Start notebook for generic choice of P

31b8ae55 · Luke Naylor · fe9bef14 · 31b8ae55
Commit 31b8ae55 authored 1 year ago by Luke Naylor
--- a/other-P-choice.ipynb
+++ b/other-P-choice.ipynb
+{
+ "cells": [
+  {
+   "cell_type": "code",
+   "execution_count": 1,
+   "id": "2c1f46cb",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "from pseudowalls import *\n",
+    "%display latex"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "48112244",
+   "metadata": {},
+   "source": [
+    "# Initialize Cherns"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 2,
+   "id": "7a52d103",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "var(\"R C D r c d A B\", domain=\"real\")\n",
+    "P = A, B"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 3,
+   "id": "cfde4b23",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}$"
+      ],
+      "text/plain": [
+       "<pseudowalls.chern_character.Chern_Char object at 0x7f7bcab93ad0>"
+      ]
+     },
+     "execution_count": 3,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "v = Chern_Char(R,C,D)\n",
+    "v"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 4,
+   "id": "f015f4ab",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle \\text{ Twisted Chern Character for $\\beta={ B }$ } \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = {\\mathrm{ch}_1^B(v)} \\ell^{1} \\\\ \\mathrm{ch}_{2} = {\\mathrm{ch}_2^B(v)} \\ell^{2} \\end{array}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle \\text{ Twisted Chern Character for $\\beta={ B }$ } \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = {\\mathrm{ch}_1^B(v)} \\ell^{1} \\\\ \\mathrm{ch}_{2} = {\\mathrm{ch}_2^B(v)} \\ell^{2} \\end{array}$"
+      ],
+      "text/plain": [
+       "<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf22f110>"
+      ]
+     },
+     "execution_count": 4,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "twisted_v = Twisted_Chern_Char(B,\n",
+    "            R,\n",
+    "            var(\"twisted_v1\", latex_name = r\"\\mathrm{ch}_1^B(v)\", domain=\"real\"),\n",
+    "            var(\"twisted_v2\", latex_name = r\"\\mathrm{ch}_2^B(v)\", domain=\"real\"),\n",
+    ")\n",
+    "twisted_v"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 5,
+   "id": "711e4205",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle \\text{ Twisted Chern Character for $\\beta={ B }$ } \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = {\\mathrm{ch}_1^B(u)} \\ell^{1} \\\\ \\mathrm{ch}_{2} = {\\mathrm{ch}_2^B(u)} \\ell^{2} \\end{array}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle \\text{ Twisted Chern Character for $\\beta={ B }$ } \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = {\\mathrm{ch}_1^B(u)} \\ell^{1} \\\\ \\mathrm{ch}_{2} = {\\mathrm{ch}_2^B(u)} \\ell^{2} \\end{array}$"
+      ],
+      "text/plain": [
+       "<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf0399d0>"
+      ]
+     },
+     "execution_count": 5,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "twisted_u = Twisted_Chern_Char(B,\n",
+    "            r,\n",
+    "            var(\"twisted_u1\", latex_name = r\"\\mathrm{ch}_1^B(u)\", domain=\"real\"),\n",
+    "            var(\"twisted_u2\", latex_name = r\"\\mathrm{ch}_2^B(u)\", domain=\"real\"),\n",
+    ")\n",
+    "twisted_u"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "5f3e6e12",
+   "metadata": {},
+   "source": [
+    "# Numerical Conditions"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "1206d912",
+   "metadata": {},
+   "source": [
+    "Condition of $P = (A,B)$ being on $\\Theta_v$ (i.e. $ch_2^{A,B}(v) = 0$) expressed in terms of twisted Chern character for $v$ at $\\beta=B$:"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": 7,
+   "id": "17c390cd",
+   "metadata": {},
+   "outputs": [
+    {
+     "data": {
+      "text/html": [
+       "<html>\\(\\displaystyle A^{2} = \\frac{2 \\, {\\mathrm{ch}_2^B(v)}}{R}\\)</html>"
+      ],
+      "text/latex": [
+       "$\\displaystyle A^{2} = \\frac{2 \\, {\\mathrm{ch}_2^B(v)}}{R}$"
+      ],
+      "text/plain": [
+       "A^2 == 2*twisted_v2/R"
+      ]
+     },
+     "execution_count": 7,
