 4): -174*y1*y2 + 43*t2*y1 - 43*t1*y2 - 63*y1 + 37*y2 - 86*t1 - 69*t2 + 97, (0, 3, 5): -112*y1*y2 + 225*t2*y1 - 225*t1*y2 - 63*y1 + 81*y2 - 86*t1 - 114*t2 + 97, (0, 4, 5): 124*y1*y2 + 3*t2*y1 - 3*t1*y2 + 37*y1 + 81*y2 - 69*t1 - 114*t2 + 97, (1, 2, 5): 160*y1*y2 + 83*t2*y1 - 83*t1*y2 + 20*y1 + 57*y2 + 64*t1 + 64*t2 - 81, (1, 3, 4): 14*y1*y2 + 9*t2*y1 - 9*t1*y2 - 94*y1 - 28*y2 + 92*t1 + 109*t2 - 81, (3, 4, 5): 146*y1*y2 - 179*t2*y1 + 179*t1*y2 + 57*y1 - 81*y2 + 17*t1 - 28*t2 + 11, (0, 1): 18*y1 - 178*t1 + 97, (0, 3): -63*y1 - 86*t1 + 97, (0, 4): 37*y1 - 69*t1 + 97, (0, 5): 81*y1 - 114*t1 + 97, (1, 2): 20*y1 + 64*t1 - 81, (1, 3): -94*y1 + 92*t1 - 81, (1, 4): -28*y1 + 109*t1 - 81, (1, 5): 57*y1 + 64*t1 - 81, (2, 5): -46*y1 - 17, (3, 4): 57*y1 + 17*t1 + 11, (3, 5): -81*y1 - 28*t1 + 11, (4, 5): 41*y1 - 45*t1 + 28, (0,): 97, (1,): -81, (2,): -17, (3,): 11, (4,): 28, (5,): -17}
iteration: 1 num_vertices: 5 max_num_facets: 2 facets: {(0, 2, 3, 4), (1, 3, 4)}
{(0, 2, 3, 4): 6*y1*y2*y3 + 506*t3*y1*y2 - 506*t2*y1*y3 + 506*t1*y2*y3 - 196*y1*y2 + 6*y1*y3 - 106*y2*y3 - 56*t2*y1 - 75*t3*y1 + 56*t1*y2 - 23*t3*y2 + 75*t1*y3 + 23*t2*y3 - 38*y1 - 86*y2 - 16*y3 - 50*t1 - 54*t2 - 3*t3 + 65, (0, 2, 3): -196*y1*y2 - 56*t2*y1 + 56*t1*y2 - 38*y1 - 86*y2 - 50*t1 - 54*t2 + 65, (0, 2, 4): 6*y1*y2 - 75*t2*y1 + 75*t1*y2 - 38*y1 - 16*y2 - 50*t1 - 3*t2 + 65, (0, 3, 4): -106*y1*y2 - 23*t2*y1 + 23*t1*y2 - 86*y1 - 16*y2 - 54*t1 - 3*t2 + 65, (1, 3, 4): -2*y1*y2 + 4*t2*y1 - 4*t1*y2 - 37*y1 + 60*y2 - 13*t1 + 38*t2 + 24, (2, 3, 4): 198*y1*y2 - 4*t2*y1 + 4*t1*y2 + 8*y1 + 97*y2 - 4*t1 + 47*t2 + 15, (0, 2): -38*y1 - 50*t1 + 65, (0, 3): -86*y1 - 54*t1 + 65, (0, 4): -16*y1 - 3*t1 + 65, (1, 3): -37*y1 - 13*t1 + 24, (1, 4): 60*y1 + 38*t1 + 24, (2, 3): 8*y1 - 4*t1 + 15, (2, 4): 97*y1 + 47*t1 + 15, (3, 4): 93*y1 + 51*t1 + 11, (0,): 65, (1,): 24, (2,): 15, (3,): 11, (4,): 62}
iteration: 2 num_vertices: 3 max_num_facets: 2 facets: {(0, 2), (1,)}
{(0, 2): 89*y1 + 2*t1 - 26, (0,): -26, (1,): -60, (2,): -24}
iteration: 3 num_vertices: 6 max_num_facets: 4 facets: {(1, 3), (2, 3), (0, 3, 5), (0, 1, 2, 4, 5)}
