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luke naylor latex documents
research
Max Destabilizer Rank
Commits
8ba0438a
Commit
8ba0438a
authored
1 year ago
by
Luke Naylor
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Extend numerical formulations to rank 0
parent
a500675b
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#29360
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1 year ago
Stage: test
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main.tex
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8ba0438a
...
@@ -506,7 +506,8 @@ semistabilizing sequence.
...
@@ -506,7 +506,8 @@ semistabilizing sequence.
\begin{lemma}
[Numerical tests for left-wall pseudo-semistabilizers]
\begin{lemma}
[Numerical tests for left-wall pseudo-semistabilizers]
\label
{
lem:pseudo
_
wall
_
numerical
_
tests
}
\label
{
lem:pseudo
_
wall
_
numerical
_
tests
}
Let
$
v
$
and
$
u
$
be Chern characters with
$
\Delta
(
v
)
,
Let
$
v
$
and
$
u
$
be Chern characters with
$
\Delta
(
v
)
,
\Delta
(
u
)
\geq
0
$
, and
$
v
$
has non-negative rank. Let
$
P
$
be a point on
$
\Theta
_
v
^
-
$
.
\Delta
(
u
)
\geq
0
$
, and
$
v
$
has non-negative rank (and
$
\chern
_
1
(
v
)
>
0
$
if rank 0).
Let
$
P
$
be a point on
$
\Theta
_
v
^
-
$
.
\noindent
\noindent
The following conditions:
The following conditions:
...
@@ -635,7 +636,8 @@ are trying to solve for.
...
@@ -635,7 +636,8 @@ are trying to solve for.
\begin{problem}
[sufficiently large `left' pseudo-walls]
\begin{problem}
[sufficiently large `left' pseudo-walls]
\label
{
problem:problem-statement-1
}
\label
{
problem:problem-statement-1
}
Fix a Chern character
$
v
$
with non-negative rank, and
$
\Delta
(
v
)
\geq
0
$
.
Fix a Chern character
$
v
$
with non-negative rank (and
$
\chern
_
1
(
v
)
>
0
$
if rank 0),
and
$
\Delta
(
v
)
\geq
0
$
.
The goal is to find all pseudo-semistabilizers
$
u
$
The goal is to find all pseudo-semistabilizers
$
u
$
which give circular pseudo-walls containing some fixed point
which give circular pseudo-walls containing some fixed point
$
P
\in\Theta
_
v
^
-
$
.
$
P
\in\Theta
_
v
^
-
$
.
...
@@ -670,8 +672,8 @@ $v-u$ for each solution $u$ of the problem.
...
@@ -670,8 +672,8 @@ $v-u$ for each solution $u$ of the problem.
\begin{problem}
[all `left' pseudo-walls]
\begin{problem}
[all `left' pseudo-walls]
\label
{
problem:problem-statement-2
}
\label
{
problem:problem-statement-2
}
Fix a Chern character
$
v
$
with non-negative rank
,
$
\Delta
(
v
)
\geq
0
$
,
Fix a Chern character
$
v
$
with non-negative rank
(and
$
\chern
_
1
(
v
)
>
0
$
if rank
0
)
,
and
$
\beta
_{
-
}
(
v
)
\in
\QQ
$
.
$
\Delta
(
v
)
\geq
0
$
,
and
$
\beta
_{
-
}
(
v
)
\in
\QQ
$
.
The goal is to find all pseudo-semistabilizers
$
u
$
which give circular
The goal is to find all pseudo-semistabilizers
$
u
$
which give circular
pseudo-walls on the left side of
$
V
_
v
$
.
pseudo-walls on the left side of
$
V
_
v
$
.
\end{problem}
\end{problem}
...
@@ -697,7 +699,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
...
@@ -697,7 +699,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\begin{lemma}
[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
\begin{lemma}
[Numerical Tests for Sufficiently Large `left' Pseudo-walls]
\label
{
lem:num
_
test
_
prob1
}
\label
{
lem:num
_
test
_
prob1
}
Given a Chern character
$
v
$
with positive rank and
$
\Delta
(
v
)
\geq
0
$
,
Given a Chern character
$
v
$
with non-negative rank
(with
$
\chern
_
1
(
v
)
>
0
$
if rank 0)
and
$
\Delta
(
v
)
\geq
0
$
,
and a choice of point
$
P
$
on
$
\Theta
_
v
^
-
$
.
and a choice of point
$
P
$
on
$
\Theta
_
v
^
-
$
.
Solutions
$
u
=(
r,c
\ell
,
\frac
{
e
}{
\lcm
(
m,
2
)
}
\ell
^
2
)
$
Solutions
$
u
=(
r,c
\ell
,
\frac
{
e
}{
\lcm
(
m,
2
)
}
\ell
^
2
)
$
to problem
\ref
{
problem:problem-statement-1
}
.
to problem
\ref
{
problem:problem-statement-1
}
.
...
@@ -718,7 +722,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
...
@@ -718,7 +722,9 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\end{lemma}
\end{lemma}
\begin{proof}
\begin{proof}
Consider the context of
$
v
$
being a Chern character with positive rank and
Consider the context of
$
v
$
being a Chern character with non-negative rank
(and
$
\chern
_
1
(
v
)
>
0
$
if rank 0)
and
$
\Delta
\geq
0
$
, and
$
u
$
being a Chern character with
$
\Delta
(
u
)
\geq
0
$
.
$
\Delta
\geq
0
$
, and
$
u
$
being a Chern character with
$
\Delta
(
u
)
\geq
0
$
.
Lemma
\ref
{
lem:pseudo
_
wall
_
numerical
_
tests
}
gives that the remaining
Lemma
\ref
{
lem:pseudo
_
wall
_
numerical
_
tests
}
gives that the remaining
conditions for
$
u
$
being a solution to problem
conditions for
$
u
$
being a solution to problem
...
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