updates
This commit is contained in:
William Ball
parent
495139741f
commit
d740a18fe1
1 changed files with 123 additions and 120 deletions
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@ -16,7 +16,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 1,
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"execution_count": 3,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -30,7 +30,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 2,
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"execution_count": 4,
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"metadata": {},
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"outputs": [
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{
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@ -42,7 +42,7 @@
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"[ 0 0 0 1]"
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]
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},
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"execution_count": 2,
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"execution_count": 4,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -62,7 +62,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 3,
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"execution_count": 5,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -99,7 +99,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 4,
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"execution_count": 6,
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"metadata": {
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"scrolled": true
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},
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@ -113,7 +113,7 @@
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"[-2 -2 -2 -4]"
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]
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},
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"execution_count": 4,
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"execution_count": 6,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -140,7 +140,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 5,
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"execution_count": 7,
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"metadata": {
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"scrolled": true
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},
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@ -154,7 +154,7 @@
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"[-1/4 -1/4 -1/4 1/8]"
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]
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},
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"execution_count": 5,
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"execution_count": 7,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -165,7 +165,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 6,
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"execution_count": 8,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -197,7 +197,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 7,
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"execution_count": 9,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -206,7 +206,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 8,
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"execution_count": 10,
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"metadata": {},
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"outputs": [
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{
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@ -218,7 +218,7 @@
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"[ 0 0 0 1]"
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]
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},
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"execution_count": 8,
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"execution_count": 10,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -229,7 +229,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 9,
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"execution_count": 11,
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"metadata": {},
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"outputs": [
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{
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@ -238,7 +238,7 @@
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"b1^2 + b2^2 + b3^2 - b1*b_avg - b2*b_avg - b3*b_avg + 1/4*b_avg^2"
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]
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},
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"execution_count": 9,
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"execution_count": 11,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -269,20 +269,21 @@
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},
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{
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"cell_type": "code",
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"execution_count": 10,
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"execution_count": 12,
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"metadata": {},
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"outputs": [],
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"source": [
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"def weyl_generators(matrix, alphas):\n",
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" retval = []\n",
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" for alpha in alphas:\n",
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" retval.append(identity_matrix(len(alphas)) - 2 * alpha * alpha.transpose() * matrix)\n",
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" scale_factor = (alpha.transpose() * matrix * alpha)[0][0]\n",
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" retval.append(identity_matrix(len(alphas)) - 2 * alpha * alpha.transpose() * matrix / scale_factor)\n",
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" return retval"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 11,
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"execution_count": 13,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -292,33 +293,56 @@
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},
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{
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"cell_type": "code",
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"execution_count": 12,
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"execution_count": 14,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[ 2 0 0 -1]\n",
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"[ 0 2 0 -1]\n",
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"[ 0 0 2 -1]\n",
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"[ -1 -1 -1 1/2]"
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]
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},
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"execution_count": 14,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"4 * M.inverse()"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[\n",
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"[-1 0 0 1] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
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"[ 0 1 0 0] [ 0 -1 0 1] [ 0 1 0 0] [ 0 1 0 0]\n",
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"[ 0 0 1 0] [ 0 0 1 0] [ 0 0 -1 1] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 1 1 1 1/2]\n",
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"[-1 0 0 1] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
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"[ 0 1 0 0] [ 0 -1 0 1] [ 0 1 0 0] [ 0 1 0 0]\n",
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"[ 0 0 1 0] [ 0 0 1 0] [ 0 0 -1 1] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 4 4 4 -1]\n",
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"]"
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]
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},
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"execution_count": 12,
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"execution_count": 15,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"S_i = weyl_generators(2 * M.inverse(), standard_basis(4))\n",
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"S_i = weyl_generators(4 * M.inverse(), standard_basis(4))\n",
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"S_i"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 13,
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"execution_count": 16,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -337,7 +361,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 14,
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"execution_count": 17,
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"metadata": {},
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"outputs": [
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{
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@ -346,7 +370,7 @@
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"(7, 2, 4, 6)"
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]
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},
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"execution_count": 14,
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"execution_count": 17,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -358,7 +382,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 15,
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"execution_count": 18,
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"metadata": {},
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"outputs": [
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{
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@ -367,7 +391,7 @@
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"(-1, 4, 4, 6)"
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]
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},
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"execution_count": 15,
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"execution_count": 18,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -378,7 +402,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 16,
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"execution_count": 19,
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"metadata": {},
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"outputs": [
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{
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@ -387,7 +411,7 @@
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"(-1, 2, 2, 6)"
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]
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},
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"execution_count": 16,
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"execution_count": 19,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -396,36 +420,9 @@
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"S3 * root"
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]
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},
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{
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"cell_type": "markdown",
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"metadata": {},
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"source": [
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"For some reason, I have no idea why, $s_4$ doesn't seem to work. Instead this weird matrix does appear to work."
