From 272f39df0e3e1b85804c3b51c177874215227e41 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu, 27 Jul 2023 15:30:04 +0100
Subject: [PATCH] Refactor sagesilent content into separate notebook
Squashed commit of the following:
commit 1646c3092b7f6e2d1fc1a73e2569ffc3e897f194
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu Jul 27 15:27:24 2023 +0100
Adjust formatting in secondary notebooks
commit 3381ee2d703d9fc37f4321330072b6458106d529
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu Jul 27 15:19:26 2023 +0100
Adjust titling and formatting in main notebook
commit 2fb2e99ff7a614678cdd06b2891bdc9d52c405ad
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Thu Jul 27 13:47:24 2023 +0100
Change tabs to spaces in notebooks
commit e9dc188d0ec808ddac3a92e23cd7d8c7cb63ee7e
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed Jul 26 17:11:01 2023 +0100
Import charact curves plot from new notebook and remove redundant from old
commit ef7d3ae41b754015bfd02cc2d7e8a8384baaca8d
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed Jul 26 17:04:51 2023 +0100
Copy characteristic curves plots to new notebook and integrate into build'
commit 98a0dd217647905bfc07d9387c16d85991190447
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed Jul 26 16:57:56 2023 +0100
Remove redundant example code in main notebook
commit 99976ab19f4daa643112d6964dae33bb82951f26
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed Jul 26 16:52:53 2023 +0100
Import examples from new notebook instead
commit eac837ddeeb2495bc2e04a46e18043fff542d3e8
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed Jul 26 16:47:53 2023 +0100
Copy recurring examples to separate notebook and integrate into build
commit 1abd1b8f60ef8205d7481670feb40cfc02eb29d1
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed Jul 26 16:33:14 2023 +0100
Small reorganisation of notebook
commit 2aff290881266938668c4d59f08030b81199697e
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed Jul 26 14:57:22 2023 +0100
Correct missing imports from previous commits'
commit 48f747d5a0f400a6d7b8dc26f30042a66cf05e31
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed Jul 26 14:11:25 2023 +0100
Import sage expressions from external script
commit ee88d79f6967fb0227ea2cb960283df26e7770af
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue Jul 25 23:03:48 2023 +0100
Refactor extravagant example into notebook
commit 96b9b38dad93e24df078ba10c0b6a6ea126b7335
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue Jul 25 22:56:42 2023 +0100
Refactor recurring example into notebook
commit 5be4f14f3a8f6aba3123379200607eac56010ccb
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue Jul 25 21:54:24 2023 +0100
Use imports from notebook pt4
commit 92295236a823db4143237a019fae9a04ccf8b0f2
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue Jul 25 21:51:32 2023 +0100
Use imports from notebook pt3
commit 1eeb885eaf0bf5b2abd80240593f4d198eea7321
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue Jul 25 21:48:06 2023 +0100
Use imports from notebook pt2
commit 93b6dcbf6e95cc2e57c0e5c7084bbc4ea2024a02
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue Jul 25 21:45:43 2023 +0100
Use imports from notebook pt1
commit efccff7ff195147171003c922be0cf5f8814c640
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue Jul 25 17:06:28 2023 +0100
Adjust Makefile to create correct python library
commit 75552ca3b9ba74ccc4893a48d63eb8ed309d48db
Author: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Tue Jul 25 16:47:37 2023 +0100
Add jupyter notebook with most of sagetex scripts
---
.gitignore | 8 +
Makefile | 28 +-
characteristic_curves.ipynb | 300 ++++++++
examples.ipynb | 323 +++++++++
main.tex | 598 ++--------------
plots_and_expressions.ipynb | 1307 +++++++++++++++++++++++++++++++++++
6 files changed, 2027 insertions(+), 537 deletions(-)
create mode 100644 characteristic_curves.ipynb
create mode 100644 examples.ipynb
create mode 100644 plots_and_expressions.ipynb
diff --git a/.gitignore b/.gitignore
index 3348276..2b7bf79 100644
--- a/.gitignore
+++ b/.gitignore
@@ -1,7 +1,15 @@
main.*
!main.tex
+plots_and_expressions.*
+!plots_and_expressions.ipynb
+examples.*
+!examples.ipynb
+characteristic_curves.*
+!characteristic_curves.ipynb
filtered_sage.txt
_minted-main/*
sage-plots-for-main.tex
.vscode/*
sagetex.sty
+.ipynb_checkpoints
+__pycache__
diff --git a/Makefile b/Makefile
index 1a65c88..62560da 100644
--- a/Makefile
+++ b/Makefile
@@ -5,17 +5,35 @@ MAINTEXFILE = main.tex
TEXFILES = ${MAINTEXFILE}
SAGETEXSCRIPT = main.sagetex.sage
-main.pdf: ${TEXFILES} main.sagetex.sout.tmp filtered_sage.txt
+main.pdf: ${TEXFILES} main.sagetex.sout.tmp
latexmk
-main.sagetex.sout.tmp: ${SAGETEXSCRIPT}
+main.sagetex.sout.tmp: ${SAGETEXSCRIPT} plots_and_expressions.py examples.py characteristic_curves.py
PYTHONPATH=./sagetexscripts/ sage ${SAGETEXSCRIPT}
-${SAGETEXSCRIPT}: ${TEXFILES} filtered_sage.txt
+${SAGETEXSCRIPT}: ${TEXFILES}
latexmk || echo this shoud fail
-filtered_sage.txt: ${MAINTEXFILE} filter_sage.sed
- ./filter_sage.sed ${MAINTEXFILE} > $@
+plots_and_expressions.py: plots_and_expressions.ipynb
+ jupyter nbconvert --to script plots_and_expressions.ipynb
+ mv plots_and_expressions.py plots_and_expressions.sage
+ sed -e "/get_ipython/d" -i plots_and_expressions.sage
+ sage --preparse plots_and_expressions.sage
+ mv plots_and_expressions.sage.py plots_and_expressions.py
+
+examples.py: plots_and_expressions.py examples.ipynb
+ jupyter nbconvert --to script examples.ipynb
+ mv examples.py examples.sage
+ sed -e "/get_ipython/d" -i examples.sage
+ sage --preparse examples.sage
+ mv examples.sage.py examples.py
+
+characteristic_curves.py: characteristic_curves.ipynb
+ jupyter nbconvert --to script characteristic_curves.ipynb
+ mv characteristic_curves.py characteristic_curves.sage
+ sed -e "/get_ipython/d" -i characteristic_curves.sage
+ sage --preparse characteristic_curves.sage
+ mv characteristic_curves.sage.py characteristic_curves.py
.PHONY: clean nosage noappendix
clean:
diff --git a/characteristic_curves.ipynb b/characteristic_curves.ipynb
new file mode 100644
index 0000000..990b233
--- /dev/null
+++ b/characteristic_curves.ipynb
@@ -0,0 +1,300 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "39a3ff68",
+ "metadata": {},
+ "source": [
+ "# Utilities"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "b87a49bc",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Requires extra package:\n",
+ "#! sage -pip install \"pseudowalls==0.0.3\" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple\n",
+ "%display latex\n",
+ "\n",
+ "from pseudowalls import *\n",
+ "\n",
+ "Δ = lambda v: v.Q_tilt()\n",
+ "mu = stability.Mumford().slope\n",
+ "ts = stability.Tilt\n",
+ "\n",
+ "var(\"beta\", domain=\"real\")\n",
+ "\n",
+ "def beta_minus(v):\n",
+ " beta = stability.Tilt().beta\n",
+ " solutions = solve(\n",
+ " stability.Tilt(alpha=0).degree(v)==0,\n",
+ " beta)\n",
+ " return min(map(lambda s: s.rhs(), solutions))\n",
+ "\n",
+ "class Object(object):\n",
+ " pass"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "5da7dba0",
+ "metadata": {},
+ "source": [
+ "# Characteristic Curves Plots"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "5e309df3",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def charact_curves(v):\n",
+ " alpha = stability.Tilt().alpha\n",
+ " beta = stability.Tilt().beta\n",
+ " coords_range = (beta, -4, 5), (alpha, 0, 4)\n",
+ " text_args = {\"fontsize\":\"xx-large\", \"clip\":True}\n",
+ " black_text_args = {\"rgbcolor\": \"black\", **text_args}\n",
+ " p = (\n",
+ " implicit_plot(stability.Tilt().degree(v), *coords_range )\n",
+ " + line([(mu(v),0),(mu(v),5)], linestyle = \"dashed\")\n",
+ " + text(r\"$\\Theta_v^+$\",[3.5, 2], rotation=45, **text_args)\n",
+ " + text(r\"$V_v$\", [0.43, 1.5], rotation=90, **text_args)\n",
+ " + text(r\"$\\Theta_v^-$\", [-2.2, 2], rotation=-45, **text_args)\n",
+ " + text(r\"$\\nu_{\\alpha, \\beta}(v)>0$\", [-3, 1], **black_text_args)\n",
+ " + text(r\"$\\nu_{\\alpha, \\beta}(v)<0$\", [-1, 3], **black_text_args)\n",
+ " + text(r\"$\\nu_{\\alpha, \\beta}(-v)>0$\", [2, 3], **black_text_args)\n",
+ " + text(r\"$\\nu_{\\alpha, \\beta}(-v)<0$\", [4, 1], **black_text_args)\n",
+ " )\n",
+ " p.xmax(5)\n",
+ " p.xmin(-4)\n",
+ " p.ymax(4)\n",
+ " p.axes_labels([r\"$\\beta$\", r\"$\\alpha$\"])\n",
+ " return p\n",
+ "\n",
+ "v1 = Chern_Char(3, 2, -2)\n",
+ "v2 = Chern_Char(3, 2, 2/3)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "00428146",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "Graphics object consisting of 9 graphics primitives"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "typical_characteristic_curves = charact_curves(v1)\n",
+ "typical_characteristic_curves"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "id": "51d49b55",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "Graphics object consisting of 9 graphics primitives"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "degenerate_characteristic_curves = charact_curves(v2)\n",
