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luke naylor latex documents
research
Max Destabilizer Rank
Commits
de19dccf
Commit
de19dccf
authored
1 year ago
by
Luke Naylor
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Extend radius condition to problem 1
parent
960ae699
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#29375
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1 year ago
Stage: test
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main.tex
+24
-2
24 additions, 2 deletions
main.tex
other_P_choice.ipynb
+141
-13
141 additions, 13 deletions
other_P_choice.ipynb
with
165 additions
and
15 deletions
main.tex
+
24
−
2
View file @
de19dccf
...
...
@@ -1014,10 +1014,32 @@ amounts to:
\begin{align}
\label
{
eqn:radius-cond-betamin
}
\chern
_
2
^{
\beta
_{
-
}}
(u)
&
\geq
0
\\
d
&
\geq
\beta
_{
-
}
q +
\frac
{
1
}{
2
}
\beta
_{
-
}^
2r
\chern
_
2
^{
\beta
_{
-
}}
(u)
&
>
0
\\
d
&
>
\beta
_{
-
}
q +
\frac
{
1
}{
2
}
\beta
_{
-
}^
2r
\end{align}
\begin{sagesilent}
import other
_
P
_
choice as problem1
\end{sagesilent}
In the case where we are tackling problem
\ref
{
problem:problem-statement-1
}
,
with some Chern character
$
v
$
with positive rank, and some choice of point
$
P
=(
A,B
)
\in
\Theta
_
v
^
-
$
.
Then
$
\sage
{
problem
1
.A
2
_
subs
}$
follows from
$
\chern
_
2
^
P
(
v
)=
0
$
. Using this substitution into the
condition
$
\chern
_
2
^
P
(
u
)
>
0
$
yields:
\begin{equation}
\sage
{
problem1.radius
_
condition
}
\end{equation}
\noindent
Expressing this as a bound on
$
d
$
, then yields:
\begin{equation}
\sage
{
problem1.radius
_
condition
_
d
_
bound
}
\end{equation}
\subsubsection
{
Semistability of the Semistabilizer:
\texorpdfstring
{
...
...
This diff is collapsed.
Click to expand it.
other_P_choice.ipynb
+
141
−
13
View file @
de19dccf
...
...
@@ -45,7 +45,7 @@
"$\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = R \\\\ \\mathrm{ch}_{1} = C \\ell^{1} \\\\ \\mathrm{ch}_{2} = D \\ell^{2} \\end{array}$"
],
"text/plain": [
"
<pseudowalls.chern_character.Chern_Char
object
at
0
x7f
7bcab93ad
0
>
"
"
<pseudowalls.chern_character.Chern_Char
object
at
0
x7f
56b7c8274
0
>
"
]
},
"execution_count": 3,
...
...
@@ -73,7 +73,7 @@
"$\\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}$"
],
"text/plain": [
"
<pseudowalls.chern_character.Twisted_Chern_Char
object
at
0
x7f
7bbf22f11
0
>
"
"
<pseudowalls.chern_character.Twisted_Chern_Char
object
at
0
x7f
56b0027e8
0
>
"
]
},
"execution_count": 4,
...
...
@@ -90,6 +90,34 @@
"twisted_v"
]
},
{
"cell_type": "code",
"execution_count": 12,
"id": "6fced6d0",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"
<html>
\\(\\displaystyle \\text{Chern Character:} \\\\ \\begin{array}{l} \\mathrm{ch}_{0} = r \\\\ \\mathrm{ch}_{1} = c \\ell^{1} \\\\ \\mathrm{ch}_{2} = d \\ell^{2} \\end{array}\\)
</html>
"
],
"text/latex": [
"$\\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}$"
],
"text/plain": [
"
<pseudowalls.chern_character.Chern_Char
object
at
0
x7f56afe16140
>
"
]
},
"execution_count": 12,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"u = Chern_Char(r,c,d)\n",
"u"
]
},
{
"cell_type": "code",
"execution_count": 5,
...
...
@@ -105,7 +133,7 @@
"$\\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}$"
],
"text/plain": [
"
<pseudowalls.chern_character.Twisted_Chern_Char
object
at
0
x7f
7bbf0399d
0
>
"
"
<pseudowalls.chern_character.Twisted_Chern_Char
object
at
0
x7f
56b9a5fb2
0
>
"
]
},
"execution_count": 5,
...
...
@@ -140,7 +168,7 @@
},
{
"cell_type": "code",
"execution_count":
7
,
"execution_count":
6
,
"id": "17c390cd",
"metadata": {},
"outputs": [
...
...
@@ -156,7 +184,7 @@
"A^2 == 2*twisted_v2/R"
]
},
"execution_count":
7
,
"execution_count":
6
,
"metadata": {},
"output_type": "execute_result"
}
...