{(0, 1, 2, 4, 5): -1704*y1*y2*y3*y4 - 534*t4*y1*y2*y3 + 534*t3*y1*y2*y4 - 534*t2*y1*y3*y4 + 534*t1*y2*y3*y4 - 558*y1*y2*y3 - 444*y1*y2*y4 + 192*y1*y3*y4 - 132*y2*y3*y4 - 90*t3*y1*y2 + 238*t4*y1*y2 + 90*t2*y1*y3 + 236*t4*y1*y3 - 90*t1*y2*y3 + 198*t4*y2*y3 - 238*t2*y1*y4 - 236*t3*y1*y4 + 238*t1*y2*y4 - 198*t3*y2*y4 + 236*t1*y3*y4 + 198*t2*y3*y4 - 112*y1*y2 - 36*y1*y3 + 8*y2*y3 + 146*y1*y4 + 94*y2*y4 - 24*y3*y4 - 55*t2*y1 - 41*t3*y1 + 23*t4*y1 + 55*t1*y2 - 121*t3*y2 - 60*t4*y2 + 41*t1*y3 + 121*t2*y3 - 23*t4*y3 - 23*t1*y4 + 60*t2*y4 + 23*t3*y4 + 38*y1 + 73*y2 + 31*y3 + 85*y4 - 88*t1 - 115*t2 - 74*t3 - 63*t4 + 85, (0, 1, 2, 4): -558*y1*y2*y3 - 90*t3*y1*y2 + 90*t2*y1*y3 - 90*t1*y2*y3 - 112*y1*y2 - 36*y1*y3 + 8*y2*y3 - 55*t2*y1 - 41*t3*y1 + 55*t1*y2 - 121*t3*y2 + 41*t1*y3 + 121*t2*y3 + 38*y1 + 73*y2 + 31*y3 - 88*t1 - 115*t2 - 74*t3 + 85, (0, 1, 2, 5): -444*y1*y2*y3 + 238*t3*y1*y2 - 238*t2*y1*y3 + 238*t1*y2*y3 - 112*y1*y2 + 146*y1*y3 + 94*y2*y3 - 55*t2*y1 + 23*t3*y1 + 55*t1*y2 - 60*t3*y2 - 23*t1*y3 + 60*t2*y3 + 38*y1 + 73*y2 + 85*y3 - 88*t1 - 115*t2 - 63*t3 + 85, (0, 1, 4, 5): 192*y1*y2*y3 + 236*t3*y1*y2 - 236*t2*y1*y3 + 236*t1*y2*y3 - 36*y1*y2 + 146*y1*y3 - 24*y2*y3 - 41*t2*y1 + 23*t3*y1 + 41*t1*y2 - 23*t3*y2 - 23*t1*y3 + 23*t2*y3 + 38*y1 + 31*y2 + 85*y3 - 88*t1 - 74*t2 - 63*t3 + 85, (0, 2, 4, 5): -132*y1*y2*y3 + 198*t3*y1*y2 - 198*t2*y1*y3 + 198*t1*y2*y3 + 8*y1*y2 + 94*y1*y3 - 24*y2*y3 - 121*t2*y1 - 60*t3*y1 + 121*t1*y2 - 23*t3*y2 + 60*t1*y3 + 23*t2*y3 + 73*y1 + 31*y2 + 85*y3 - 115*t1 - 74*t2 - 63*t3 + 85, (1, 2, 4, 5): 324*y1*y2*y3 + 290*t3*y1*y2 - 290*t2*y1*y3 + 290*t1*y2*y3 - 158*y1*y2 + 74*y1*y3 + 30*y2*y3 - 135*t2*y1 - 138*t3*y1 + 135*t1*y2 - 87*t3*y2 + 138*t1*y3 + 87*t2*y3 + 90*y1 + 34*y2 + 24*y3 - 27*t1 + 14*t2 + 25*t3 - 3, (0, 1, 2): -112*y1*y2 - 55*t2*y1 + 55*t1*y2 + 38*y1 + 73*y2 - 88*t1 - 115*t2 + 85, (0, 1, 4): -36*y1*y2 - 41*t2*y1 + 41*t1*y2 + 38*y1 + 31*y2 - 88*t1 - 74*t2 + 85, (0, 1, 5): 146*y1*y2 + 23*t2*y1 - 23*t1*y2 + 38*y1 + 85*y2 - 88*t1 - 63*t2 + 85, (0, 2, 