+ "degenerate_characteristic_curves"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "4f77e7d6",
+ "metadata": {},
+ "source": [
+ "## Characteristic Curves for Pseudo-semistabilizers"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "87b4eeeb",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "def hyperbola_intersection_plot():\n",
+ " var(\"alpha beta\", domain=\"real\")\n",
+ " coords_range = (beta, -3, -1/2), (alpha, 0, 2.5)\n",
+ " delta1 = -sqrt(2)+1/100\n",
+ " delta2 = 1/2\n",
+ " pbeta=-1.5\n",
+ " text_args = {\"fontsize\":\"large\", \"clip\":True}\n",
+ " black_text_args = {\"rgbcolor\":\"black\", **text_args}\n",
+ " p = (\n",
+ " implicit_plot( beta^2 - alpha^2 == 2,\n",
+ " *coords_range , rgbcolor = \"black\", legend_label=r\"a\")\n",
+ " + implicit_plot( (beta+4)^2 - (alpha)^2 == 2,\n",
+ " *coords_range , rgbcolor = \"red\")\n",
+ " + implicit_plot( (beta+delta1)^2 - alpha^2 == (delta1-2)^2-2,\n",
+ " *coords_range , rgbcolor = \"blue\")\n",
+ " + implicit_plot( (beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,\n",
+ " *coords_range , rgbcolor = \"green\")\n",
+ " + point([-2, sqrt(2)], size=50, rgbcolor=\"black\", zorder=50)\n",
+ " + text(\"Q\",[-2, sqrt(2)+0.1], **black_text_args)\n",
+ " + point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor=\"black\", zorder=50)\n",
+ " + text(\"P\",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)\n",
+ " + circle((-2,0),sqrt(2), linestyle=\"dashed\", rgbcolor=\"purple\")\n",
+ " # dummy lines to add legends (circumvent bug in implicit_plot)\n",
+ " + line([(2,0),(2,0)] , rgbcolor = \"purple\", linestyle=\"dotted\",\n",
+ " legend_label=r\"pseudo-wall\")\n",
+ " + line([(2,0),(2,0)] , rgbcolor = \"black\",\n",
+ " legend_label=r\"$\\Theta_v^-$\")\n",
+ " + line([(2,0),(2,0)] , rgbcolor = \"red\", legend_label=r\"$\\Theta_u$ case 1\")\n",
+ " + line([(2,0),(2,0)] , rgbcolor = \"blue\", legend_label=r\"$\\Theta_u$ case 2\")\n",
+ " + line([(2,0),(2,0)] , rgbcolor = \"green\", legend_label=r\"$\\Theta_u$ case 3\")\n",
+ " )\n",
+ " p.set_legend_options(loc=\"upper right\", font_size=\"x-large\",\n",
+ " font_family=\"serif\")\n",
+ " p.xmax(coords_range[0][2])\n",
+ " p.xmin(coords_range[0][1])\n",
+ " p.ymax(coords_range[1][2])\n",
+ " p.ymin(coords_range[1][1])\n",
+ " p.axes_labels([r\"$\\beta$\", r\"$\\alpha$\"])\n",
+ " return p\n",
+ "\n",
+ "def correct_hyperbola_intersection_plot():\n",
+ " var(\"alpha beta\", domain=\"real\")\n",
+ " coords_range = (beta, -2.5, 0.5), (alpha, 0, 3)\n",
+ " delta2 = 1/2\n",
+ " pbeta=-1.5\n",
+ " text_args = {\"fontsize\":\"large\", \"clip\":True}\n",
+ " black_text_args = {\"rgbcolor\":\"black\", **text_args}\n",
+ " p = (\n",
+ " implicit_plot( beta^2 - alpha^2 == 2,\n",
+ " *coords_range , rgbcolor = \"black\", legend_label=r\"a\")\n",
+ " + implicit_plot((beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,\n",
+ " *coords_range , rgbcolor = \"green\")\n",
+ " + point([-2, sqrt(2)], size=50, rgbcolor=\"black\", zorder=50)\n",
+ " + text(\"Q\",[-2, sqrt(2)+0.1], **black_text_args)\n",
+ " + point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor=\"black\", zorder=50)\n",
+ " + text(\"P\",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)\n",
+ " + circle((-2,0),sqrt(2), linestyle=\"dashed\", rgbcolor=\"purple\")\n",
+ " # dummy lines to add legends (circumvent bug in implicit_plot)\n",
+ " + line([(2,0),(2,0)] , rgbcolor = \"purple\", linestyle=\"dotted\",\n",
+ " legend_label=r\"pseudo-wall\")\n",
+ " + line([(2,0),(2,0)] , rgbcolor = \"black\",\n",
+ " legend_label=r\"$\\Theta_v^-$\")\n",
+ " + line([(2,0),(2,0)] , rgbcolor = \"green\",\n",
+ " legend_label=r\"$\\Theta_u^-$\")\n",
+ " # vertical characteristic lines\n",
+ " + line([(0,0),(0,coords_range[1][2])],\n",
+ " rgbcolor=\"black\", linestyle=\"dashed\",\n",
+ " legend_label=r\"$V_v$\")\n",
+ " + line([(-delta2,0),(-delta2,coords_range[1][2])],\n",
+ " rgbcolor=\"green\", linestyle=\"dashed\",\n",
+ " legend_label=r\"$V_u$\")\n",
+ " + line([(-delta2,0),(-delta2-coords_range[1][2],coords_range[1][2])],\n",
+ " rgbcolor=\"green\", linestyle=\"dotted\",\n",
+ " legend_label=r\"$\\Theta_u^-$ assymptote\")\n",
+ " + line([(0,0),(-coords_range[1][2],coords_range[1][2])],\n",
+ " rgbcolor=\"black\", linestyle=\"dotted\",\n",
+ " legend_label=r\"$\\Theta_v^-$ assymptote\")\n",
+ " )\n",
+ " p.set_legend_options(loc=\"upper right\", font_size=\"x-large\",\n",
+ " font_family=\"serif\")\n",
+ " p.xmax(coords_range[0][2])\n",
+ " p.xmin(coords_range[0][1])\n",
+ " p.ymax(coords_range[1][2])\n",
+ " p.ymin(coords_range[1][1])\n",
+ " p.axes_labels([r\"$\\beta$\", r\"$\\alpha$\"])\n",
+ " return p"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "id": "ff33752c",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "Graphics object consisting of 14 graphics primitives"
+ ]
+ },
+ "execution_count": 6,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "hyperbola_intersection_plot()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "6068850f",
+ "metadata": {
+ "scrolled": true
+ },
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "Graphics object consisting of 14 graphics primitives"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "correct_hyperbola_intersection_plot()"
+ ]
+ }
+ ],
+ "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
+}
diff --git a/examples.ipynb b/examples.ipynb
new file mode 100644
index 0000000..db3873b
--- /dev/null
+++ b/examples.ipynb
@@ -0,0 +1,323 @@
+{
+ "cells": [
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "abe149f0",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Requires extra package:\n",
+ "#! sage -pip install \"pseudowalls==0.0.3\" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple\n",
+ "%display latex\n",
+ "\n",
+ "from pseudowalls import *\n",
+ "\n",
+ "Δ = lambda v: v.Q_tilt()\n",
+ "mu = stability.Mumford().slope\n",
+ "ts = stability.Tilt\n",
+ "\n",
+ "var(\"beta\", domain=\"real\")\n",
+ "\n",
+ "def beta_minus(v):\n",
+ " beta = stability.Tilt().beta\n",
+ " solutions = solve(\n",
+ " stability.Tilt(alpha=0).degree(v)==0,\n",
+ " beta)\n",
+ " return min(map(lambda s: s.rhs(), solutions))\n",
+ "\n",
+ "class Object(object):\n",
+ " pass"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "77fff07c",
+ "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": 3,
+ "id": "9edbb954",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle 144\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle 144$"
+ ],
+ "text/plain": [
+ "144"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "recurring.loose_bound"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "id": "21b52c4d",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle 215296\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle 215296$"
+ ],
+ "text/plain": [
+ "215296"
+ ]
+ },
+ "execution_count": 4,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "extravagant.loose_bound"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "712b7324",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "from plots_and_expressions import r_upper_bound_all_q, Delta, nu, n, R"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "id": "ab9a828f",
+ "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": 7,
+ "id": "2965357d",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "extravagant.n = extravagant.b.denominator()\n",
+ "extravagant.bgmlv = extravagant.chern.Q_tilt()\n",
+ "extravagant.corrolary_bound = (r_upper_bound_all_q\n",
+ " .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": 8,
+ "id": "5baed51c",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle 53838.5009765625\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle 53838.5009765625$"
+ ],
+ "text/plain": [
+ "53838.5009765625"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "float(extravagant.corrolary_bound)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "id": "fcc15f60",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle 37.515625\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle 37.515625$"
+ ],
+ "text/plain": [
+ "37.515625"
+ ]
+ },
+ "execution_count": 9,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "float(recurring.corrolary_bound)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "id": "ef8f09ff",
+ "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",
+ " 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"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "id": "9cd102ac",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\begin{array}{l}\n",
+ "\\verb|[[0|\\verb| |\\verb|1/3|\\verb| |\\verb|2/3|\\verb| |\\verb|1|\\verb| |\\verb|4/3|\\verb| |\\verb|5/3|\\verb| |\\verb|2|\\verb| |\\verb|7/3|\\verb| |\\verb|8/3|\\verb| |\\verb|3|\\verb| |\\verb|10/3|\\verb| |\\verb|11/3|\\verb| |\\verb|4]|\\\\\n",
+ "\\verb| |\\verb|[0|\\verb| |\\verb|1|\\verb| |\\verb|4|\\verb| |\\verb|9|\\verb| |\\verb|16|\\verb| |\\verb|25|\\verb| |\\verb|36|\\verb| |\\verb|28|\\verb| |\\verb|19|\\verb| |\\verb|12|\\verb| |\\verb|7|\\verb| |\\verb|4|\\verb| |\\verb|3]|\\\\\n",
