184 lines
5.2 KiB
C++
184 lines
5.2 KiB
C++
// Copyright 2004, Ondra Kamenik
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// Symmetry.
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/* Symmetry is an abstraction for a term of the form $y^3u^2$. It manages
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only indices, not the variable names. So if one uses this
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abstraction, he must keep in mind that $y$ is the first, and $u$ is
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the second.
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In fact, the symmetry is a special case of equivalence, but its
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implementation is much simpler. We do not need an abstraction for the
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term $yyuyu$ but due to Green theorem we can have term $y^3u^2$. That
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is why the equivalence is too general for our purposes.
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One of a main purposes of the tensor library is to calculate something like:
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$$\left[B_{y^2u^3}\right]_{\alpha_1\alpha_2\beta_1\beta_2\beta_3}
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=\left[g_{y^l}\right]_{\gamma_1\ldots\gamma_l}
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\left(\sum_{c\in M_{l,5}}
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\prod_{m=1}^l\left[g_{c_m}\right]^{\gamma_m}_{c_m(\alpha,\beta)}\right)$$
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If, for instance, $l=3$, and $c=\{\{0,4\},\{1,2\},\{3\}\}$, then we
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have to calculate
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$$\left[g_{y^3}\right]_{\gamma_1\gamma_2\gamma_3}
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\left[g_{yu}\right]^{\gamma_1}_{\alpha_1\beta_3}
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\left[g_{yu}\right]^{\gamma_2}_{\alpha_2\beta_1}
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\left[g_u\right]^{\gamma_3}_{\beta_2}
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$$
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We must be able to calculate a symmetry induced by symmetry $y^2u^3$
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and by an equivalence class from equivalence $c$. For equivalence
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class $\{0,4\}$ the induced symmetry is $yu$, since we pick first and
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fifth variable from $y^2u^3$. For a given outer symmetry, the class
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|InducedSymmetries| does this for all classes of a given equivalence.
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We need also to cycle through all possible symmetries yielding the
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given dimension. For this purpose we define classes |SymmetrySet| and
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|symiterator|.
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The symmetry is implemented as |IntSequence|, in fact, it inherits
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from it. */
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#ifndef SYMMETRY_H
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#define SYMMETRY_H
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#include "equivalence.hh"
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#include "int_sequence.hh"
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#include <list>
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#include <vector>
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#include <initializer_list>
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#include <utility>
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#include <memory>
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/* Clear. The method |isFull| returns true if and only if the symmetry
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allows for any permutation of indices.
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WARNING: Symmetry(n) and Symmetry{n} are not the same. The former
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initializes a symmetry of n elements, while the latter is a full symmetry of
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order n. This is similar to the behaviour of std::vector. */
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class Symmetry : public IntSequence
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{
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public:
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// Constructor allocating a given length of (zero-initialized) data
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explicit Symmetry(int len)
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: IntSequence(len, 0)
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{
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}
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/* Constructor using an initializer list, that gives the contents of the
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Symmetry. Used for symmetries of the form $y^n$, $y^n u^m$, $y^nu^m\sigma^k$ */
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Symmetry(std::initializer_list<int> init)
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: IntSequence(std::move(init))
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{
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}
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// Constructor of implied symmetry for a symmetry and an equivalence class
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Symmetry(const Symmetry &s, const OrdSequence &cl);
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/* Subsymmetry, which takes the given length of symmetry from the end (shares
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data pointer) */
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Symmetry(Symmetry &s, int len)
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: IntSequence(s, s.size()-len, s.size())
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{
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}
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int
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num() const
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{
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return size();
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}
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int
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dimen() const
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{
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return sum();
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}
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int findClass(int i) const;
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bool isFull() const;
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};
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/* This is an iterator that iterates over all symmetries of given length and
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dimension (see the SymmetrySet class for details).
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The beginning iterator is (0, …, 0, dim).
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Increasing it gives (0, … , 1, dim-1)
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The just-before-end iterator is (dim, 0, …, 0)
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The past-the-end iterator is (dim+1, 0, …, 0)
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The constructor creates the iterator which starts from the given symmetry
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symmetry (beginning). */
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class symiterator
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{
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const int dim;
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Symmetry run;
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public:
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symiterator(int dim_arg, Symmetry run_arg);
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~symiterator() = default;
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symiterator &operator++();
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const Symmetry &
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operator*() const
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{
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return run;
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}
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bool
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operator=(const symiterator &it)
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{
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return dim == it.dim && run == it.run;
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}
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bool
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operator!=(const symiterator &it)
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{
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return !operator=(it);
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}
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};
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/* The class |SymmetrySet| defines a set of symmetries of the given length
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having given dimension (i.e. it represents all the lists of integers of
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length "len" and of sum equal to "dim"). It does not store all the
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symmetries, it is just a convenience class for iteration.
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The typical usage of the abstractions for |SymmetrySet| and
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|symiterator| is as follows:
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for (auto &si : SymmetrySet(6, 4))
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It goes through all symmetries of lenght 4 having dimension 6. One can use
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"si" as the symmetry in the body. */
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class SymmetrySet
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{
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public:
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const int len;
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const int dim;
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SymmetrySet(int dim_arg, int len_arg)
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: len(len_arg), dim(dim_arg)
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{
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}
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symiterator
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begin() const
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{
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Symmetry run(len);
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run[len-1] = dim;
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return { dim, run };
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}
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symiterator
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end() const
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{
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Symmetry run(len);
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run[0] = dim+1;
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return { dim, run };
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}
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};
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/* This simple abstraction just constructs a vector of induced
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symmetries from the given equivalence and outer symmetry. A
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permutation might optionally permute the classes of the equivalence. */
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class InducedSymmetries : public std::vector<Symmetry>
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{
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public:
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InducedSymmetries(const Equivalence &e, const Symmetry &s);
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InducedSymmetries(const Equivalence &e, const Permutation &p, const Symmetry &s);
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void print() const;
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};
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#endif
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