diff --git a/plots_and_expressions.ipynb b/plots_and_expressions.ipynb
index 48c42eac5864d6217272970f0c3a661f21a5cd13..940bd859c7668f1d82b21f35b36cf19180850b13 100644
--- a/plots_and_expressions.ipynb
+++ b/plots_and_expressions.ipynb
@@ -337,96 +337,6 @@
"## SUBSECTION: Bound on $\\operatorname{ch}_0(u)$ for Semistabilizers"
]
},
- {
- "cell_type": "code",
- "execution_count": 8,
- "id": "2b641339",
- "metadata": {},
- "outputs": [],
- "source": [
- "var(\"m\") # Initialize symbol for variety parameter\n",
- "recurring = Object()\n",
- "recurring.chern = Chern_Char(3, 2, -2)\n",
- "recurring.b = beta_minus(recurring.chern)\n",
- "recurring.twisted = recurring.chern.twist(recurring.b)\n",
- "# RENDERED TO LATEX: recurring.b\n",
- "# RENDERED TO LATEX: recurring.b.denominator()\n",
- "# RENDERED TO LATEX: recurring.twisted.ch[1]\n",
- "n = recurring.b.denominator()\n",
- "m = 2\n",
- "recurring.loose_bound = (\n",
- " m*n^2*recurring.twisted.ch[1]^2\n",
- ") / gcd(m, 2*n^2)\n",
- "# RENDERED TO LATEX: loose_bound\n",
- "extravagant = Object()\n",
- "extravagant.chern = Chern_Char(29, 13, -3/2)\n",
- "extravagant.b = beta_minus(extravagant.chern)\n",
- "extravagant.twisted = extravagant.chern.twist(extravagant.b)\n",
- "extravagant.actual_rmax = 49313\n",
- "# RENDERED TO LATEX: extravagant.b\n",
- "# RENDERED TO LATEX: extravagant.b.denominator()\n",
- "# RENDERED TO LATEX: extravagant.twisted.ch[1]\n",
- "n = extravagant.b.denominator()\n",
- "m = 2\n",
- "extravagant.loose_bound = (\n",
- " m*n^2*extravagant.twisted.ch[1]^2\n",
- ") / gcd(m, 2*n^2)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 9,
- "id": "f38c4896",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 215296\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 215296$"
- ],
- "text/plain": [
- "215296"
- ]
- },
- "execution_count": 9,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "recurring.loose_bound"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 10,
- "id": "c2e307ec",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 49313\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 49313$"
- ],
- "text/plain": [
- "49313"
- ]
- },
- "execution_count": 10,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "extravagant.actual_rmax"
- ]
- },
{
"cell_type": "markdown",
"id": "4982f084",
@@ -1494,231 +1404,6 @@
"r_upper_bound_all_q.expand().subs([nu==1,Delta==psi^2])"
]
},
- {
- "cell_type": "code",
- "execution_count": 45,
- "id": "8ba2ed91",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle -\\frac{2}{3}\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle -\\frac{2}{3}$"
- ],
- "text/plain": [
- "-2/3"
- ]
- },
- "execution_count": 45,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "recurring.b"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 46,
- "id": "fcfae8a1",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 3\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 3$"
- ],
- "text/plain": [
- "3"
- ]
- },
- "execution_count": 46,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "recurring.b.denominator()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 47,
- "id": "36d896f4",
- "metadata": {},
- "outputs": [],
- "source": [
- "recurring.n = recurring.b.denominator()\n",
- "recurring.bgmlv = recurring.chern.Q_tilt()\n",
- "recurring.corrolary_bound = (\n",
- " r_upper_bound_all_q.expand()\n",
- " .subs(Delta==recurring.bgmlv)\n",
- " .subs(nu==1) ## \\ell^2=1 on P^2\n",
- " .subs(R==recurring.chern.ch[0])\n",
- " .subs(n==recurring.n)\n",
- ")"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 48,
- "id": "b3f47ade",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 37.515625\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 37.515625$"
- ],
- "text/plain": [
- "37.515625"
- ]
- },
- "execution_count": 48,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "float(recurring.corrolary_bound)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 49,
- "id": "b52a3a56",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle -\\frac{3}{29}\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle -\\frac{3}{29}$"
- ],
- "text/plain": [
- "-3/29"
- ]
- },