+ "\\verb| |\\verb|[0|\\verb| |\\verb|0|\\verb| |\\verb|4|\\verb| |\\verb|3|\\verb| |\\verb|8|\\verb| |\\verb|25|\\verb| |\\verb|12|\\verb| |\\verb|15|\\verb| |\\verb|19|\\verb| |\\verb|6|\\verb| |\\verb|5|\\verb| |\\verb|4|\\verb| |\\verb|3]]|\n",
+ "\\end{array}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\begin{array}{l}\n",
+ "\\verb|[[0|\\verb| |\\verb|1/3|\\verb| |\\verb|2/3|\\verb| |\\verb|1|\\verb| |\\verb|4/3|\\verb| |\\verb|5/3|\\verb| |\\verb|2|\\verb| |\\verb|7/3|\\verb| |\\verb|8/3|\\verb| |\\verb|3|\\verb| |\\verb|10/3|\\verb| |\\verb|11/3|\\verb| |\\verb|4]|\\\\\n",
+ "\\verb| |\\verb|[0|\\verb| |\\verb|1|\\verb| |\\verb|4|\\verb| |\\verb|9|\\verb| |\\verb|16|\\verb| |\\verb|25|\\verb| |\\verb|36|\\verb| |\\verb|28|\\verb| |\\verb|19|\\verb| |\\verb|12|\\verb| |\\verb|7|\\verb| |\\verb|4|\\verb| |\\verb|3]|\\\\\n",
+ "\\verb| |\\verb|[0|\\verb| |\\verb|0|\\verb| |\\verb|4|\\verb| |\\verb|3|\\verb| |\\verb|8|\\verb| |\\verb|25|\\verb| |\\verb|12|\\verb| |\\verb|15|\\verb| |\\verb|19|\\verb| |\\verb|6|\\verb| |\\verb|5|\\verb| |\\verb|4|\\verb| |\\verb|3]]|\n",
+ "\\end{array}$"
+ ],
+ "text/plain": [
+ "array([[0, 1/3, 2/3, 1, 4/3, 5/3, 2, 7/3, 8/3, 3, 10/3, 11/3, 4],\n",
+ " [0, 1, 4, 9, 16, 25, 36, 28, 19, 12, 7, 4, 3],\n",
+ " [0, 0, 4, 3, 8, 25, 12, 15, 19, 6, 5, 4, 3]], dtype=object)"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "np.array(bound_comparisons(recurring))"
+ ]
+ }
+ ],
+ "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
+}
diff --git a/main.tex b/main.tex
index 52636ef..0f6fef2 100644
--- a/main.tex
+++ b/main.tex
@@ -45,25 +45,7 @@ sorting=ynt
\begin{document}
-\begin{sagesilent}
-# Requires extra package:
-#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple
-
-from pseudowalls import *
-
-Δ = lambda v: v.Q_tilt()
-mu = stability.Mumford().slope
-def beta_minus(v):
- beta = stability.Tilt().beta
- solutions = solve(
- stability.Tilt(alpha=0).degree(v)==0,
- beta)
- return min(map(lambda s: s.rhs(), solutions))
-
-class Object(object):
- pass
-\end{sagesilent}
\title{Tighter Bounds for Ranks of Tilt Semistabilizers on Picard Rank 1 Surfaces
\\[1em] \large
@@ -330,46 +312,26 @@ Because of this, when using these characteristic curves, only positive ranks are
considered, as negative rank objects are implicitly considered on the right hand
side of $V_v$.
+
+
\begin{sagesilent}
-def charact_curves(v):
- alpha = stability.Tilt().alpha
- beta = stability.Tilt().beta
- coords_range = (beta, -4, 5), (alpha, 0, 4)
- text_args = {"fontsize":"xx-large", "clip":True}
- black_text_args = {"rgbcolor": "black", **text_args}
- p = (
- implicit_plot(stability.Tilt().degree(v), *coords_range )
- + line([(mu(v),0),(mu(v),5)], linestyle = "dashed")
- + text(r"$\Theta_v^+$",[3.5, 2], rotation=45, **text_args)
- + text(r"$V_v$", [0.43, 1.5], rotation=90, **text_args)
- + text(r"$\Theta_v^-$", [-2.2, 2], rotation=-45, **text_args)
- + text(r"$\nu_{\alpha, \beta}(v)>0$", [-3, 1], **black_text_args)
- + text(r"$\nu_{\alpha, \beta}(v)<0$", [-1, 3], **black_text_args)
- + text(r"$\nu_{\alpha, \beta}(-v)>0$", [2, 3], **black_text_args)
- + text(r"$\nu_{\alpha, \beta}(-v)<0$", [4, 1], **black_text_args)
- )
- p.xmax(5)
- p.xmin(-4)
- p.ymax(4)
- p.axes_labels([r"$\beta$", r"$\alpha$"])
- return p
-
-v1 = Chern_Char(3, 2, -2)
-v2 = Chern_Char(3, 2, 2/3)
+from characteristic_curves import \
+typical_characteristic_curves, \
+degenerate_characteristic_curves
\end{sagesilent}
\begin{figure}
\centering
\begin{subfigure}{.49\textwidth}
\centering
- \sageplot[width=\textwidth]{charact_curves(v1)}
+ \sageplot[width=\textwidth]{typical_characteristic_curves}
\caption{$\Delta(v)>0$}
\label{fig:charact_curves_vis_bgmvlPos}
\end{subfigure}%
\hfill
\begin{subfigure}{.49\textwidth}
\centering
- \sageplot[width=\textwidth]{charact_curves(v2)}
+ \sageplot[width=\textwidth]{degenerate_characteristic_curves}
\caption{
$\Delta(v)=0$: hyperbola collapses
}
@@ -484,92 +446,9 @@ Recalling how the sign of $\nu_{\alpha,\beta}(\pm u)$ changes (illustrated in
Fig \ref{fig:charact_curves_vis}), we can eliminate cases 1 and 2.
\begin{sagesilent}
-def hyperbola_intersection_plot():
- var("alpha beta", domain="real")
- coords_range = (beta, -3, -1/2), (alpha, 0, 2.5)
- delta1 = -sqrt(2)+1/100
- delta2 = 1/2
- pbeta=-1.5
- text_args = {"fontsize":"large", "clip":True}
- black_text_args = {"rgbcolor":"black", **text_args}
- p = (
- implicit_plot( beta^2 - alpha^2 == 2,
- *coords_range , rgbcolor = "black", legend_label=r"a")
- + implicit_plot( (beta+4)^2 - (alpha)^2 == 2,
- *coords_range , rgbcolor = "red")
- + implicit_plot( (beta+delta1)^2 - alpha^2 == (delta1-2)^2-2,
- *coords_range , rgbcolor = "blue")
- + implicit_plot( (beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,
- *coords_range , rgbcolor = "green")
- + point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
- + text("Q",[-2, sqrt(2)+0.1], **black_text_args)
- + point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
- + text("P",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)
- + circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
- # dummy lines to add legends (circumvent bug in implicit_plot)
- + line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted",
- legend_label=r"pseudo-wall")
- + line([(2,0),(2,0)] , rgbcolor = "black",
- legend_label=r"$\Theta_v^-$")
- + line([(2,0),(2,0)] , rgbcolor = "red", legend_label=r"$\Theta_u$ case 1")
- + line([(2,0),(2,0)] , rgbcolor = "blue", legend_label=r"$\Theta_u$ case 2")
- + line([(2,0),(2,0)] , rgbcolor = "green", legend_label=r"$\Theta_u$ case 3")
- )
- p.set_legend_options(loc="upper right", font_size="x-large",
- font_family="serif")
- p.xmax(coords_range[0][2])
- p.xmin(coords_range[0][1])
- p.ymax(coords_range[1][2])
- p.ymin(coords_range[1][1])
- p.axes_labels([r"$\beta$", r"$\alpha$"])
- return p
-
-def correct_hyperbola_intersection_plot():
- var("alpha beta", domain="real")
- coords_range = (beta, -2.5, 0.5), (alpha, 0, 3)
- delta2 = 1/2
- pbeta=-1.5
- text_args = {"fontsize":"large", "clip":True}
- black_text_args = {"rgbcolor":"black", **text_args}
- p = (
- implicit_plot( beta^2 - alpha^2 == 2,
- *coords_range , rgbcolor = "black", legend_label=r"a")
- + implicit_plot((beta+delta2)^2 - alpha^2 == (delta2-2)^2-2,
- *coords_range , rgbcolor = "green")
- + point([-2, sqrt(2)], size=50, rgbcolor="black", zorder=50)
- + text("Q",[-2, sqrt(2)+0.1], **black_text_args)
- + point([pbeta, sqrt(pbeta^2-2)], size=50, rgbcolor="black", zorder=50)
- + text("P",[pbeta+0.1, sqrt(pbeta^2-2)], **black_text_args)
- + circle((-2,0),sqrt(2), linestyle="dashed", rgbcolor="purple")
- # dummy lines to add legends (circumvent bug in implicit_plot)
- + line([(2,0),(2,0)] , rgbcolor = "purple", linestyle="dotted",
- legend_label=r"pseudo-wall")
- + line([(2,0),(2,0)] , rgbcolor = "black",
- legend_label=r"$\Theta_v^-$")
- + line([(2,0),(2,0)] , rgbcolor = "green",
- legend_label=r"$\Theta_u^-$")
- # vertical characteristic lines
- + line([(0,0),(0,coords_range[1][2])],
- rgbcolor="black", linestyle="dashed",
- legend_label=r"$V_v$")
- + line([(-delta2,0),(-delta2,coords_range[1][2])],
- rgbcolor="green", linestyle="dashed",
- legend_label=r"$V_u$")
- + line([(-delta2,0),(-delta2-coords_range[1][2],coords_range[1][2])],
- rgbcolor="green", linestyle="dotted",
- legend_label=r"$\Theta_u^-$ assymptote")
- + line([(0,0),(-coords_range[1][2],coords_range[1][2])],
- rgbcolor="black", linestyle="dotted",
- legend_label=r"$\Theta_v^-$ assymptote")
- )
- p.set_legend_options(loc="upper right", font_size="x-large",
- font_family="serif")
- p.xmax(coords_range[0][2])
- p.xmin(coords_range[0][1])
- p.ymax(coords_range[1][2])
- p.ymin(coords_range[1][1])
- p.axes_labels([r"$\beta$", r"$\alpha$"])
- return p
+from characteristic_curves import \
+hyperbola_intersection_plot, \
+correct_hyperbola_intersection_plot
\end{sagesilent}
\begin{figure}
@@ -805,9 +684,7 @@ The restrictions on $\chern^{\beta_-}_0(E)$ and $\chern^{\beta_-}_2(E)$
is best seen with the following graph:
% TODO: hyperbola restriction graph (shaded)
-\begin{sagesilent}
-var("m") # Initialize symbol for variety parameter
-\end{sagesilent}
+
This is where the rationality of $\beta_{-}$ comes in. If
$\beta_{-} = \frac{a_v}{n}$ for some $a_v,n \in \ZZ$. Then
@@ -826,56 +703,37 @@ bound for the rank of $E$:
\end{proof}
-\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
-\label{exmpl:recurring-first}
\begin{sagesilent}
-recurring = Object()
-recurring.chern = Chern_Char(3, 2, -2)
-recurring.b = beta_minus(recurring.chern)
-recurring.twisted = recurring.chern.twist(recurring.b)
+from examples import recurring
\end{sagesilent}
+
+\begin{example}[$v=(3, 2\ell, -2)$ on $\PP^2$]
+\label{exmpl:recurring-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta_-=\sage{recurring.b}$,
giving $n=\sage{recurring.b.denominator()}$ and
$\chern_1^{\sage{recurring.b}}(F) = \sage{recurring.twisted.ch[1]}$.