4): 8*y1*y2 - 121*t2*y1 + 121*t1*y2 + 73*y1 + 31*y2 - 115*t1 - 74*t2 + 85, (0, 2, 5): 94*y1*y2 - 60*t2*y1 + 60*t1*y2 + 73*y1 + 85*y2 - 115*t1 - 63*t2 + 85, (0, 3, 5): -146*y1*y2 + 167*t2*y1 - 167*t1*y2 - 46*y1 + 85*y2 - 13*t1 - 63*t2 + 85, (0, 4, 5): -24*y1*y2 - 23*t2*y1 + 23*t1*y2 + 31*y1 + 85*y2 - 74*t1 - 63*t2 + 85, (1, 2, 4): -158*y1*y2 - 135*t2*y1 + 135*t1*y2 + 90*y1 + 34*y2 - 27*t1 + 14*t2 - 3, (1, 2, 5): 74*y1*y2 - 138*t2*y1 + 138*t1*y2 + 90*y1 + 24*y2 - 27*t1 + 25*t2 - 3, (1, 4, 5): 30*y1*y2 - 87*t2*y1 + 87*t1*y2 + 34*y1 + 24*y2 + 14*t1 + 25*t2 - 3, (2, 4, 5): 88*y1*y2 - 84*t2*y1 + 84*t1*y2 + 79*y1 + 72*y2 + 41*t1 + 52*t2 - 30, (0, 1): 38*y1 - 88*t1 + 85, (0, 2): 73*y1 - 115*t1 + 85, (0, 3): -46*y1 - 13*t1 + 85, (0, 4): 31*y1 - 74*t1 + 85, (0, 5): 85*y1 - 63*t1 + 85, (1, 2): 90*y1 - 27*t1 - 3, (1, 3): -88*y1 + 75*t1 - 3, (1, 4): 34*y1 + 14*t1 - 3, (1, 5): 24*y1 + 25*t1 - 3, (2, 3): -68*y1 + 102*t1 - 30, (2, 4): 79*y1 + 41*t1 - 30, (2, 5): 72*y1 + 52*t1 - 30, (3, 5): -36*y1 - 50*t1 + 72, (4, 5): 77*y1 + 11*t1 + 11, (0,): 85, (1,): -3, (2,): -30, (3,): 72, (4,): 11, (5,): 22}
iteration: 4 num_vertices: 4 max_num_facets: 3 facets: {(0, 1, 2), (0, 3)}
{(0, 1, 2): -128*y1*y2 + 102*t2*y1 - 102*t1*y2 - 83*y1 - 17*y2 - 3*t1 + 23*t2 + 1, (0, 1): -83*y1 - 3*t1 + 1, (0, 2): -17*y1 + 23*t1 + 1, (0, 3): -98*y1 - 21*t1 + 1, (1, 2): -36*y1 + 26*t1 - 2, (0,): 1, (1,): -2, (2,): 24, (3,): -20}
iteration: 5 num_vertices: 3 max_num_facets: 2 facets: {(0, 2), (1,)}
{(0, 2): 91*y1 - 45*t1 - 46, (0,): -46, (1,): -75, (2,): -91}
iteration: 6 num_vertices: 4 max_num_facets: 1 facets: {(0, 1, 2)}
{(0, 1, 2): 106*y1*y2 + 100*t2*y1 - 100*t1*y2 - 83*y1 + 50*y2 - 60*t1 + 7*t2 - 20, (0, 1): -83*y1 - 60*t1 - 20, (0, 2): 50*y1 + 7*t1 - 20, (1, 2): 33*y1 + 67*t1 - 80, (0,): -20, (1,): -80, (2,): -13}
iteration: 7 num_vertices: 5 max_num_facets: 1 facets: {(0, 1, 4), (1, 3)}
{(0, 1, 4): -36*y1*y2 + 72*t2*y1 - 72*t1*y2 - 88*y1 + 25*y2 - 12*t1 + 77*t2 - 1, (0, 1): -88*y1 - 12*t1 - 1, (0, 4): 25*y1 + 77*t1 - 1, (1, 3): 52*y1 + 43*t1 - 13, (1, 4): 41*y1 + 89*t1 - 13, (0,): -1, (1,): -13, (3,): 30, (4,): 76}
