Added tensor_product and hom_module to algebra/module.h.
This commit is contained in:
Cotton Seed
parent
b337bd36ac
commit
9bed7282ac
@ -145,6 +145,17 @@ class linear_combination
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unsigned card () const { return v.card (); }
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linear_combination<R> tensor (const linear_combination<R> &lc) const
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{
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linear_combination<R> r (m->tensor (lc.m));
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for (linear_combination_const_iter<R> i = *this; i; i ++)
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for (linear_combination_const_iter<R> j = lc; j; j ++)
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{
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r.muladd (i.val () * j.val (), m->tensor_generators (i.key (), lc.m, j.key ()));
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}
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return r;
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}
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#ifndef NDEBUG
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void check () const
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{
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@ -292,7 +303,10 @@ linear_combination<R>::show_self () const
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else
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printf (" + ");
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show (i.val ());
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printf ("*%d", i.key ());
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// printf ("*%d", i.key ());
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printf ("*");
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m->show_generator (i.key ());
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}
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}
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@ -445,7 +459,8 @@ linear_combination<Z2>::show_self () const
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first = 0;
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else
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printf ("+");
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printf ("%d", i.val ());
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// printf ("%d", i.val ());
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m->show_generator (i.val ());
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}
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}
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366
algebra/module.h
366
algebra/module.h
@ -1,15 +1,38 @@
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template<class R> class mod_map;
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template<class R> class mod_span;
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template<class R> class free_submodule;
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template<class R> class quotient_module;
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template<class R> class tensor_product;
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template<class R> class hom_module;
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/* `module' is a bigraded module over a ring R. */
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template<class R>
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class module : public refcounted
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{
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private:
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unsigned id;
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static unsigned id_counter;
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#if 0
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static map<basedvector<unsigned, 1>,
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ptr<const direct_sum<R> > direct_sum_idx;
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#endif
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static map<basedvector<unsigned, 1>,
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ptr<const tensor_product<R> > > tensor_product_idx;
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static map<pair<unsigned, unsigned>,
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ptr<const hom_module<R> > > hom_module_idx;
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public:
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module () { }
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module ()
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{
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id = id_counter;
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id_counter ++;
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}
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module (const module &); // doesn't exist
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virtual ~module () { }
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@ -28,6 +51,8 @@ class module : public refcounted
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// r < i <= n
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virtual R generator_ann (unsigned i) const = 0;
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bool is_free () const { return dim () == free_rank (); }
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bool is_zero (R c, unsigned i) const
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{
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if (i <= free_rank ())
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@ -53,10 +78,278 @@ class module : public refcounted
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multivariate_laurentpoly<Z> free_poincare_polynomial () const;
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multivariate_laurentpoly<Z> free_delta_poincare_polynomial () const;
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ptr<const tensor_product<R> > tensor (ptr<const module<R> > m) const
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{
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basedvector<ptr<const module<R> >, 1> factors (2);
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factors[1] = this;
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factors[2] = m;
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return tensor (factors);
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}
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ptr<const hom_module<R> > hom (ptr<const module<R> > to) const;
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pair<unsigned, unsigned>
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generator_factors (ptr<const module<R> > m, unsigned g) const
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{
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pair<unsigned, unsigned> p ((g - 1) % dim () + 1,
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(g - 1) / dim () + 1);
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assert (g == tensor_generators (p.first, m, p.second));
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return p;
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}
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static ptr<const tensor_product<R> > tensor (basedvector<ptr<const module<R> >, 1> compound_factors);
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unsigned tensor_generators (unsigned i, ptr<const module<R> > m, unsigned j) const
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{
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return (i - 1) + (j - 1) * dim () + 1;
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}
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virtual void append_tensor_factors (basedvector<ptr<const module<R> >, 1> &factors) const
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{
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factors.append (this);
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}
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void show_self () const;
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void display_self () const;
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};
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template<class R> unsigned module<R>::id_counter = 1;
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template<class R> map<basedvector<unsigned, 1>,
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ptr<const tensor_product<R> > > module<R>::tensor_product_idx;
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template<class R> map<pair<unsigned, unsigned>,
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ptr<const hom_module<R> > > module<R>::hom_module_idx;
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#if 0
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template<class R>
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class direct_sum : public module<R>
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{
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unsigned n;
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basedvector<ptr<const module<R> >, 1> summands;
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public:
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direct_sum (basedvector<ptr<const module<R> >, 1> summands_)
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: n(0),
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summands(summands_)
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{
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#ifndef NDEBUG
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for (unsigned i = 1; i <= terms.size (); i ++)
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assert (terms[i]->is_free ());
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#endif
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for (unsigned i = 1; i <= terms.size (); i ++)
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n += terms[i]->dim ();
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}
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unsigned dim () const { return n; }
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unsigned free_rank () const { return n; }
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// ???