-\begin{sagesilent}
-n = recurring.b.denominator()
-m = 2
-loose_bound = (
- m*n^2*recurring.twisted.ch[1]^2
-) / gcd(m, 2*n^2)
-\end{sagesilent}
Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
-tilt semistabilizers for $v$ are bounded above by $\sage{loose_bound}$.
+tilt semistabilizers for $v$ are bounded above by $\sage{recurring.loose_bound}$.
However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be 25. This will be a recurring example to
illustrate the performance of later theorems about rank bounds
\end{example}
-\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
-\label{exmpl:extravagant-first}
\begin{sagesilent}
-extravagant = Object()
-extravagant.chern = Chern_Char(29, 13, -3/2)
-extravagant.b = beta_minus(extravagant.chern)
-extravagant.twisted = extravagant.chern.twist(extravagant.b)
-extravagant.actual_rmax = 49313
+from examples import extravagant
\end{sagesilent}
+
+\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
+\label{exmpl:extravagant-first}
Taking $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta_-=\sage{extravagant.b}$,
giving $n=\sage{extravagant.b.denominator()}$ and
$\chern_1^{\sage{extravagant.b}}(F) = \sage{extravagant.twisted.ch[1]}$.
-\begin{sagesilent}
-n = extravagant.b.denominator()
-m = 2
-loose_bound = (
- m*n^2*extravagant.twisted.ch[1]^2
-) / gcd(m, 2*n^2)
-\end{sagesilent}
Using the above theorem \ref{thm:loose-bound-on-r}, we get that the ranks of
-tilt semistabilizers for $v$ are bounded above by $\sage{loose_bound}$.
+tilt semistabilizers for $v$ are bounded above by $\sage{extravagant.loose_bound}$.
However, when computing all tilt semistabilizers for $v$ on $\PP^2$, the maximum
rank that appears turns out to be $\sage{extravagant.actual_rmax}$.
\end{example}
@@ -984,17 +842,7 @@ Take $\beta = \beta(P)$ where $P\in\Theta_v^-$ is the choice made in problem
&& \text{where $r,c,2d\in \ZZ$}
\end{align}
-\begin{sagesilent}
-# Requires extra package:
-#! sage -pip install "pseudowalls==0.0.3" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple
-
-from pseudowalls import *
-v = Chern_Char(*var("R C D", domain="real"))
-u = Chern_Char(*var("r c d", domain="real"))
-
-Δ = lambda v: v.Q_tilt()
-\end{sagesilent}
Recall from condition \ref{item:chern1bound:lem:num_test_prob1} in
lemma \ref{lem:num_test_prob1}
@@ -1002,17 +850,10 @@ lemma \ref{lem:num_test_prob1}
that $\chern_1^{\beta}(u)$ has fixed bounds in terms of $\chern_1^{\beta}(v)$,
and so we can write:
-\begin{sagesilent}
-ts = stability.Tilt
-var("beta", domain="real")
-c_lower_bound = -(
- ts(beta=beta).rank(u)
- /ts().alpha
-).expand() + c
-var("q", domain="real")
-c_in_terms_of_q = c_lower_bound + q
+\begin{sagesilent}
+from plots_and_expressions import c_in_terms_of_q
\end{sagesilent}
\begin{equation}
@@ -1066,21 +907,13 @@ from lemma \ref{lem:num_test_prob1}
(or corollary \ref{cor:num_test_prob2}).
-\begin{sagesilent}
-# First Bogomolov-Gieseker form expression that must be non-negative:
-bgmlv2 = Δ(u)
-\end{sagesilent}
-
\noindent
Expressing $\Delta(u)\geq 0$ in terms of $q$ as defined in eqn \ref{eqn-cintermsofm}
we get the following:
+
\begin{sagesilent}
-bgmlv2_with_q = (
- bgmlv2
- .expand()
- .subs(c == c_in_terms_of_q)
-)
+from plots_and_expressions import bgmlv2_with_q
\end{sagesilent}
\begin{equation}
@@ -1094,45 +927,21 @@ This can be rearranged to express a bound on $d$ as follows
in lemma \ref{lem:num_test_prob1} or corollary
\ref{cor:num_test_prob2} that $r>0$):
-\begin{sagesilent}
-bgmlv2_d_ineq = (
- (0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0
- + d # Rearrange for d
-).expand()
-# Keep hold of lower bound for d
-bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs()
+\begin{sagesilent}
+from plots_and_expressions import bgmlv2_d_ineq
\end{sagesilent}
-
\begin{equation}
\label{eqn-bgmlv2_d_upperbound}
\sage{bgmlv2_d_ineq}
\end{equation}
\begin{sagesilent}
-# Seperate out the terms of the lower bound for d
-
-bgmlv2_d_upperbound_without_hyp = (
- bgmlv2_d_upperbound
- .subs(1/r == 0)
-)
-
-bgmlv2_d_upperbound_const_term = (
- bgmlv2_d_upperbound_without_hyp
- .subs(r==0)
-)
-
-bgmlv2_d_upperbound_linear_term = (
- bgmlv2_d_upperbound_without_hyp
- - bgmlv2_d_upperbound_const_term
-).expand()
-
-bgmlv2_d_upperbound_exp_term = (
- bgmlv2_d_upperbound
- - bgmlv2_d_upperbound_without_hyp
-).expand()
+from plots_and_expressions import \
+bgmlv2_d_upperbound_const_term, \
+bgmlv2_d_upperbound_linear_term, \
+bgmlv2_d_upperbound_exp_term
\end{sagesilent}
-
Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
of $r$ again, there is a constant term
$\sage{bgmlv2_d_upperbound_const_term}$,
@@ -1163,99 +972,12 @@ from lemma \ref{lem:num_test_prob1}
Expressing $\Delta(v-u)\geq 0$ in term of $q$ and rearranging as a bound on
$d$ yields:
+
\begin{sagesilent}
-# Third Bogomolov-Gieseker form expression that must be non-negative:
-bgmlv3 = Δ(v-u)
-bgmlv3_with_q = (
- bgmlv3
- .expand()
- .subs(c == c_in_terms_of_q)
-)
-var("r_alt",domain="real") # r_alt = r - R temporary substitution
-
-bgmlv3_with_q_reparam = (
- bgmlv3_with_q
- .subs(r == r_alt + R)
- /r_alt # This operation assumes r_alt > 0
-).expand()
-
-bgmlv3_d_ineq = (
- ((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d
- .subs(r_alt == r - R) # Resubstitute r back in
- .expand()
-)
-
-# Check that this equation represents a bound for d
-assert bgmlv3_d_ineq.lhs() == d
-
-bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d
-
-# Seperate out the terms of the lower bound for d
-
-bgmlv3_d_upperbound_without_hyp = (
- bgmlv3_d_upperbound
- .subs(1/(R-r) == 0)
-)
-
-bgmlv3_d_upperbound_const_term = (
- bgmlv3_d_upperbound_without_hyp
- .subs(r==0)
-)
-
-bgmlv3_d_upperbound_linear_term = (
- bgmlv3_d_upperbound_without_hyp
- - bgmlv3_d_upperbound_const_term
-).expand()
-
-bgmlv3_d_upperbound_exp_term = (
- bgmlv3_d_upperbound
- - bgmlv3_d_upperbound_without_hyp
-).expand()
-
-# Verify the simplified forms of the terms that will be mentioned in text
-
-var("chb1v chb2v",domain="real") # symbol to represent ch_1^\beta(v)
-var("psi phi", domain="real") # symbol to represent ch_1^\beta(v) and
-# ch_2^\beta(v)
-
-assert bgmlv3_d_upperbound_const_term == (
- (
- # keep hold of this alternative expression:
- bgmlv3_d_upperbound_const_term_alt := (
- phi
- + beta*q
- )
- )
- .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
- .expand()
-)
-
-assert bgmlv3_d_upperbound_exp_term == (
- (
- # Keep hold of this alternative expression:
- bgmlv3_d_upperbound_exp_term_alt :=
- (
- R*phi
- + (C - q)^2/2
- + R*beta*q
- - D*R
- )/(r-R)
- )
- .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\beta(v)
- .expand()
-)
-
-assert bgmlv3_d_upperbound_exp_term == (
- (
- # Keep hold of this alternative expression:
- bgmlv3_d_upperbound_exp_term_alt2 :=
- (
- (psi - q)^2/2/(r-R)
- )
- )
- .subs(psi == v.twist(beta).ch[1]) # subs real val of ch_1^\beta(v)
- .expand()
-)
+from plots_and_expressions import \
+bgmlv3_d_upperbound_const_term_alt, \
+bgmlv3_d_upperbound_linear_term, \
+bgmlv3_d_upperbound_exp_term_alt2
\end{sagesilent}
\bgroup