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]
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},
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{
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"cell_type": "code",
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"execution_count": 17,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"(-1, 2, 4, 8)"
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]
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},
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"execution_count": 17,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"S4 * root"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 18,
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"execution_count": 20,
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"metadata": {},
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"outputs": [
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{
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@ -434,18 +431,13 @@
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"(-1, 2, 4, 14)"
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]
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},
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"execution_count": 18,
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"execution_count": 20,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"matrix([\n",
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" [1, 0, 0, 0],\n",
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" [0, 1, 0, 0],\n",
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" [0, 0, 1, 0],\n",
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" [4, 4, 4, -1]\n",
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"]) * root"
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"S4 * root"
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]
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},
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{
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@ -459,7 +451,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 19,
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"execution_count": 21,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -483,7 +475,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 20,
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"execution_count": 22,
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"metadata": {},
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"outputs": [
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{
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@ -548,7 +540,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 21,
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"execution_count": 23,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -565,7 +557,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 22,
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"execution_count": 24,
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"metadata": {},
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"outputs": [
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{
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@ -577,7 +569,7 @@
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"[ 0 0 0 1 -1/2 1/2 1/2 1/2]"
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]
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},
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"execution_count": 22,
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"execution_count": 24,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -594,7 +586,7 @@
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" [0, 2, 0, -1],\n",
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"])\n",
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"\n",
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"Wc.transpose().echelon_form()"
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"Wc.transpose().rref()"
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]
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},
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{
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@ -621,7 +613,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 23,
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"execution_count": 25,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -635,7 +627,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 24,
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"execution_count": 26,
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"metadata": {},
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"outputs": [
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{
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@ -647,7 +639,7 @@
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"[-3 -3 -3 1]"
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]
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},
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"execution_count": 24,
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"execution_count": 26,
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"metadata": {},
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"output_type": "execute_result"
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}
