Evgenij-Gr/equivariance-check.ipynb

## equivariance-check.ipynb
{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {
    "collapsed": false
   },
   "source": [
    "Let's consider the following vector field in $\\mathbb{R}^3$:\n",
    "\n",
    "$$\\left \\lbrace \\begin{array}{ccc}\n",
    "\\dot{x} & = & y, \\\\\n",
    "\\dot{y} & = & -x, \\\\\n",
    "\\dot{z} & = & -z\n",
    "\\end{array} \\right .$$\n",
    "\n",
    "It's quite obvious that $(x, y, z) \\mapsto (x \\cos \\varphi - y \\sin \\varphi, x \\sin \\varphi + y \\cos \\varphi, z)$ is a symmetry of this system for any real $\\varphi$ in a sense that it maps a solution into solution. Recall that $g \\in O(3)$ is a symmetry of a vector field $\\dot{\\mathbf{x}} = v (\\mathbf{x})$ iff\n",
    "\n",
    "$$g \\cdot v(\\mathbf{x}) \\equiv v(g \\cdot \\mathbf{x}).$$\n",
    "\n",
    "This is an \"equivarince condition\". That's exactly what we can check using CAS like SymPy. Let's illustrate this:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 51,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "import sympy as sp\n",
    "\n",
    "sp.init_printing()\n",
    "\n",
    "x, y, z, fi = sp.symbols('x y z varphi')\n",
    "\n",
    "R_fi = sp.Matrix([[sp.cos(fi), -sp.sin(fi), 0],[sp.sin(fi), sp.cos(fi), 0],[0,0,1]])\n",
    "\n",
    "u, v, w = sp.symbols('u, v, w')\n",
    "vector_field = sp.Matrix([[v], [-u], [-w]])\n",
    "X = sp.Matrix([[x], [y], [z]])\n",
    "gX = R_fi * X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The method is very crude and ineffective, but might work for small groups: just check if equivariance condition holds for all elements of group $G$:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 52,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRInN3SJm77tsdo1uFAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAMtJREFUOBHtVdEW\ngyAIxSTXMrPG///rBDaTM7fnnVO8ZPeKIF4FHIkNUC0pAuDIY7GxMpD5fyCm3IEeo2yo4CP69zxL\nLWXRkFb1NFTeGIxLh9olyZGCcMaLhJpJc22pQJ6nz4QfXitFxpx+oPX6QQWd3lsQNNbYSQPSzrFy\nL3mULfveliFxobZXEdsMCxzLWfTLy6GqWa8K8+D01GkkepvuevJWoo+pucuXRM3z9Sc35TQSrT3F\nStS8UZdERaJfW3bgBo04H9qVlo0IT25QGoNPPQEEAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 52,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "R_fi * vector_field.subs([(u, x), (v, y), (w, z)]) - vector_field.subs([(u, gX[0]), (v, gX[1]), (w, gX[2])])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let's also consider this system:\n",
    "\n",
    "$$\\left \\lbrace \\begin{array}{ccc}\n",
    "\\dot{x} & = & Ax -6Bx^2 y^2 + By^4 + Bx^4, \\\\\n",
    "\\dot{y} & = & Ay + 4Bx^3y - 4 Bxy^3, \\\\\n",
    "\\dot{z} & = & -z\n",
    "\\end{array} \\right .$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It's not obvious from the beginning whether it has some symmetry or not, but I've constructed this example having discrete $\\mathbb{Z}_3$-symmetry in mind, i.e. \n",
    "$(x, y, z) \\mapsto (x \\cos \\varphi - y \\sin \\varphi, x \\sin \\varphi + y \\cos \\varphi, z)$ is a symmetry of this system when $\\varphi = \\frac{2 \\pi}{3}$. Let's check it:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 53,
   "metadata": {
    "collapsed": true
   },
   "outputs": [],
   "source": [
    "R_fi = R_fi.subs([(fi, 2*sp.pi/3)])\n",
    "A, B = sp.symbols('A B')\n",
    "vector_field = sp.Matrix([[A*u-6*B*u**2*v**2+B*u**4+B*v**4], [A*v + 4*B*u**3*v-4*B*u*v**3], [-w]])\n",
    "X = sp.Matrix([[x], [y], [z]])\n",
    "gX = R_fi * X"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "What happens when we check equivariance condition?"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 54,
   "metadata": {
    "collapsed": false
   },
   "outputs": [],
   "source": [
    "res = R_fi * vector_field.subs([(u, x), (v, y), (w, z)]) - vector_field.subs([(u, gX[0]), (v, gX[1]), (w, gX[2])])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Well, without simplification it looks a bit ugly"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false,
    "scrolled": true
   },
   "outputs": [
    {
     "data": {