@@ -1312,6 +1034,9 @@ These give bounds with the same assymptotes when we take $r\to\infty$
\let\originalbeta\beta
\renewcommand\beta{{\originalbeta_{-}}}
+\begin{sagesilent}
+from plots_and_expressions import phi
+\end{sagesilent}
\bgroup
% redefine \psi in sage expressions (placeholder for ch_1^\beta(F)
\def\psi{\chern_1^{\beta}(F)}
@@ -1340,111 +1065,25 @@ These give bounds with the same assymptotes when we take $r\to\infty$
\end{align}
\egroup
-\begin{sagesilent}
-positive_radius_condition = (
- (
- (0 > - u.twist(beta).ch[2])
- + d # rearrange for d
- )
- .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q
- .expand()
-)
-
-def beta_min(chern):
- ts = stability.Tilt()
- return min(
- map(
- lambda soln: soln.rhs(),
- solve(
- (ts.degree(chern))
- .expand()
- .subs(ts.alpha == 0),
- beta
- )
- )
- )
-
-v_example = Chern_Char(3,2,-2)
-q_example = 7/3
-
-def plot_d_bound(
- v_example,
- q_example,
- ymax=5,
- ymin=-2,
- xmax=20,
- aspect_ratio=None
-):
-
- # Equations to plot imminently representing the bounds on d:
- eq2 = (
- bgmlv2_d_upperbound
- .subs(R == v_example.ch[0])
- .subs(C == v_example.ch[1])
- .subs(D == v_example.ch[2])
- .subs(beta = beta_min(v_example))
- .subs(q == q_example)
- )
-
- eq3 = (
- bgmlv3_d_upperbound
- .subs(R == v_example.ch[0])
- .subs(C == v_example.ch[1])
- .subs(D == v_example.ch[2])
- .subs(beta = beta_min(v_example))
- .subs(q == q_example)
- )
-
- eq4 = (
- positive_radius_condition.rhs()
- .subs(q == q_example)
- .subs(beta = beta_min(v_example))
- )
-
- example_bounds_on_d_plot = (
- plot(
- eq3,
- (r,v_example.ch[0],xmax),
- color='green',
- linestyle = "dashed",
- legend_label=r"upper bound: $\Delta(v-u) \geq 0$",
- )
- + plot(
- eq2,
- (r,0,xmax),
- color='blue',
- linestyle = "dashed",
- legend_label=r"upper bound: $\Delta(u) \geq 0$"
- )
- + plot(
- eq4,
- (r,0,xmax),
- color='orange',
- linestyle = "dotted",
- legend_label=r"lower bound: $\mathrm{ch}_2^{\beta_{-}}(u)>0$"
- )
- )
- example_bounds_on_d_plot.ymin(ymin)
- example_bounds_on_d_plot.ymax(ymax)
- example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])
- if aspect_ratio:
- example_bounds_on_d_plot.set_aspect_ratio(aspect_ratio)
- return example_bounds_on_d_plot
+\begin{sagesilent}
+from plots_and_expressions import \
+bounds_on_d_qmin, \
+bounds_on_d_qmax
\end{sagesilent}
\begin{figure}
\centering
\begin{subfigure}{.45\textwidth}
\centering
- \sageplot[width=\linewidth]{plot_d_bound(v_example, 0, ymin=-0.5)}
+ \sageplot[width=\linewidth]{bounds_on_d_qmin}
\caption{$q = 0$ (all bounds other than green coincide on line)}
\label{fig:d_bounds_xmpl_min_q}
\end{subfigure}%
\hfill
\begin{subfigure}{.45\textwidth}
\centering
- \sageplot[width=\linewidth]{plot_d_bound(v_example, 4, ymin=-3, ymax=3)}
+ \sageplot[width=\linewidth]{bounds_on_d_qmax}
\caption{$q = \chern^{\beta}(F)$ (all bounds other than blue coincide on line)}
\label{fig:d_bounds_xmpl_max_q}
\end{subfigure}
@@ -1488,11 +1127,13 @@ As mentioned in the introduction (sec \ref{sec:intro}), the finiteness of these
solutions is entirely determined by whether $\beta$ is rational or irrational.
Some of the details around the associated numerics are explored next.
+\begin{sagesilent}
+from plots_and_expressions import typical_bounds_on_d
+\end{sagesilent}
+
\begin{figure}
\centering
-\sageplot[
- width=\linewidth
-]{plot_d_bound(v_example, 2, ymax=4, ymin=-2, aspect_ratio=1)}
+\sageplot[width=\linewidth]{typical_bounds_on_d}
\caption{
Bounds on $d\coloneqq\chern_2(E)$ in terms of $r\coloneqq \chern_0(E)$ for a fixed
value $\chern_1^{\beta}(F)/2$ of $q\coloneqq\chern_1^{\beta}(E)$.
@@ -1512,11 +1153,6 @@ $u=(r,c\ell,d\ell^2)$ to problem \ref{problem:problem-statement-2}.
The strategy here is similar to what was shown in theorem
\ref{thm:loose-bound-on-r}.
-\begin{sagesilent}
-var("a_v b_q n") # Define symbols introduce for values of beta and q
-beta_value_expr = (beta == a_v/n)
-q_value_expr = (q == b_q/n)
-\end{sagesilent}
\renewcommand{\aa}{{a_v}}
\newcommand{\bb}{{b_q}}
@@ -1537,6 +1173,12 @@ Substituting the current values of $q$ and $\beta$ into the condition for the
radius of the pseudo-wall being positive
(eqn \ref{eqn:radiuscond_d_bound_betamin}) we get:
+\begin{sagesilent}
+from plots_and_expressions import \
+positive_radius_condition, \
+q_value_expr, \
+beta_value_expr
+\end{sagesilent}
\begin{equation}
\label{eqn:positive_rad_condition_in_terms_of_q_beta}
\frac{1}{2}\ZZ
@@ -1548,28 +1190,10 @@ radius of the pseudo-wall being positive
\frac{1}{2n^2}\ZZ
\end{equation}
-\begin{sagesilent}
-# placeholder for the specific values of k (start with 1):
-var("kappa", domain="real")
-
-assymptote_gap_condition1 = (kappa/(2*n^2) < bgmlv2_d_upperbound_exp_term)
-assymptote_gap_condition2 = (kappa/(2*n^2) < bgmlv3_d_upperbound_exp_term_alt2)
-
-r_upper_bound1 = (
- assymptote_gap_condition1
- * r * 2*n^2 / kappa
-)
-
-assert r_upper_bound1.lhs() == r
-
-r_upper_bound2 = (
- assymptote_gap_condition2
- * (r-R) * 2*n^2 / kappa + R
-)
-assert r_upper_bound2.lhs() == r
+\begin{sagesilent}
+from plots_and_expressions import r_upper_bound1, r_upper_bound2, kappa
\end{sagesilent}
-
\begin{theorem}[Bound on $r$ \#1]
\label{thm:rmax_with_uniform_eps}
Let $v = (R,C\ell,D\ell^2)$ be a fixed Chern character. Then the ranks of the
@@ -1618,21 +1242,9 @@ considering equations
\let\originalepsilon\epsilon
\renewcommand\epsilon{{\originalepsilon_{v}}}
-\begin{sagesilent}
-var("epsilon")
-var("chbv") # symbol to represent \chern_1^{\beta}(v)
-
-# Tightness conditions:
-
-bounds_too_tight_condition1 = (
- bgmlv2_d_upperbound_exp_term
- < epsilon
-)
-bounds_too_tight_condition2 = (
- bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)
- < epsilon
-)
+\begin{sagesilent}
+from plots_and_expressions import bounds_too_tight_condition1, bounds_too_tight_condition2
\end{sagesilent}
\bgroup
@@ -1669,20 +1281,9 @@ This is equivalent to:
\end{proof}
+
\begin{sagesilent}
-var("Delta nu", domain="real")
-q_sol = solve(
- r_upper_bound1.subs(kappa==1).rhs()
- == r_upper_bound2.subs(kappa==1).rhs()
-, q)[0].rhs()
-r_upper_bound_all_q = (
- r_upper_bound1.rhs()
- .expand()
- .subs(q==q_sol)
- .subs(kappa==1)
- .subs(psi**2 == Delta/nu^2)
- .subs(1/psi**2 == nu^2/Delta)
-)
+from plots_and_expressions import r_upper_bound_all_q, q_sol, nu, Delta, psi
\end{sagesilent}
\begin{corollary}[Bound on $r$ \#2]
@@ -1740,22 +1341,12 @@ $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{recurring.b}$,
giving $n=\sage{recurring.b.denominator()}$.
-\begin{sagesilent}
-recurring.n = recurring.b.denominator()
-recurring.bgmlv = recurring.chern.Q_tilt()
-corrolary_bound = (
- r_upper_bound_all_q.expand()
- .subs(Delta==recurring.bgmlv)
- .subs(nu==1) ## \ell^2=1 on P^2
- .subs(R==recurring.chern.ch[0])
- .subs(n==recurring.n)
-)
-\end{sagesilent}
Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
-$\sage{corrolary_bound} \approx \sage{float(corrolary_bound)}$,
+$\sage{recurring.corrolary_bound} \approx \sage{float(recurring.corrolary_bound)}$,
which is much closer to real maximum 25 than the original bound 144.