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grading generator_grading (unsigned i) const { abort (); }
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void show_generator (unsigned i) const { abort (); }
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};
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#endif
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template<class R>
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class tensor_product : public module<R>
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{
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unsigned n;
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basedvector<ptr<const module<R> >, 1> factors;
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basedvector<unsigned, 1> generator_factors (unsigned g) const;
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public:
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tensor_product (basedvector<ptr<const module<R> >, 1> factors_)
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: n(1),
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factors(factors_)
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{
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#ifndef NDEBUG
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for (unsigned i = 1; i <= factors.size (); i ++)
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assert (factors[i]->is_free ());
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#endif
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for (unsigned i = 1; i <= factors.size (); i ++)
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n *= factors[i]->dim ();
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}
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~tensor_product () { }
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unsigned dim () const { return n; }
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unsigned free_rank () const { return n; }
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grading generator_grading (unsigned i) const;
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void show_generator (unsigned i) const;
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R generator_ann (unsigned i) const { return R (0); }
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unsigned tensor_generators (basedvector<unsigned, 1> gs) const;
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void append_tensor_factors (basedvector<ptr<const module<R> >, 1> &pfactors) const
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{
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for (unsigned i = 1; i <= factors.size (); i ++)
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pfactors.append (factors[i]);
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}
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};
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template<class R> ptr<const tensor_product<R> >
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module<R>::tensor (basedvector<ptr<const module<R> >, 1> compound_factors)
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{
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basedvector<ptr<const module<R> >, 1> factors;
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for (unsigned i = 1; i <= compound_factors.size (); i ++)
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compound_factors[i]->append_tensor_factors (factors);
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basedvector<unsigned, 1> factor_ids (factors.size ());
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for (unsigned i = 1; i <= factors.size (); i ++)
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factor_ids[i] = factors[i]->id;
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pair<ptr<const tensor_product<R> > &, bool> p = tensor_product_idx.find (factor_ids);
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if (!p.second)
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p.first = new tensor_product<R> (factors);
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return p.first;
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}
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template<class R> grading
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tensor_product<R>::generator_grading (unsigned i) const
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{
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basedvector<unsigned, 1> gs = generator_factors (i);
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assert (gs.size () == factors.size ());
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grading gr;
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for (unsigned i = 1; i <= factors.size (); i ++)
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gr += factors[i]->generator_grading (gs[i]);
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return gr;
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}
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template<class R> void
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tensor_product<R>::show_generator (unsigned i) const
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{
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basedvector<unsigned, 1> gs = generator_factors (i);
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assert (gs.size () == factors.size ());
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printf ("o(");
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for (unsigned i = 1; i <= factors.size (); i ++)
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{
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if (i > 1)
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printf (",");
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factors[i]->show_generator (gs[i]);
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}
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printf (")");
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}
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template<class R> unsigned
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tensor_product<R>::tensor_generators (basedvector<unsigned, 1> gs) const
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{
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assert (gs.size () == factors.size ());
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unsigned r = gs[gs.size ()] - 1;
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for (unsigned i = gs.size () - 1; i >= 1; i --)
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{
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r *= factors[i]->dim ();
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r += gs[i] - 1;
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}
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r ++;
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return r;
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}
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template<class R> basedvector<unsigned, 1>
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tensor_product<R>::generator_factors (unsigned g) const
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{
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basedvector<unsigned, 1> r (factors.size ());
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unsigned g0 = g;
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g --;
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for (unsigned i = 1; i <= factors.size (); i ++)
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{
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r[i] = (g % factors[i]->dim ()) + 1;
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g /= factors[i]->dim ();
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}
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assert (g == 0);
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assert (tensor_generators (r) == g0);
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return r;
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}
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template<class R>