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},
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{
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"cell_type": "code",
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"execution_count": 25,
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"execution_count": 27,
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"metadata": {},
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"outputs": [
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{
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"data": {
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"text/plain": [
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"[ 5 -3 -3 -3]\n",
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"[-3 5 -3 -3]\n",
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"[-3 -3 5 -3]\n",
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"[-3 -3 -3 5]"
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"[ 2 -6/5 -6/5 -6/5]\n",
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"[-6/5 2 -6/5 -6/5]\n",
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"[-6/5 -6/5 2 -6/5]\n",
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"[-6/5 -6/5 -6/5 2]"
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]
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},
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"execution_count": 25,
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"execution_count": 27,
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"metadata": {},
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"output_type": "execute_result"
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}
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],
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"source": [
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"32 * m.inverse()"
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"32 * m.inverse() * 2/5"
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]
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},
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{
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"cell_type": "code",
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"execution_count": 26,
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"execution_count": 28,
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"metadata": {},
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"outputs": [
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{
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"[bt4 b4 h14 h24]"
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]
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},
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"execution_count": 26,
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"execution_count": 28,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -713,7 +705,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 27,
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"execution_count": 29,
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"metadata": {},
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"outputs": [],
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"source": [
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@ -722,7 +714,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 28,
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"execution_count": 30,
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"metadata": {},
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"outputs": [
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{
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@ -731,7 +723,7 @@
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"5*b1^2 - 6*b1*b2 + 5*b2^2 - 6*b1*b3 - 6*b2*b3 + 5*b3^2 - 6*b1*b4 - 6*b2*b4 - 6*b3*b4 + 5*b4^2"
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]
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},
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"execution_count": 28,
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"execution_count": 30,
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"metadata": {},
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"output_type": "execute_result"
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}
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@ -758,7 +750,7 @@
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},
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{
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"cell_type": "code",
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"execution_count": 29,
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"execution_count": 31,
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"metadata": {
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"scrolled": true
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},
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@ -767,26 +759,31 @@
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"data": {
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"text/plain": [
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"[\n",
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"[-1 6 6 6] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
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"[ 0 1 0 0] [ 6 -1 6 6] [ 0 1 0 0] [ 0 1 0 0]\n",