\end{example}
+
\begin{example}[extravagant example: $v=(29, 13\ell, -3/2)$ on $\PP^2$]
\label{exmpl:extravagant-second}
Just like in example \ref{exmpl:extravagant-first}, take
@@ -1763,20 +1354,9 @@ $\ell=c_1(\mathcal{O}(1))$ as the standard polarization on $\PP^2$, so
that $m=2$, $\beta=\sage{extravagant.b}$,
giving $n=\sage{extravagant.b.denominator()}$.
-\begin{sagesilent}
-extravagant.n = extravagant.b.denominator()
-extravagant.bgmlv = extravagant.chern.Q_tilt()
-corrolary_bound = (
- r_upper_bound_all_q.expand()
- .subs(Delta==extravagant.bgmlv)
- .subs(nu==1) ## \ell^2=1 on P^2
- .subs(R==extravagant.chern.ch[0])
- .subs(n==extravagant.n)
-)
-\end{sagesilent}
Using the above corollary \ref{cor:direct_rmax_with_uniform_eps}, we get that
the ranks of tilt semistabilizers for $v$ are bounded above by
-$\sage{corrolary_bound} \approx \sage{float(corrolary_bound)}$,
+$\sage{extravagant.corrolary_bound} \approx \sage{float(extravagant.corrolary_bound)}$,
which is much closer to real maximum $\sage{extravagant.actual_rmax}$ than the
original bound 215296.
\end{example}
@@ -1814,15 +1394,6 @@ integral:
That is, $r \equiv -\aa^{-1}\bb$ mod $n$ ($\aa$ is coprime to
$n$, and so invertible mod $n$).
-\begin{sagesilent}
- rhs_numerator = (
- positive_radius_condition
- .rhs()
- .subs([q_value_expr,beta_value_expr])
- .factor()
- .numerator()
- )
-\end{sagesilent}
\noindent
Let $\aa^{'}$ be an integer representative of $\aa^{-1}$ in $\ZZ/n\ZZ$.
@@ -1921,10 +1492,6 @@ $\epsilon_{v,q}\geq\epsilon_v$, with equality when $k_{v,q}=1$.
$\chern_1^\beta(u) = q = \frac{b_q}{n}$
are bounded above by the following expression:
-\begin{sagesilent}
-var("delta", domain="real") # placeholder symbol to be replaced by k_{q,i}
-\end{sagesilent}
-
\bgroup
\def\kappa{k_{v,q}}
\def\psi{\chern_1^{\beta}(F)}
@@ -1956,40 +1523,7 @@ The (non-exclusive) upper bounds for $r\coloneqq\chern_0(u)$ of a tilt semistabi
in terms of the possible values for $q\coloneqq\chern_1^{\beta}(u)$ are as follows:
\begin{sagesilent}
-import numpy as np
-
-def bound_comparisons(example):
- n = example.b.denominator()
- a_v = example.b.numerator()
-
- def theorem_bound(v_twisted, q_val, k):
- return int(min(
- n^2*q_val^2/k
- ,
- v_twisted.ch[0]
- + n^2*(v_twisted.ch[1] - q_val)^2/k
- ))
-
- def k(n, a_v, b_q):
- n = int(n)
- a_v = int(a_v)
- b_q = int(b_q)
- k = -a_v*b_q % n
- return k if k > 0 else k + n
-
- b_qs = list(range(example.twisted.ch[1]*n+1))
- qs = list(map(lambda x: x/n,b_qs))
- ks = list(map(lambda b_q: k(n, a_v, b_q), b_qs))
- theorem2_bounds = [
- theorem_bound(example.twisted, q_val, 1)
- for q_val in qs
- ]
- theorem3_bounds = [
- theorem_bound(example.twisted, q_val, k)
- for q_val, k in zip(qs,ks)
- ]
- return qs, theorem2_bounds, theorem3_bounds
-
+from examples import bound_comparisons
qs, theorem2_bounds, theorem3_bounds = bound_comparisons(recurring)
\end{sagesilent}
diff --git a/plots_and_expressions.ipynb b/plots_and_expressions.ipynb
new file mode 100644
index 0000000..78c6aef
--- /dev/null
+++ b/plots_and_expressions.ipynb
@@ -0,0 +1,1307 @@
+{
+ "cells": [
+ {
+ "cell_type": "markdown",
+ "id": "8ebd0216",
+ "metadata": {},
+ "source": [
+ "# Utilities"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 1,
+ "id": "b87a49bc",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Requires extra package:\n",
+ "#! sage -pip install \"pseudowalls==0.0.3\" --extra-index-url https://gitlab.com/api/v4/projects/43962374/packages/pypi/simple\n",
+ "%display latex\n",
+ "\n",
+ "from pseudowalls import *\n",
+ "\n",
+ "Δ = lambda v: v.Q_tilt()\n",
+ "mu = stability.Mumford().slope\n",
+ "ts = stability.Tilt\n",
+ "\n",
+ "var(\"beta\", domain=\"real\")\n",
+ "\n",
+ "def beta_minus(v):\n",
+ " beta = stability.Tilt().beta\n",
+ " solutions = solve(\n",
+ " stability.Tilt(alpha=0).degree(v)==0,\n",
+ " beta)\n",
+ " return min(map(lambda s: s.rhs(), solutions))\n",
+ "\n",
+ "class Object(object):\n",
+ " pass"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "cb9c11e7",
+ "metadata": {},
+ "source": [
+ "# Bounds on $\\operatorname{ch}_2(u)=d$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 2,
+ "id": "4cacebc7",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "v = Chern_Char(*var(\"R C D\", domain=\"real\"))\n",
+ "u = Chern_Char(*var(\"r c d\", domain=\"real\"))\n",
+ "\n",
+ "alpha = ts().alpha\n",
+ "beta = ts().beta\n",
+ "\n",
+ "\n",
+ "c_lower_bound = -(\n",
+ " ts(beta=beta).rank(u)\n",
+ " /alpha\n",
+ ").expand() + c\n",
+ "\n",
+ "var(\"q\", domain=\"real\")\n",
+ "c_in_terms_of_q = c_lower_bound + q"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 3,
+ "id": "f53cee90",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\beta r + q\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\beta r + q$"
+ ],
+ "text/plain": [
+ "beta*r + q"
+ ]
+ },
+ "execution_count": 3,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "c_in_terms_of_q"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "900f332b",
+ "metadata": {},
+ "source": [
+ "## $\\Delta(u) \\geq 0$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 4,
+ "id": "6ae4f2e7",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# First Bogomolov-Gieseker form expression that must be non-negative:\n",
+ "bgmlv2 = Δ(u)\n",
+ "bgmlv2_with_q = (\n",
+ " bgmlv2\n",
+ " .expand()\n",
+ " .subs(c == c_in_terms_of_q)\n",
+ ")\n",
+ "# RENDERED TO LATEX: 0 <= bgmlv2_with_q\n",
+ "bgmlv2_d_ineq = (\n",
+ " (0 <= bgmlv2_with_q)/2/r # rescale assuming r > 0\n",
+ " + d # Rearrange for d\n",
+ ").expand()\n",
+ "\n",
+ "# Keep hold of lower bound for d\n",
+ "bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 5,
+ "id": "c0c12e43",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle d \\leq \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle d \\leq \\frac{1}{2} \\, \\beta^{2} r + \\beta q + \\frac{q^{2}}{2 \\, r}$"
+ ],
+ "text/plain": [
+ "d <= 1/2*beta^2*r + beta*q + 1/2*q^2/r"
+ ]
+ },
+ "execution_count": 5,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv2_d_ineq"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 6,
+ "id": "653a3340",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Seperate out the terms of the lower bound for d\n",
+ "\n",
+ "bgmlv2_d_upperbound_without_hyp = (\n",
+ " bgmlv2_d_upperbound\n",
+ " .subs(1/r == 0)\n",
+ ")\n",
+ "\n",
+ "bgmlv2_d_upperbound_const_term = (\n",
+ " bgmlv2_d_upperbound_without_hyp\n",
+ " .subs(r==0)\n",
+ ")\n",
+ "\n",
+ "bgmlv2_d_upperbound_linear_term = (\n",
+ " bgmlv2_d_upperbound_without_hyp\n",
+ " - bgmlv2_d_upperbound_const_term\n",
+ ").expand()\n",
+ "\n",
+ "bgmlv2_d_upperbound_exp_term = (\n",
+ " bgmlv2_d_upperbound\n",
+ " - bgmlv2_d_upperbound_without_hyp\n",
+ ").expand()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 7,
+ "id": "326bb656",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\beta q\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\beta q$"
+ ],
+ "text/plain": [
+ "beta*q"
+ ]
+ },
+ "execution_count": 7,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv2_d_upperbound_const_term"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 8,
+ "id": "fcd9fa2a",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
+ ],
+ "text/plain": [
+ "1/2*beta^2*r"
+ ]
+ },
+ "execution_count": 8,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv2_d_upperbound_linear_term"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 9,
+ "id": "ade0f7b2",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
+ ],
+ "text/plain": [
+ "1/2*q^2/r"
+ ]
+ },
+ "execution_count": 9,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv2_d_upperbound_exp_term"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "024e8c41",
+ "metadata": {},
+ "source": [
+ "## $\\Delta(v-u) \\geq 0$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 10,
+ "id": "b7799156",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "# Third Bogomolov-Gieseker form expression that must be non-negative:\n",
+ "bgmlv3 = Δ(v-u)\n",
+ "bgmlv3_with_q = (\n",
+ " bgmlv3\n",
+ " .expand()\n",
+ " .subs(c == c_in_terms_of_q)\n",
+ ")\n",
+ "var(\"r_alt\",domain=\"real\") # r_alt = r - R temporary substitution\n",
+ "\n",
+ "bgmlv3_with_q_reparam = (\n",
+ " bgmlv3_with_q\n",
+ " .subs(r == r_alt + R)\n",
+ " /r_alt # This operation assumes r_alt > 0\n",
+ ").expand()\n",
+ "\n",
+ "bgmlv3_d_ineq = (\n",
+ " ((0 <= bgmlv3_with_q_reparam)/2 + d) # Rearrange for d\n",
+ " .subs(r_alt == r - R) # Resubstitute r back in\n",
+ " .expand()\n",
+ ")\n",
+ "\n",
+ "# Check that this equation represents a bound for d\n",
+ "assert bgmlv3_d_ineq.lhs() == d\n",
+ "\n",
+ "bgmlv3_d_upperbound = bgmlv3_d_ineq.rhs() # Keep hold of lower bound for d\n",