- "execution_count": 49,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "extravagant.b"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 50,
- "id": "cdf1726c",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 29\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 29$"
- ],
- "text/plain": [
- "29"
- ]
- },
- "execution_count": 50,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "extravagant.b.denominator()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 51,
- "id": "8f5ea43e",
- "metadata": {},
- "outputs": [],
- "source": [
- "extravagant.n = extravagant.b.denominator()\n",
- "extravagant.bgmlv = extravagant.chern.Q_tilt()\n",
- "extravagant.corrolary_bound = (\n",
- " r_upper_bound_all_q.expand()\n",
- " .subs(Delta==extravagant.bgmlv)\n",
- " .subs(nu==1) ## \\ell^2=1 on P^2\n",
- " .subs(R==extravagant.chern.ch[0])\n",
- " .subs(n==extravagant.n)\n",
- ")"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 52,
- "id": "bc7f5e56",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 53838.5009765625\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 53838.5009765625$"
- ],
- "text/plain": [
- "53838.5009765625"
- ]
- },
- "execution_count": 52,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "float(extravagant.corrolary_bound)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 53,
- "id": "00a2355a",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 49313\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 49313$"
- ],
- "text/plain": [
- "49313"
- ]
- },
- "execution_count": 53,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "extravagant.actual_rmax"
- ]
- },
{
"cell_type": "code",
"execution_count": 54,
@@ -1869,158 +1554,6 @@
"source": [
"r_upper_bound2.rhs()"
]
- },
- {
- "cell_type": "code",
- "execution_count": 60,
- "id": "50a94cf4",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 3\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 3$"
- ],
- "text/plain": [
- "3"
- ]
- },
- "execution_count": 60,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "recurring.b.denominator()"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 61,
- "id": "3171ad07",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 4\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 4$"
- ],
- "text/plain": [
- "4"
- ]
- },
- "execution_count": 61,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "recurring.twisted.ch[1]"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 62,
- "id": "16b28a02",
- "metadata": {},
- "outputs": [],
- "source": [
- "import numpy as np\n",
- "\n",
- "def bound_comparisons(example):\n",
- " n = example.b.denominator()\n",
- " a_v = example.b.numerator()\n",
- "\n",
- " def theorem_bound(v_twisted, q_val, k):\n",
- " return int(min(\n",
- " n^2*q_val^2/k\n",
- " ,\n",
- " v_twisted.ch[0]\n",
- " + n^2*(v_twisted.ch[1] - q_val)^2/k\n",
- " ))\n",
- "\n",
- " def k(n, a_v, b_q):\n",
- " n = int(n)\n",
- " a_v = int(a_v)\n",
- " b_q = int(b_q)\n",
- " k = -a_v*b_q % n\n",
- " return k if k > 0 else k + n\n",
- "\n",
- " b_qs = list(range(example.twisted.ch[1]*n+1))\n",
- " qs = list(map(lambda x: x/n,b_qs))\n",
- " ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs))\n",
- " theorem2_bounds = [\n",
- " theorem_bound(example.twisted, q_val, 1)\n",
- " for q_val in qs\n",
- " ]\n",
- " theorem3_bounds = [\n",
- " theorem_bound(example.twisted, q_val, k)\n",
- " for q_val, k in zip(qs,ks)\n",
- " ]\n",
- " return qs, theorem2_bounds, theorem3_bounds\n",
- "\n",
- "qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 63,
- "id": "bb236b7b",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 36\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 36$"
- ],
- "text/plain": [
- "36"
- ]
- },
- "execution_count": 63,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "max(theorem2_bounds)"
- ]
- },
- {
- "cell_type": "code",
- "execution_count": 64,
- "id": "7205efcd",
- "metadata": {},
- "outputs": [
- {
- "data": {
- "text/html": [
- "<html>\\(\\displaystyle 25\\)</html>"
- ],
- "text/latex": [
- "$\\displaystyle 25$"
- ],
- "text/plain": [
- "25"
- ]
- },
- "execution_count": 64,
- "metadata": {},
- "output_type": "execute_result"
- }
- ],
- "source": [
- "max(theorem3_bounds)"
- ]
}
],
"metadata": {