...
@@ -179,26 +207,126 @@
},
{
"cell_type": "code",
"execution_count":
null
,
"execution_count":
7
,
"id": "47b34ed7",
"metadata": {},
"outputs": [],
"outputs": [
{
"data": {
"text/html": [
"
<html>
\\(\\displaystyle -\\frac{1}{2} \\, A^{2} r + {\\mathrm{ch}_2^B(u)} > 0\\)
</html>
"
],
"text/latex": [
"$\\displaystyle -\\frac{1}{2} \\, A^{2} r + {\\mathrm{ch}_2^B(u)} > 0$"
],
"text/plain": [
"-1/2*A^2*r + twisted_u2 > 0"
]
},
"execution_count": 7,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"stability.Tilt(*P).degree(twisted_u) > 0"
]
},
{
"cell_type": "code",
"execution_count":
null
,
"execution_count":
11
,
"id": "d8abf566",
"metadata": {},
"outputs": [],
"outputs": [
{
"data": {
"text/html": [
"
<html>
\\(\\displaystyle R {\\mathrm{ch}_2^B(u)} - r {\\mathrm{ch}_2^B(v)} > 0\\)
</html>
"
],
"text/latex": [
"$\\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}$"
],
"text/plain": [
"d > -1/2*B^2*r + B*c + r*twisted_v2/R"
]
},
"execution_count": 28,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(\n",
" A2_subs\n",
"radius_condition_d_bound = (\n",
" radius_condition\n",
" .subs(twisted_u.ch[2] == u.twist(B).ch[2])\n",
" .expand()\n",
" .add_to_both_sides(B*R*c - B^2*R*r/2 + r*twisted_v.ch[2])\n",
" .divide_both_sides(R)\n",
" .expand()\n",
")"
]
},
{
"cell_type": "code",
"execution_count": 16,
"id": "e4fc4758",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"
<html>
\\(\\displaystyle \\frac{1}{2} \\, B^{2} r - B c + d\\)
</html>
"
],
"text/latex": [
"$\\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",
...
...
@@ -218,7 +346,7 @@
],
"metadata": {
"kernelspec": {
"display_name": "SageMath 9.
8
",
"display_name": "SageMath 9.
7
",
"language": "sage",
"name": "sagemath"
},
...
...
@@ -232,7 +360,7 @@
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.1
1.3
"
"version": "3.1
0.8
"
}
},
"nbformat": 4,
...
...
%% 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 0x7f
7bcab93ad
0>
<pseudowalls.chern_character.Chern_Char object at 0x7f
56b7c8274
0>
%% 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>
<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f56b0027e80>
%% Cell type:code id:6fced6d0 tags:
```
sage
u = Chern_Char(r,c,d)
u
```
%% 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 0x7f56afe16140>
%% 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 0x7f
7bbf0399d
0>
<pseudowalls.chern_character.Twisted_Chern_Char object at 0x7f
56b9a5fb2
0>
%% 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
```
%% Output
$\displaystyle -\frac{1}{2} \, A^{2} r + {\mathrm{ch}_2^B(u)} > 0$
-1/2*A^2*r + twisted_u2 > 0
%% Cell type:code id:d8abf566 tags:
```
sage
(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(
A2_subs
radius_condition = expand(
(stability.Tilt(*P).degree(twisted_u) / r > 0).expand().subs(
A2_subs
) * r * R
)
radius_condition
```
%% Output
$\displaystyle R {\mathrm{ch}_2^B(u)} - r {\mathrm{ch}_2^B(v)} > 0$
R*twisted_u2 - r*twisted_v2 > 0
%% Cell type:code id:194e313d tags:
```
sage
radius_condition_d_bound = (
radius_condition
.subs(twisted_u.ch[2] == u.twist(B).ch[2])
.expand()
.add_to_both_sides(B*R*c - B^2*R*r/2 + r*twisted_v.ch[2])
.divide_both_sides(R)
.expand()
)
```
%% Output
$\displaystyle d > -\frac{1}{2} \, B^{2} r + B c + \frac{r {\mathrm{ch}_2^B(v)}}{R}$
d > -1/2*B^2*r + B*c + r*twisted_v2/R
%% Cell type:code id:e4fc4758 tags:
```
sage
u.twist(B).ch[2]
```
%% Output
$\displaystyle \frac{1}{2} \, B^{2} r - B c + d$
1/2*B^2*r - B*c + d
%% Cell type:markdown id:fe9fe5b8 tags:
## Condition: $\Delta(u) \geq 0$
%% Cell type:code id:f09514cb tags:
```
sage
```
...
...
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