+ "\n",
+ "# Seperate out the terms of the lower bound for d\n",
+ "\n",
+ "bgmlv3_d_upperbound_without_hyp = (\n",
+ " bgmlv3_d_upperbound\n",
+ " .subs(1/(R-r) == 0)\n",
+ ")\n",
+ "\n",
+ "bgmlv3_d_upperbound_const_term = (\n",
+ " bgmlv3_d_upperbound_without_hyp\n",
+ " .subs(r==0)\n",
+ ")\n",
+ "\n",
+ "bgmlv3_d_upperbound_linear_term = (\n",
+ " bgmlv3_d_upperbound_without_hyp\n",
+ " - bgmlv3_d_upperbound_const_term\n",
+ ").expand()\n",
+ "\n",
+ "bgmlv3_d_upperbound_exp_term = (\n",
+ " bgmlv3_d_upperbound\n",
+ " - bgmlv3_d_upperbound_without_hyp\n",
+ ").expand()\n",
+ "\n",
+ "# Verify the simplified forms of the terms that will be mentioned in text\n",
+ "\n",
+ "var(\"chb1v chb2v\",domain=\"real\") # symbol to represent ch_1^\\beta(v)\n",
+ "var(\"psi phi\", domain=\"real\") # symbol to represent ch_1^\\beta(v) and\n",
+ "# ch_2^\\beta(v)\n",
+ "\n",
+ "assert bgmlv3_d_upperbound_const_term == ( \n",
+ " (\n",
+ " # keep hold of this alternative expression:\n",
+ " bgmlv3_d_upperbound_const_term_alt := (\n",
+ " phi\n",
+ " + beta*q\n",
+ " )\n",
+ " )\n",
+ " .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\\beta(v)\n",
+ " .expand()\n",
+ ")\n",
+ "\n",
+ "assert bgmlv3_d_upperbound_exp_term == (\n",
+ " (\n",
+ " # Keep hold of this alternative expression:\n",
+ " bgmlv3_d_upperbound_exp_term_alt :=\n",
+ " (\n",
+ " R*phi\n",
+ " + (C - q)^2/2\n",
+ " + R*beta*q\n",
+ " - D*R\n",
+ " )/(r-R)\n",
+ " )\n",
+ " .subs(phi == v.twist(beta).ch[2]) # subs real val of ch_1^\\beta(v)\n",
+ " .expand()\n",
+ ")\n",
+ "\n",
+ "assert bgmlv3_d_upperbound_exp_term == (\n",
+ " (\n",
+ " # Keep hold of this alternative expression:\n",
+ " bgmlv3_d_upperbound_exp_term_alt2 :=\n",
+ " (\n",
+ " (psi - q)^2/2/(r-R)\n",
+ " )\n",
+ " )\n",
+ " .subs(psi == v.twist(beta).ch[1]) # subs real val of ch_1^\\beta(v)\n",
+ " .expand()\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 11,
+ "id": "38bbbcd2",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
+ ],
+ "text/plain": [
+ "1/2*beta^2*r"
+ ]
+ },
+ "execution_count": 11,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv3_d_upperbound_linear_term"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 12,
+ "id": "6701d8a7",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\beta q + \\phi\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\beta q + \\phi$"
+ ],
+ "text/plain": [
+ "beta*q + phi"
+ ]
+ },
+ "execution_count": 12,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv3_d_upperbound_const_term_alt"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 13,
+ "id": "7a82d747",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
+ ],
+ "text/plain": [
+ "-1/2*(psi - q)^2/(R - r)"
+ ]
+ },
+ "execution_count": 13,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv3_d_upperbound_exp_term_alt2"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 14,
+ "id": "927e6929",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
+ ],
+ "text/plain": [
+ "1/2*beta^2*r"
+ ]
+ },
+ "execution_count": 14,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv2_d_upperbound_linear_term"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 15,
+ "id": "ff25eea1",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\beta q\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\beta q$"
+ ],
+ "text/plain": [
+ "beta*q"
+ ]
+ },
+ "execution_count": 15,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv2_d_upperbound_const_term"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 16,
+ "id": "e8359c0f",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
+ ],
+ "text/plain": [
+ "1/2*q^2/r"
+ ]
+ },
+ "execution_count": 16,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv2_d_upperbound_exp_term"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 17,
+ "id": "6823c5b8",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{1}{2} \\, \\beta^{2} r\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{1}{2} \\, \\beta^{2} r$"
+ ],
+ "text/plain": [
+ "1/2*beta^2*r"
+ ]
+ },
+ "execution_count": 17,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv3_d_upperbound_linear_term"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 18,
+ "id": "160871f2",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\beta q\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\beta q$"
+ ],
+ "text/plain": [
+ "beta*q"
+ ]
+ },
+ "execution_count": 18,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv3_d_upperbound_const_term_alt.subs(phi == 0)"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 19,
+ "id": "1da23b29",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
+ ],
+ "text/plain": [
+ "-1/2*(psi - q)^2/(R - r)"
+ ]
+ },
+ "execution_count": 19,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv3_d_upperbound_exp_term_alt2"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "2997ec1a",
+ "metadata": {},
+ "source": [
+ "## Plots for all Bounds on $d$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 20,
+ "id": "d7235dc3",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "positive_radius_condition = (\n",
+ " (\n",
+ " (0 > - u.twist(beta).ch[2])\n",
+ " + d # rearrange for d\n",
+ " )\n",
+ " .subs(solve(q == u.twist(beta).ch[1], c)[0]) # express c in term of q\n",
+ " .expand()\n",
+ ")\n",
+ "\n",
+ "\n",
+ "v_example = Chern_Char(3,2,-2)\n",
+ "q_example = 7/3\n",
+ "\n",
+ "def plot_d_bound(\n",
+ " v_example,\n",
+ " q_example,\n",
+ " ymax=5,\n",
+ " ymin=-2,\n",
+ " xmax=20,\n",
+ " aspect_ratio=None):\n",
+ "\n",
+ " # Equations to plot imminently representing the bounds on d:\n",
+ " eq2 = (bgmlv2_d_upperbound\n",
+ " .subs(R == v_example.ch[0])\n",
+ " .subs(C == v_example.ch[1])\n",
+ " .subs(D == v_example.ch[2])\n",
+ " .subs(beta = beta_minus(v_example))\n",
+ " .subs(q == q_example)\n",
+ " )\n",
+ "\n",
+ " eq3 = (bgmlv3_d_upperbound\n",
+ " .subs(R == v_example.ch[0])\n",
+ " .subs(C == v_example.ch[1])\n",
+ " .subs(D == v_example.ch[2])\n",
+ " .subs(beta = beta_minus(v_example))\n",
+ " .subs(q == q_example)\n",
+ " )\n",
+ "\n",
+ " eq4 = (positive_radius_condition.rhs()\n",
+ " .subs(q == q_example)\n",
+ " .subs(beta = beta_minus(v_example))\n",
+ " )\n",
+ "\n",
+ " example_bounds_on_d_plot = (\n",
+ " plot(\n",
+ " eq3,\n",
+ " (r,v_example.ch[0],xmax),\n",
+ " color='green',\n",
+ " linestyle = \"dashed\",\n",
+ " legend_label=r\"upper bound: $\\Delta(v-u) \\geq 0$\",\n",
+ " )\n",
+ " + plot(\n",
+ " eq2,\n",
+ " (r,0,xmax),\n",
+ " color='blue',\n",
+ " linestyle = \"dashed\",\n",
+ " legend_label=r\"upper bound: $\\Delta(u) \\geq 0$\"\n",
+ " )\n",
+ " + plot(\n",
+ " eq4,\n",
+ " (r,0,xmax),\n",
+ " color='orange',\n",
+ " linestyle = \"dotted\",\n",
+ " legend_label=r\"lower bound: $\\mathrm{ch}_2^{\\beta_{-}}(u)>0$\"\n",
+ " )\n",
+ " )\n",
+ " example_bounds_on_d_plot.ymin(ymin)\n",
+ " example_bounds_on_d_plot.ymax(ymax)\n",
+ " example_bounds_on_d_plot.axes_labels(['$r$', '$d$'])\n",
+ " if aspect_ratio:\n",
+ " example_bounds_on_d_plot.set_aspect_ratio(aspect_ratio)\n",
+ " return example_bounds_on_d_plot"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "683ac3f7",
+ "metadata": {},
+ "source": [
+ "### Bounds on $d$ with Minimal $q=\\operatorname{ch}^{\\beta}_1(u)$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 21,
+ "id": "f5b1d9bf",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "Graphics object consisting of 3 graphics primitives"
+ ]
+ },
+ "execution_count": 21,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bounds_on_d_qmin = plot_d_bound(v_example, 0, ymin=-0.5)\n",
+ "bounds_on_d_qmin"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "24dd62c1",
+ "metadata": {},
+ "source": [
+ "### Bounds on $d$ with Maximal $q=\\operatorname{ch}^{\\beta}_1(u)$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 22,
+ "id": "47b30d7e",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "Graphics object consisting of 3 graphics primitives"
+ ]
+ },
+ "execution_count": 22,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bounds_on_d_qmax = plot_d_bound(v_example, 4, ymin=-3, ymax=3)\n",
+ "bounds_on_d_qmax"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "133ccbe7",
+ "metadata": {},
+ "source": [
+ "### Bounds on $d$ with Mid-way $q=\\operatorname{ch}^{\\beta}_1(u)$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 23,
+ "id": "25e4850b",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "image/png": 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\n",
+ "text/plain": [
+ "Graphics object consisting of 3 graphics primitives"
+ ]
+ },
+ "execution_count": 23,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "typical_bounds_on_d = plot_d_bound(v_example, 2, ymax=4, ymin=-2, aspect_ratio=1)\n",