+     "metadata": {},
+     "output_type": "execute_result"
+    }
+   ],
+   "source": [
+    "A2_subs = solve(\n",
+    "    stability.Tilt(*P).degree(twisted_v) == 0,\n",
+    "    A^2)[0]\n",
+    "\n",
+    "A2_subs"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "11b9c67b",
+   "metadata": {},
+   "source": [
+    "## Condition: $ch_2^{P}(u) > 0$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "47b34ed7",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "stability.Tilt(*P).degree(twisted_u) > 0"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "d8abf566",
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(\n",
+    "    A2_subs\n",
+    ")"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "id": "fe9fe5b8",
+   "metadata": {},
+   "source": [
+    "## Condition: $\\Delta(u) \\geq 0$"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "id": "f09514cb",
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  }
+ ],
+ "metadata": {
+  "kernelspec": {
+   "display_name": "SageMath 9.8",
+   "language": "sage",
+   "name": "sagemath"
+  },
+  "language_info": {
+   "codemirror_mode": {
+    "name": "ipython",
+    "version": 3
+   },
+   "file_extension": ".py",
+   "mimetype": "text/x-python",
+   "name": "python",
+   "nbconvert_exporter": "python",
+   "pygments_lexer": "ipython3",
+   "version": "3.11.3"
+  }
+ },
+ "nbformat": 4,
+ "nbformat_minor": 5
+}
+%% Cell type:code id:2c1f46cb tags:
+
+``` sage
+from pseudowalls import *
+%display latex
+```
+
+%% Cell type:markdown id:48112244 tags:
+
+# Initialize Cherns
+
+%% Cell type:code id:7a52d103 tags:
+
+``` sage
+var("R C D r c d A B", domain="real")
+P = A, B
+```
+
+%% Cell type:code id:cfde4b23 tags:
+
+``` sage
+v = Chern_Char(R,C,D)
+v
+```
+
+%% Output
+
+    $\displaystyle \text{Chern Character:} \\ \begin{array}{l} \mathrm{ch}_{0} = R \\ \mathrm{ch}_{1} = C \ell^{1} \\ \mathrm{ch}_{2} = D \ell^{2} \end{array}$
+    <pseudowalls.chern_character.Chern_Char object at 0x7f7bcab93ad0>
+
+%% Cell type:code id:f015f4ab tags:
+
+``` sage
+twisted_v = Twisted_Chern_Char(B,
+            R,
+            var("twisted_v1", latex_name = r"\mathrm{ch}_1^B(v)", domain="real"),
+            var("twisted_v2", latex_name = r"\mathrm{ch}_2^B(v)", domain="real"),
+)
+twisted_v
+```
+
+%% Output
+
+    $\displaystyle \text{ Twisted Chern Character for $\beta={ B }$ } \\ \begin{array}{l} \mathrm{ch}_{0} = R \\ \mathrm{ch}_{1} = {\mathrm{ch}_1^B(v)} \ell^{1} \\ \mathrm{ch}_{2} = {\mathrm{ch}_2^B(v)} \ell^{2} \end{array}$
+    <pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf22f110>
+
+%% Cell type:code id:711e4205 tags:
+
+``` sage
+twisted_u = Twisted_Chern_Char(B,
+            r,
+            var("twisted_u1", latex_name = r"\mathrm{ch}_1^B(u)", domain="real"),
+            var("twisted_u2", latex_name = r"\mathrm{ch}_2^B(u)", domain="real"),
+)
+twisted_u
+```
+
+%% Output
+
+    $\displaystyle \text{ Twisted Chern Character for $\beta={ B }$ } \\ \begin{array}{l} \mathrm{ch}_{0} = r \\ \mathrm{ch}_{1} = {\mathrm{ch}_1^B(u)} \ell^{1} \\ \mathrm{ch}_{2} = {\mathrm{ch}_2^B(u)} \ell^{2} \end{array}$
+    <pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f7bbf0399d0>
+
+%% Cell type:markdown id:5f3e6e12 tags:
+
+# Numerical Conditions
+
+%% Cell type:markdown id:1206d912 tags:
+
+Condition of $P = (A,B)$ being on $\Theta_v$ (i.e. $ch_2^{A,B}(v) = 0$) expressed in terms of twisted Chern character for $v$ at $\beta=B$:
+
+%% Cell type:code id:17c390cd tags:
+
+``` sage
+A2_subs = solve(
+    stability.Tilt(*P).degree(twisted_v) == 0,
+    A^2)[0]
+
+A2_subs
+```
+
+%% Output
+
+    $\displaystyle A^{2} = \frac{2 \, {\mathrm{ch}_2^B(v)}}{R}$
+    A^2 == 2*twisted_v2/R
+
+%% Cell type:markdown id:11b9c67b tags:
+
+## Condition: $ch_2^{P}(u) > 0$
+
+%% Cell type:code id:47b34ed7 tags:
+
+``` sage
+stability.Tilt(*P).degree(twisted_u) > 0
+```
+
+%% Cell type:code id:d8abf566 tags:
+
+``` sage
+(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(
+    A2_subs
+)
+```
+
+%% Cell type:markdown id:fe9fe5b8 tags:
+
+## Condition: $\Delta(u) \geq 0$
+
+%% Cell type:code id:f09514cb tags:
+
+``` sage
+```