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"[ 0 0 1 0] [ 0 0 1 0] [ 6 6 -1 6] [ 0 0 1 0]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 6 6 6 -1]\n",
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"[ -1 6/5 6/5 6/5] [ 1 0 0 0] [ 1 0 0 0]\n",
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"[ 0 1 0 0] [6/5 -1 6/5 6/5] [ 0 1 0 0]\n",
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"[ 0 0 1 0] [ 0 0 1 0] [6/5 6/5 -1 6/5]\n",
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"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1],\n",
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"\n",
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"[ 1 0 0 0]\n",
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"[ 0 1 0 0]\n",
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"[ 0 0 1 0]\n",
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"[6/5 6/5 6/5 -1]\n",
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"]"
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]
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},
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"execution_count": 29,
|
||||
"execution_count": 31,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"S_i = weyl_generators(m, standard_basis(4))\n",
|
||||
"S_i = weyl_generators(m.inverse(), standard_basis(4))\n",
|
||||
"S_i"
|
||||
]
|
||||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 30,
|
||||
"execution_count": 32,
|
||||
"metadata": {
|
||||
"scrolled": true
|
||||
},
|
||||
|
|
@ -794,23 +791,23 @@
|
|||
{
|
||||
"data": {
|
||||
"text/plain": [
|
||||
"[ 1 -3 -3 -3 -1 -1 -1 -5]\n",
|
||||
"[-3 1 -3 -3 -1 -1 -5 -1]\n",
|
||||
"[-3 -3 1 -3 -1 -5 -1 -1]\n",
|
||||
"[-3 -3 -3 1 -5 -1 -1 -1]\n",
|
||||
"[-1 -1 -1 -5 1 -3 -3 -3]\n",
|
||||
"[-1 -1 -5 -1 -3 1 -3 -3]\n",
|
||||
"[-1 -5 -1 -1 -3 -3 1 -3]\n",
|
||||
"[-5 -1 -1 -1 -3 -3 -3 1]"
|
||||
"[ 1 0 0 0 1/2 1/2 1/2 -1/2]\n",
|
||||
"[ 0 1 0 0 1/2 1/2 -1/2 1/2]\n",
|
||||
"[ 0 0 1 0 1/2 -1/2 1/2 1/2]\n",
|
||||
"[ 0 0 0 1 -1/2 1/2 1/2 1/2]\n",
|
||||
"[ 0 0 0 0 0 0 0 0]\n",
|
||||
"[ 0 0 0 0 0 0 0 0]\n",
|
||||
"[ 0 0 0 0 0 0 0 0]\n",
|
||||
"[ 0 0 0 0 0 0 0 0]"
|
||||
]
|
||||
},
|
||||
"execution_count": 30,
|
||||
"execution_count": 32,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
],
|
||||
"source": [
|
||||
"Wc * P * Wc.transpose()"
|
||||
"(Wc * P * Wc.transpose()).rref()"
|
||||
]
|
||||
},
|
||||
{
|
||||
|
|
@ -823,7 +820,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 31,
|
||||
"execution_count": 33,
|
||||
"metadata": {},
|
||||
"outputs": [],
|
||||
"source": [
|
||||
|
|
@ -839,7 +836,7 @@
|
|||
" relation = root_matrix.transpose().rref() * vector([ var('b' + str(i)) for i in range(1, root_matrix.dimensions()[0] + 1)])\n",
|
||||
" \n",
|
||||
" # step 2: find matrix of quadratic form\n",
|
||||
" W = root_matrix[0:4]\n",
|
||||
" W = root_matrix[-4:]\n",
|
||||
" M = W * P * W.transpose()\n",
|
||||
" \n",
|
||||
" # step 3: repeat with arbitrary matrix\n",
|
||||
|
|
@ -858,7 +855,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 32,
|
||||
"execution_count": 34,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
|
|
@ -877,14 +874,19 @@
|
|||
"data": {
|
||||
"text/plain": [
|
||||
"[\n",
|
||||
"[-9 6 6 6] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
||||
"[ 0 1 0 0] [ 6 -9 6 6] [ 0 1 0 0] [ 0 1 0 0]\n",
|
||||
"[ 0 0 1 0] [ 0 0 1 0] [ 6 6 -9 6] [ 0 0 1 0]\n",
|
||||
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 6 6 6 -9]\n",
|
||||
"[ -1 6/5 6/5 6/5] [ 1 0 0 0] [ 1 0 0 0]\n",
|
||||
"[ 0 1 0 0] [6/5 -1 6/5 6/5] [ 0 1 0 0]\n",
|
||||
"[ 0 0 1 0] [ 0 0 1 0] [6/5 6/5 -1 6/5]\n",
|
||||
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1],\n",
|
||||
"\n",
|
||||
"[ 1 0 0 0]\n",
|
||||
"[ 0 1 0 0]\n",
|
||||
"[ 0 0 1 0]\n",
|
||||
"[6/5 6/5 6/5 -1]\n",
|
||||
"]"
|
||||
]
|
||||
},
|
||||
"execution_count": 32,
|
||||
"execution_count": 34,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
@ -900,7 +902,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 33,
|
||||
"execution_count": 35,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
|
|
@ -920,13 +922,13 @@
|
|||
"text/plain": [
|
||||
"[\n",
|
||||
"[-1 4 4 2] [ 1 0 0 0] [ 1 0 0 0] [ 1 0 0 0]\n",
|
||||
"[ 0 1 0 0] [ 4 -7 0 4] [ 0 1 0 0] [ 0 1 0 0]\n",
|
||||
"[ 0 0 1 0] [ 0 0 1 0] [ 4 0 -7 4] [ 0 0 1 0]\n",
|
||||
"[ 0 1 0 0] [ 1 -1 0 1] [ 0 1 0 0] [ 0 1 0 0]\n",
|
||||
"[ 0 0 1 0] [ 0 0 1 0] [ 1 0 -1 1] [ 0 0 1 0]\n",
|
||||
"[ 0 0 0 1], [ 0 0 0 1], [ 0 0 0 1], [ 2 4 4 -1]\n",
|
||||
"]"
|
||||
]
|
||||
},
|
||||
"execution_count": 33,
|
||||
"execution_count": 35,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
@ -949,7 +951,7 @@
|
|||
},
|
||||
{
|
||||
"cell_type": "code",
|
||||
"execution_count": 34,
|
||||
"execution_count": 36,
|
||||
"metadata": {},
|
||||
"outputs": [
|
||||
{
|
||||
|
|
@ -975,7 +977,7 @@
|
|||
"]"
|
||||
]
|
||||
},
|
||||
"execution_count": 34,
|
||||
"execution_count": 36,
|
||||
"metadata": {},
|
||||
"output_type": "execute_result"
|
||||
}
|
||||
|
|
@ -1019,7 +1021,8 @@
|
|||
"nbconvert_exporter": "python",
|
||||
"pygments_lexer": "ipython3",
|
||||
"version": "3.9.2"
|
||||
}
|
||||
},
|
||||
"name": "Apollonian Circle Packings.ipynb"
|
||||
},
|
||||
"nbformat": 4,
|
||||
"nbformat_minor": 4
|
||||
|
|
|
|||
Loading…
Reference in a new issue