+ "typical_bounds_on_d"
+ ]
+ },
+ {
+ "cell_type": "markdown",
+ "id": "1c6f5622",
+ "metadata": {},
+ "source": [
+ "# Bounds on Semistabilizer Rank $r=\\operatorname{ch}_0(u)$"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 24,
+ "id": "553bba31",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "var(\"a_v b_q n\") # Define symbols introduce for values of beta and q\n",
+ "beta_value_expr = (beta == a_v/n)\n",
+ "q_value_expr = (q == b_q/n)\n",
+ "# RENDERED TO LATEX: positive_radius_condition.subs([q_value_expr,beta_value_expr]).factor()\n",
+ "# placeholder for the specific values of k (start with 1):\n",
+ "var(\"kappa\", domain=\"real\")\n",
+ "\n",
+ "assymptote_gap_condition1 = (kappa/(2*n^2) < bgmlv2_d_upperbound_exp_term)\n",
+ "assymptote_gap_condition2 = (kappa/(2*n^2) < bgmlv3_d_upperbound_exp_term_alt2)\n",
+ "\n",
+ "r_upper_bound1 = (\n",
+ " assymptote_gap_condition1\n",
+ " * r * 2*n^2 / kappa\n",
+ ")\n",
+ "\n",
+ "assert r_upper_bound1.lhs() == r\n",
+ "\n",
+ "r_upper_bound2 = (\n",
+ " assymptote_gap_condition2\n",
+ " * (r-R) * 2*n^2 / kappa + R\n",
+ ")\n",
+ "\n",
+ "assert r_upper_bound2.lhs() == r"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 25,
+ "id": "990a2840",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle n^{2} q^{2}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle n^{2} q^{2}$"
+ ],
+ "text/plain": [
+ "n^2*q^2"
+ ]
+ },
+ "execution_count": 25,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "r_upper_bound1.subs(kappa==1).rhs()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 26,
+ "id": "5a3c7037",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle n^{2} {\\left(\\psi - q\\right)}^{2} + R\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle n^{2} {\\left(\\psi - q\\right)}^{2} + R$"
+ ],
+ "text/plain": [
+ "n^2*(psi - q)^2 + R"
+ ]
+ },
+ "execution_count": 26,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "r_upper_bound2.subs(kappa==1).rhs()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 27,
+ "id": "82add957",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "var(\"epsilon\")\n",
+ "var(\"chbv\") # symbol to represent \\chern_1^{\\beta}(v)\n",
+ "\n",
+ "# Tightness conditions:\n",
+ "\n",
+ "bounds_too_tight_condition1 = (\n",
+ " bgmlv2_d_upperbound_exp_term\n",
+ " < epsilon\n",
+ ")\n",
+ "\n",
+ "bounds_too_tight_condition2 = (\n",
+ " bgmlv3_d_upperbound_exp_term_alt.subs(chbv==0)\n",
+ " < epsilon\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 28,
+ "id": "d33e3459",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{q^{2}}{2 \\, r}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{q^{2}}{2 \\, r}$"
+ ],
+ "text/plain": [
+ "1/2*q^2/r"
+ ]
+ },
+ "execution_count": 28,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv2_d_upperbound_exp_term"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 29,
+ "id": "3728c192",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle -\\frac{{\\left(\\psi - q\\right)}^{2}}{2 \\, {\\left(R - r\\right)}}$"
+ ],
+ "text/plain": [
+ "-1/2*(psi - q)^2/(R - r)"
+ ]
+ },
+ "execution_count": 29,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "bgmlv3_d_upperbound_exp_term_alt2"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 30,
+ "id": "90149eb2",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "var(\"Delta nu\", domain=\"real\")\n",
+ "\n",
+ "q_sol = solve(\n",
+ " r_upper_bound1.subs(kappa==1).rhs()\n",
+ " == r_upper_bound2.subs(kappa==1).rhs()\n",
+ " , q\n",
+ ")[0].rhs()\n",
+ "\n",
+ "r_upper_bound_all_q = (r_upper_bound1.rhs()\n",
+ " .expand()\n",
+ " .subs(q==q_sol)\n",
+ " .subs(kappa==1)\n",
+ " .subs(psi**2 == Delta/nu^2)\n",
+ " .subs(1/psi**2 == nu^2/Delta)\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 31,
+ "id": "8b0b0bf4",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{1}{2} \\, R + \\frac{\\Delta n^{2}}{4 \\, \\nu^{2}} + \\frac{R^{2} \\nu^{2}}{4 \\, \\Delta n^{2}}$"
+ ],
+ "text/plain": [
+ "1/2*R + 1/4*Delta*n^2/nu^2 + 1/4*R^2*nu^2/(Delta*n^2)"
+ ]
+ },
+ "execution_count": 31,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "r_upper_bound_all_q.expand()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 32,
+ "id": "53bd2a9c",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle n^{2} q^{2}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle n^{2} q^{2}$"
+ ],
+ "text/plain": [
+ "n^2*q^2"
+ ]
+ },
+ "execution_count": 32,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "r_upper_bound1.subs(kappa==1).rhs()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 33,
+ "id": "cfa6e1af",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle n^{2} {\\left(\\psi - q\\right)}^{2} + R\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle n^{2} {\\left(\\psi - q\\right)}^{2} + R$"
+ ],
+ "text/plain": [
+ "n^2*(psi - q)^2 + R"
+ ]
+ },
+ "execution_count": 33,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "r_upper_bound2.subs(kappa==1).rhs()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 34,
+ "id": "801a348a",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{1}{2} \\, \\psi + \\frac{R}{2 \\, n^{2} \\psi}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{1}{2} \\, \\psi + \\frac{R}{2 \\, n^{2} \\psi}$"
+ ],
+ "text/plain": [
+ "1/2*psi + 1/2*R/(n^2*psi)"
+ ]
+ },
+ "execution_count": 34,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "q_sol.expand()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 35,
+ "id": "d4bf7486",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{1}{4} \\, n^{2} \\psi^{2} + \\frac{1}{2} \\, R + \\frac{R^{2}}{4 \\, n^{2} \\psi^{2}}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{1}{4} \\, n^{2} \\psi^{2} + \\frac{1}{2} \\, R + \\frac{R^{2}}{4 \\, n^{2} \\psi^{2}}$"
+ ],
+ "text/plain": [
+ "1/4*n^2*psi^2 + 1/2*R + 1/4*R^2/(n^2*psi^2)"
+ ]
+ },
+ "execution_count": 35,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "r_upper_bound_all_q.expand().subs([nu==1,Delta==psi^2])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 36,
+ "id": "896d26dd",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{a_{v} r}{n} + \\frac{b_{q}}{n}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{a_{v} r}{n} + \\frac{b_{q}}{n}$"
+ ],
+ "text/plain": [
+ "a_v*r/n + b_q/n"
+ ]
+ },
+ "execution_count": 36,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "c_in_terms_of_q.subs([q_value_expr,beta_value_expr])"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 37,
+ "id": "51f22f7d",
+ "metadata": {},
+ "outputs": [],
+ "source": [
+ "rhs_numerator = (positive_radius_condition\n",
+ " .rhs()\n",
+ " .subs([q_value_expr,beta_value_expr])\n",
+ " .factor()\n",
+ " .numerator()\n",
+ ")"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 38,
+ "id": "8148f5cd",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle d > \\frac{{\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}}{2 \\, n^{2}}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle d > \\frac{{\\left(a_{v} r + 2 \\, b_{q}\\right)} a_{v}}{2 \\, n^{2}}$"
+ ],
+ "text/plain": [
+ "d > 1/2*(a_v*r + 2*b_q)*a_v/n^2"
+ ]
+ },
+ "execution_count": 38,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "(positive_radius_condition\n",
+ " .subs([q_value_expr,beta_value_expr])\n",
+ " .factor())"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 39,
+ "id": "af5315c8",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\delta\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\delta$"
+ ],
+ "text/plain": [
+ "delta"
+ ]
+ },
+ "execution_count": 39,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "var(\"delta\", domain=\"real\") # placeholder symbol to be replaced by k_{q,i}"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 40,
+ "id": "e3e75309",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{n^{2} q^{2}}{\\kappa}\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{n^{2} q^{2}}{\\kappa}$"
+ ],
+ "text/plain": [
+ "n^2*q^2/kappa"
+ ]
+ },
+ "execution_count": 40,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "r_upper_bound1.rhs()"
+ ]
+ },
+ {
+ "cell_type": "code",
+ "execution_count": 41,
+ "id": "203b216b",
+ "metadata": {},
+ "outputs": [
+ {
+ "data": {
+ "text/html": [
+ "<html>\\(\\displaystyle \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R\\)</html>"
+ ],
+ "text/latex": [
+ "$\\displaystyle \\frac{n^{2} {\\left(\\psi - q\\right)}^{2}}{\\kappa} + R$"
+ ],
+ "text/plain": [
+ "n^2*(psi - q)^2/kappa + R"
+ ]
+ },
+ "execution_count": 41,
+ "metadata": {},
+ "output_type": "execute_result"
+ }
+ ],
+ "source": [
+ "r_upper_bound2.rhs()"
+ ]
+ }
+ ],
+ "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
+}
--
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