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  1. {
  2.  "cells": [
  3.   {
  4.    "cell_type": "markdown",
  5.    "metadata": {},
  6.    "source": [
  7.     "We use the harmonic extensions\n",
  8.     "\n",
  9.     "$$\n",
  10.     "H_1 = \\sum_{i=1}^3 \\frac{1}{r} R_i^T D R_i\n",
  11.     "$$\n",
  12.     "\n",
  13.     "In the case of the Sierpinski gasket, it is well known that $r=\\frac{3}{5}$, \n",
  14.     "$$\n",
  15.     "D=\\begin{pmatrix}\n",
  16.     "-2 & 1 & 1 \\\\\n",
  17.     "1 & -2 & 1 \\\\\n",
  18.     "1 & 1 & -2\n",
  19.     "\\end{pmatrix}\n",
  20.     "$$ \n",
  21.     "and $R_i: \\ell(V_1) \\to \\ell(V_0)$ is given by\n",
  22.     "\n",
  23.     "$$\n",
  24.     "(R_i f) (v_j) = f(F_i(v_j)), \\quad \\text{ where }f\\in \\ell(V_1), v_j \\in V_0.\n",
  25.     "$$\n",
  26.     "\n",
  27.     "In our case, we let $v_1, v_2, v_3$ be the left, top, and right most points on $V_0$, and $v_4, v_5, v_6$ denote the points in $V_1$ between $v_1$ and $v_2$, $v_2$ and $v_3$, and $v_1$ and $v_3$ respectively.\n",
  28.     "\n",
  29.     "We compute $H_1$."
  30.    ]
  31.   },
  32.   {
  33.    "cell_type": "code",
  34.    "execution_count": 1,
  35.    "metadata": {},
  36.    "outputs": [],
  37.    "source": [
  38.     "import numpy as np\n",
  39.     "import sympy as sym\n",
  40.     "\n",
  41.     "R_1 = np.array([[1, 0, 0, 0, 0, 0], \n",
  42.     "                [0, 0, 0, 1, 0, 0], \n",
  43.     "                [0, 0, 0, 0, 0, 1]])\n",
  44.     "\n",
  45.     "R_2 = np.array([[0, 0, 0, 1, 0, 0], \n",
  46.     "                [0, 1, 0, 0, 0, 0], \n",
  47.     "                [0, 0, 0, 0, 1, 0]])\n",
  48.     "\n",
  49.     "R_3 = np.array([[0, 0, 0, 0, 0, 1], \n",
  50.     "                [0, 0, 0, 0, 1, 0], \n",
  51.     "                [0, 0, 1, 0, 0, 0]])\n",
  52.     "\n",
  53.     "D = np.array([[-2, 1, 1], [1, -2, 1], [1, 1, -2]])\n",
  54.     "\n",
  55.     "r = sym.Rational(3, 5)"
  56.    ]
  57.   },
  58.   {
  59.    "cell_type": "code",
  60.    "execution_count": 2,
  61.    "metadata": {},
  62.    "outputs": [],
  63.    "source": [
  64.     "R = R_1 + R_2 + R_3\n",
  65.     "\n",
  66.     "H_1 = np.dot(np.dot(np.transpose(R), D), R) / r"
  67.    ]
  68.   },
  69.   {
  70.    "cell_type": "code",
  71.    "execution_count": 3,
  72.    "metadata": {},
  73.    "outputs": [
  74.     {
  75.      "name": "stdout",
  76.      "output_type": "stream",
  77.      "text": [
  78.       "[[-10/3 5/3 5/3 -5/3 10/3 -5/3]\n",
  79.       " [5/3 -10/3 5/3 -5/3 -5/3 10/3]\n",
  80.       " [5/3 5/3 -10/3 10/3 -5/3 -5/3]\n",
  81.       " [-5/3 -5/3 10/3 -10/3 5/3 5/3]\n",
  82.       " [10/3 -5/3 -5/3 5/3 -10/3 5/3]\n",
  83.       " [-5/3 10/3 -5/3 5/3 5/3 -10/3]]\n"
  84.      ]
  85.     }
  86.    ],
  87.    "source": [
  88.     "print(H_1)"
  89.    ]
  90.   },
  91.   {
  92.    "cell_type": "markdown",
  93.    "metadata": {},
  94.    "source": [
  95.     "In our case, we need to compute $G = X_1^{-1}$, where $X_1$ is given by $H_1$ restricted to $V_1 - V_0$."
  96.    ]
  97.   },
  98.   {
  99.    "cell_type": "code",
  100.    "execution_count": 4,
  101.    "metadata": {},
  102.    "outputs": [],
  103.    "source": [
  104.     "X = H_1[3:, 3:]"
  105.    ]
  106.   },
  107.   {
  108.    "cell_type": "code",
  109.    "execution_count": 5,
  110.    "metadata": {},
  111.    "outputs": [
  112.     {
  113.      "name": "stdout",
  114.      "output_type": "stream",
  115.      "text": [
  116.       "[[-10/3 5/3 5/3]\n",
  117.       " [5/3 -10/3 5/3]\n",
  118.       " [5/3 5/3 -10/3]]\n"
  119.      ]
  120.     }
  121.    ],
  122.    "source": [
  123.     "print(X)"
  124.    ]
  125.   },
  126.   {
  127.    "cell_type": "markdown",
  128.    "metadata": {},
  129.    "source": [
  130.     "Which is noninvertible. In particular, $(1, 1, 1) \\in ker(X)$."
  131.    ]
  132.   },
  133.   {
  134.    "cell_type": "code",
  135.    "execution_count": null,
  136.    "metadata": {},
  137.    "outputs": [],
  138.    "source": []
  139.   }
  140.  ],
  141.  "metadata": {
  142.   "kernelspec": {
  143.    "display_name": "Python 3",
  144.    "language": "python",
  145.    "name": "python3"
  146.   },
  147.   "language_info": {
  148.    "codemirror_mode": {
  149.     "name": "ipython",
  150.     "version": 3
  151.    },
  152.    "file_extension": ".py",
  153.    "mimetype": "text/x-python",
  154.    "name": "python",
  155.    "nbconvert_exporter": "python",
  156.    "pygments_lexer": "ipython3",
  157.    "version": "3.7.2"
  158.   }
  159.  },
  160.  "nbformat": 4,
  161.  "nbformat_minor": 2
  162. }
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