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dt+xZ1sQ3GputNOdAJyf+AmPOKUM+psRfeiEX8Hv4gYC7yv3biZ5PNDV0CiM+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\nA2/rJrv/B8zB3/aVSsNqAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}- \\frac{A x}{2} - A \\left(- \\frac{x}{2} - \\frac{\\sqrt{3} y}{2}\\right) - \\frac{B x^{4}}{2} + 3 B x^{2} y^{2} - \\frac{B y^{4}}{2} - B \\left(- \\frac{x}{2} - \\frac{\\sqrt{3} y}{2}\\right)^{4} + 6 B \\left(- \\frac{x}{2} - \\frac{\\sqrt{3} y}{2}\\right)^{2} \\left(\\frac{\\sqrt{3} x}{2} - \\frac{y}{2}\\right)^{2} - B \\left(\\frac{\\sqrt{3} x}{2} - \\frac{y}{2}\\right)^{4} - \\frac{\\sqrt{3}}{2} \\left(A y + 4 B x^{3} y - 4 B x y^{3}\\right)\\\\- \\frac{A y}{2} - A \\left(\\frac{\\sqrt{3} x}{2} - \\frac{y}{2}\\right) - 2 B x^{3} y + 2 B x y^{3} - 4 B \\left(- \\frac{x}{2} - \\frac{\\sqrt{3} y}{2}\\right)^{3} \\left(\\frac{\\sqrt{3} x}{2} - \\frac{y}{2}\\right) + 4 B \\left(- \\frac{x}{2} - \\frac{\\sqrt{3} y}{2}\\right) \\left(\\frac{\\sqrt{3} x}{2} - \\frac{y}{2}\\right)^{3} + \\frac{\\sqrt{3}}{2} \\left(A x + B x^{4} - 6 B x^{2} y^{2} + B y^{4}\\right)\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡                            4                  4                 4           \n",
       "⎢  A⋅x     ⎛  x   √3⋅y⎞   B⋅x         2  2   B⋅y      ⎛  x   √3⋅y⎞        ⎛  x\n",
       "⎢- ─── - A⋅⎜- ─ - ────⎟ - ──── + 3⋅B⋅x ⋅y  - ──── - B⋅⎜- ─ - ────⎟  + 6⋅B⋅⎜- ─\n",
       "⎢   2      ⎝  2    2  ⎠    2                  2       ⎝  2    2  ⎠        ⎝  2\n",
       "⎢                                                                             \n",
       "⎢                                                                3            \n",
       "⎢     A⋅y     ⎛√3⋅x   y⎞        3            3       ⎛  x   √3⋅y⎞  ⎛√3⋅x   y⎞ \n",
       "⎢   - ─── - A⋅⎜──── - ─⎟ - 2⋅B⋅x ⋅y + 2⋅B⋅x⋅y  - 4⋅B⋅⎜- ─ - ────⎟ ⋅⎜──── - ─⎟ \n",
       "⎢      2      ⎝ 2     2⎠                             ⎝  2    2  ⎠  ⎝ 2     2⎠ \n",
       "⎢                                                                             \n",
       "⎣                                                                         0   \n",
       "\n",
       "        2           2               4      ⎛           3            3⎞⎤\n",
       "   √3⋅y⎞  ⎛√3⋅x   y⎞      ⎛√3⋅x   y⎞    √3⋅⎝A⋅y + 4⋅B⋅x ⋅y - 4⋅B⋅x⋅y ⎠⎥\n",
       " - ────⎟ ⋅⎜──── - ─⎟  - B⋅⎜──── - ─⎟  - ──────────────────────────────⎥\n",
       "    2  ⎠  ⎝ 2     2⎠      ⎝ 2     2⎠                  2               ⎥\n",
       "                                                                      ⎥\n",
       "                             3      ⎛         4        2  2      4⎞   ⎥\n",
       "      ⎛  x   √3⋅y⎞ ⎛√3⋅x   y⎞    √3⋅⎝A⋅x + B⋅x  - 6⋅B⋅x ⋅y  + B⋅y ⎠   ⎥\n",
       "+ 4⋅B⋅⎜- ─ - ────⎟⋅⎜──── - ─⎟  + ──────────────────────────────────   ⎥\n",
       "      ⎝  2    2  ⎠ ⎝ 2     2⎠                    2                    ⎥\n",
       "                                                                      ⎥\n",
       "                                                                      ⎦"
      ]
     },
     "execution_count": 55,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "res"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "but performing simple brackets expansion and basic simplification we get the following:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 56,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAABoAAABLCAMAAABZRmeuAAAAPFBMVEX///8AAAAAAAAAAAAAAAAA\nAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAo1xBWAAAAE3RSTlMA\nMquZdlQQQOkwRInN3SJm77tsdo1uFAAAAAlwSFlzAAAOxAAADsQBlSsOGwAAAMtJREFUOBHtVdEW\ngyAIxSTXMrPG///rBDaTM7fnnVO8ZPeKIF4FHIkNUC0pAuDIY7GxMpD5fyCm3IEeo2yo4CP69zxL\nLWXRkFb1NFTeGIxLh9olyZGCcMaLhJpJc22pQJ6nz4QfXitFxpx+oPX6QQWd3lsQNNbYSQPSzrFy\nL3mULfveliFxobZXEdsMCxzLWfTLy6GqWa8K8+D01GkkepvuevJWoo+pucuXRM3z9Sc35TQSrT3F\nStS8UZdERaJfW3bgBo04H9qVlo0IT25QGoNPPQEEAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left[\\begin{matrix}0\\\\0\\\\0\\end{matrix}\\right]$$"
      ],
      "text/plain": [
       "⎡0⎤\n",
       "⎢ ⎥\n",
       "⎢0⎥\n",
       "⎢ ⎥\n",
       "⎣0⎦"
      ]
     },
     "execution_count": 56,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sp.simplify(res.expand())"
   ]
  }
 ],
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