iteration: 8 num_vertices: 5 max_num_facets: 2 facets: {(0, 3, 4), (0, 2, 4), (1, 2)}
{(0, 2, 4): 198*y1*y2 - 84*t2*y1 + 84*t1*y2 - 68*y1 - 75*y2 - 129*t1 - 5*t2 + 46, (0, 3, 4): 152*y1*y2 + 57*t2*y1 - 57*t1*y2 - 100*y1 - 75*y2 - 40*t1 - 5*t2 + 46, (0, 2): -68*y1 - 129*t1 + 46, (0, 3): -100*y1 - 40*t1 + 46, (0, 4): -75*y1 - 5*t1 + 46, (1, 2): -86*y1 - 66*t1 - 17, (2, 4): 77*y1 + 124*t1 - 83, (3, 4): -32*y1 + 35*t1 + 6, (0,): 46, (1,): -17, (2,): -83, (3,): 6, (4,): 41}
iteration: 9 num_vertices: 2 max_num_facets: 2 facets: {(1,), (0,)}
{(0,): -75, (1,): -70}
success: True
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Minimal model (sin terminar)¶
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class MinimalModel(): def __init__(self, S): self.K = S self.Phi = {} gens = self.first_non_trivial_cohomology_group() [self.cdga, self.cohomology_dict] = self.createModel(gens) def first_non_trivial_cohomology_group(self, max_iter=20): # busco el primer grupo de cohomología no trivial i=1 gens = self.cohomology_gens(i) while(len(gens)==0 and i<max_iter): i+=1 gens = self.cohomology_gens(i) return gens def lift_cohomology_generator(self, g): cycle = g.to_cycle() cochain = {} for n in range(self.K.dimension()+1): for simplex in self.K._n_cells_sorted(n): cochain[simplex.tuple()] = 0 for i in cycle: #print(type(i[0].tuple())) cochain[i[0].tuple()] = i[1] return cochain def cohomology_gens(self, n): gens = {} basis = self.K.cohomology_ring().basis() for k in basis.keys(): if k[0] == n: gens[k] = basis[k] return gens def createModel(self, gens): n = len(gens) generators = [] degrees = [] cohomology_dict = {} for k in gens.keys(): degree = k[0] index = k[1] genName = 'x'+str(index)+'d'+str(degree) generators.append(genName) degrees.append(degree) cohomology_dict[gens[k]] = genName gca = GradedCommutativeAlgebra(QQ, names=generators, degrees=degrees) dfs = {gca.gens_dict()[generators[i]]:0 for i in range(n)} return [gca.cdg_algebra(dfs), cohomology_dict]