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class hom_module : public module<R>
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{
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public:
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unsigned n;
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ptr<const module<R> > from;
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ptr<const module<R> > to;
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public:
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hom_module (ptr<const module<R> > from_,
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ptr<const module<R> > to_)
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: from(from_), to(to_)
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{
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assert (from->is_free ()
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&& to->is_free ());
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n = from->dim () * to->dim ();
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}
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~hom_module () { }
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// e_ij -> ij
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pair<unsigned, unsigned> generator_indices (unsigned g) const
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{
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unsigned d = from->dim ();
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unsigned g0 = g;
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g --;
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pair<unsigned, unsigned> p ((g % d) + 1,
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(g / d) + 1);
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assert (generator (p.first, p.second) == g0);
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return p;
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}
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// ij -> e_ij
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unsigned generator (unsigned i, unsigned j) const
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{
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return (i - 1) + (j - 1) * from->dim () + 1;
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}
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unsigned dim () const { return n; }
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unsigned free_rank () const { return n; }
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grading generator_grading (unsigned i) const;
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void show_generator (unsigned i) const;
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R generator_ann (unsigned i) const { return R (0); }
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linear_combination<R> map_as_element (const mod_map<R> &m) const;
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};
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template<class R> ptr<const hom_module<R> >
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module<R>::hom (ptr<const module<R> > to) const
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{
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pair<ptr<const hom_module<R> > &, bool> p = hom_module_idx.find (pair<unsigned, unsigned>
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(id, to->id));
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if (!p.second)
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p.first = new hom_module<R> (this, to);
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return p.first;
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}
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template<class R> grading
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hom_module<R>::generator_grading (unsigned i) const
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{
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pair<unsigned, unsigned> p = generator_indices (i);
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return (to->generator_grading (p.second)
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- from->generator_grading (p.first));
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}
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template<class R> void
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hom_module<R>::show_generator (unsigned i) const
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{
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pair<unsigned, unsigned> p = generator_indices (i);
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printf ("(");
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from->show_generator (p.first);
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printf (" -> ");
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to->show_generator (p.second);
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printf (")");
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}
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template<class R, class G>
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class base_module : public module<R>
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{
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@ -287,10 +580,10 @@ template<class R>
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class composition_impl : public map_impl<R>
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{
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// f(g(x))
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ptr<map_impl<R> > f, g;
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ptr<const map_impl<R> > f, g;
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public:
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composition_impl (ptr<map_impl<R> > f_, ptr<map_impl<R> > g_)
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composition_impl (ptr<const map_impl<R> > f_, ptr<const map_impl<R> > g_)
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: map_impl<R>(g_->from, f_->to),
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f(f_),
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g(g_)
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@ -304,6 +597,30 @@ class composition_impl : public map_impl<R>
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}
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};
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template<class R>
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class tensor_impl : public map_impl<R>
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{
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// f\otimes g
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ptr<const map_impl<R> > f, g;
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public:
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tensor_impl (ptr<const map_impl<R> > f_, ptr<const map_impl<R> > g_)
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: map_impl<R>(f_->from->tensor (g_->from),
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f_->to->tensor (g_->from)),
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f(f_),
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g(g_)
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{
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}
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linear_combination<R> column (unsigned i) const
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{
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pair<unsigned, unsigned> p = f->from->generator_factors (g->from, i);
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// ??
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return f->column (p.first).tensor (g->column (p.second));
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}
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};
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template<class R>
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class map_builder
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{
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@ -349,9 +666,12 @@ map_builder<R>::init ()
