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luke naylor latex documents
research
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31b8ae55
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31b8ae55
authored
1 year ago
by
Luke Naylor
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Start notebook for generic choice of P
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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
0
x7f7bcab93ad0
>
"
]
},
"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
0
x7f7bbf22f110
>
"
]
},
"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
0
x7f7bbf0399d0
>
"
]
},
"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 $
\T
heta_v$ (i.e. $ch_2^{A,B}(v) = 0$) expressed in terms of twisted Chern character for $v$ at $
\b
eta=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
```
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