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{
"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 0x7f56b7c82740>"
]
},
"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}$"
],
"text/plain": [
"<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f56b0027e80>"
]
},
"execution_count": 4,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"twisted_v = Twisted_Chern_Char(B,\n",
" R,\n",
" var(\"twisted_v1\", latex_name = r\"\\mathrm{ch}_1^B(v)\", domain=\"real\"),\n",
" var(\"twisted_v2\", latex_name = r\"\\mathrm{ch}_2^B(v)\", domain=\"real\"),\n",
")\n",
"twisted_v"
]
},
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{
"cell_type": "code",
"execution_count": 12,
"id": "6fced6d0",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}\\)</html>"
],
"text/latex": [
"$\\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 0x7f56afe16140>"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u = Chern_Char(r,c,d)\n",
"u"
]
},
{
"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 0x7f56b9a5fb20>"
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]
},
"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",
"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"
]
},
"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",
"id": "47b34ed7",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle -\\frac{1}{2} \\, A^{2} r + {\\mathrm{ch}_2^B(u)} > 0\\)</html>"
],
"text/latex": [
"$\\displaystyle -\\frac{1}{2} \\, A^{2} r + {\\mathrm{ch}_2^B(u)} > 0$"
],
"text/plain": [
"-1/2*A^2*r + twisted_u2 > 0"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"stability.Tilt(*P).degree(twisted_u) > 0"
]
},
{
"cell_type": "code",
"id": "d8abf566",
"metadata": {},
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"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle R {\\mathrm{ch}_2^B(u)} - r {\\mathrm{ch}_2^B(v)} > 0\\)</html>"
],
"text/latex": [
"$\\displaystyle R {\\mathrm{ch}_2^B(u)} - r {\\mathrm{ch}_2^B(v)} > 0$"
],
"text/plain": [
"R*twisted_u2 - r*twisted_v2 > 0"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"radius_condition = expand(\n",
" (stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(\n",
" A2_subs\n",
" ) * r * R\n",
")\n",
"radius_condition"
]
},
{
"cell_type": "code",
"execution_count": 28,
"id": "194e313d",
"metadata": {
"collapsed": true
},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle d > -\\frac{1}{2} \\, B^{2} r + B c + \\frac{r {\\mathrm{ch}_2^B(v)}}{R}\\)</html>"
],
"text/latex": [
"$\\displaystyle d > -\\frac{1}{2} \\, B^{2} r + B c + \\frac{r {\\mathrm{ch}_2^B(v)}}{R}$"
],
"text/plain": [
"d > -1/2*B^2*r + B*c + r*twisted_v2/R"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"radius_condition_d_bound = (\n",
" radius_condition\n",
" .subs(twisted_u.ch[2] == u.twist(B).ch[2])\n",
" .expand()\n",
" .add_to_both_sides(B*R*c - B^2*R*r/2 + r*twisted_v.ch[2])\n",
" .divide_both_sides(R)\n",
" .expand()\n",
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{
"cell_type": "code",
"execution_count": 16,
"id": "e4fc4758",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<html>\\(\\displaystyle \\frac{1}{2} \\, B^{2} r - B c + d\\)</html>"
],
"text/latex": [
"$\\displaystyle \\frac{1}{2} \\, B^{2} r - B c + d$"
],
"text/plain": [
"1/2*B^2*r - B*c + d"
]
},
"execution_count": 16,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u.twist(B).ch[2]"
]
},
{
"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": {
"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",
}
},
"nbformat": 4,
"nbformat_minor": 5
}