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template<class R>
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class mod_map
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{
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ptr<map_impl<R> > impl;
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// ???
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enum impl_ctor { IMPL };
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mod_map (ptr<map_impl<R> > impl_) : impl(impl_) { }
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ptr<const map_impl<R> > impl;
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mod_map (impl_ctor, ptr<const map_impl<R> > impl_) : impl(impl_) { }
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public:
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mod_map () { }
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@ -389,13 +709,16 @@ class mod_map
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mod_map &operator = (const mod_map &m) { impl = m.impl; return *this; }
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ptr<const module<R> > domain () const { return impl->from; }
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ptr<const module<R> > codomain () const { return impl->to; }
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bool operator == (const mod_map &m) const
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{
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assert (impl->from == m.impl->from);
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assert (impl->to == m.impl->to);
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for (unsigned i = 1; i <= impl->from->dim (); i ++)
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{
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if (impl->columns (i) != m.impl->columns (i))
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if (impl->column (i) != m.impl->column (i))
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return 0;
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}
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return 1;
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@ -420,7 +743,17 @@ class mod_map
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linear_combination<R> operator [] (unsigned i) const { return impl->column (i); }
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linear_combination<R> map (const linear_combination<R> &lc) const { return impl->map (lc); }
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mod_map compose (const mod_map &m) const { return new composition_impl<R> (impl, m.impl); }
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mod_map compose (const mod_map &m) const
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{
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return mod_map (IMPL,
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new composition_impl<R> (impl, m.impl));
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}
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mod_map tensor (const mod_map &m) const
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{
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return mod_map (IMPL,
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new tensor_impl<R> (impl, m.impl));
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}
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// ?? add and other map operations should not be explicit
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mod_map operator + (const mod_map &m) const;
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@ -457,6 +790,21 @@ class mod_map
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void display_self () const;
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};
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template<class R> linear_combination<R>
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hom_module<R>::map_as_element (const mod_map<R> &m) const
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{
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assert (from == m.domain ()
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&& to == m.codomain ());
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linear_combination<R> r (this);
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for (unsigned i = 1; i <= from->dim (); i ++)
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{
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for (linear_combination_const_iter<R> j = m.column (i); j; j ++)
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r.muladd (j.val (), generator (i, j.key ()));
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}
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return r;
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}
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template<class R>
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class mod_span
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{
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@ -1319,7 +1667,9 @@ mod_map<R>::display_self () const
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show_self (); newline ();
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for (unsigned i = 1; i <= impl->from->dim (); i ++)
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{
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printf (" %d: ", i);
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printf (" %d ", i);
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impl->from->show_generator (i);
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printf (": ");
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show (column (i));
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newline ();
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}
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50
main.cpp
50
main.cpp
@ -165,6 +165,55 @@ check (const dt_code &dt)
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int
|
||||
main ()
|
||||
{
|
||||
ptr<const explicit_module<Q> > A
|
||||
= new explicit_module<Q> (2, basedvector<Q, 1> (), basedvector<grading, 1> (2));
|
||||
ptr<const explicit_module<Q> > B
|
||||
= new explicit_module<Q> (3, basedvector<Q, 1> (), basedvector<grading, 1> (3));
|
||||
ptr<const explicit_module<Q> > C
|
||||
= new explicit_module<Q> (3, basedvector<Q, 1> (), basedvector<grading, 1> (3));
|
||||
ptr<const explicit_module<Q> > D
|
||||
= new explicit_module<Q> (2, basedvector<Q, 1> (), basedvector<grading, 1> (2));
|
||||
ptr<const explicit_module<Q> > E
|
||||
= new explicit_module<Q> (2, basedvector<Q, 1> (), basedvector<grading, 1> (3));
|
||||
ptr<const explicit_module<Q> > F
|
||||
= new explicit_module<Q> (2, basedvector<Q, 1> (), basedvector<grading, 1> (2));
|
||||
|
||||
map_builder<Q> fb (A, B);
|
||||
fb[1].muladd (2, 1);
|
||||
fb[1].muladd (3, 2);
|
||||
fb[2].muladd (-5, 2);
|
||||
fb[2].muladd (4, 3);
|
||||
mod_map<Q> f (fb);
|
||||
display ("f:\n", f);
|
||||
|
||||
map_builder<Q> gb (C, D);
|
||||
gb[1].muladd (1, 1);
|
||||
gb[2].muladd (3, 1);
|
||||
gb[2].muladd (-2, 2);
|
||||
gb[3].muladd (-6, 2);
|
||||
mod_map<Q> g (gb);
|
||||
display ("g:\n", g);
|
||||
|
||||
map_builder<Q> hb (E, F);
|
||||
hb[1].muladd (3, 2);
|
||||
hb[2].muladd (-3, 1);
|
||||
mod_map<Q> h (hb);
|
||||
display ("h:\n", h);
|
||||
|
||||
mod_map<Q> fg = f.tensor (g);
|
||||
display ("fg:\n", fg);
|
||||
|
||||
ptr<const module<Q> > AB_C = (A->tensor (B))->tensor (C),
|
||||
A_BC = A->tensor (B->tensor (C));
|
||||
assert (AB_C == A_BC);
|
||||
|
||||
assert ((f.tensor (g)).tensor (h) == f.tensor (g.tensor (h)));
|
||||
|
||||
ptr<const hom_module<Q> > homAB = A->hom (B);
|
||||
linear_combination<Q> x = homAB->map_as_element (f);
|
||||
display ("x:\n", x);
|
||||
|
||||
#if 0
|
||||
for (unsigned i = 1; i <= 14; i ++)
|
||||
{
|
||||
for (unsigned j = 1; j <= mt_links (i, 0); j ++)
|
||||
@ -173,6 +222,7 @@ main ()
|
||||
for (unsigned j = 1; j <= mt_links (i, 1); j ++)
|
||||
check (mt_link (i, 1, j));
|
||||
}
|
||||
#endif
|
||||
|
||||
#if 0
|
||||
knot_diagram kd (rolfsen_knot (8, 19));
|
||||
|
||||
Loading…
Reference in New Issue
Block a user