Skip to content
GitLab
Explore
Sign in
Primary navigation
Search or go to…
Project
M
Max Destabilizer Rank
Manage
Activity
Members
Labels
Plan
Issues
Issue boards
Milestones
Wiki
Code
Merge requests
Repository
Branches
Commits
Tags
Repository graph
Compare revisions
Snippets
Build
Pipelines
Jobs
Pipeline schedules
Artifacts
Deploy
Releases
Package Registry
Container Registry
Model registry
Operate
Environments
Terraform modules
Monitor
Incidents
Analyze
Value stream analytics
Contributor analytics
CI/CD analytics
Repository analytics
Model experiments
Help
Help
Support
GitLab documentation
Compare GitLab plans
Community forum
Contribute to GitLab
Provide feedback
Keyboard shortcuts
?
Snippets
Groups
Projects
Show more breadcrumbs
luke naylor latex documents
research
Max Destabilizer Rank
Commits
9a91722e
Commit
9a91722e
authored
1 year ago
by
Luke Naylor
Browse files
Options
Downloads
Patches
Plain Diff
Correct corrolary to corollary
parent
695797b0
No related branches found
No related tags found
No related merge requests found
Changes
1
Hide whitespace changes
Inline
Side-by-side
Showing
1 changed file
main.tex
+14
-14
14 additions, 14 deletions
main.tex
with
14 additions
and
14 deletions
main.tex
+
14
−
14
View file @
9a91722e
...
...
@@ -33,7 +33,7 @@ sorting=ynt
\newcommand
{
\minorheading
}
[1]
{{
\noindent\normalfont\normalsize\bfseries
#1
}}
\newtheorem
{
theorem
}{
Theorem
}
[section]
\newtheorem
{
cor
r
olary
}{
Cor
r
olary
}
[section]
\newtheorem
{
coro
l
lary
}{
Coro
l
lary
}
[section]
\newtheorem
{
lemma
}{
Lemma
}
[section]
\newtheorem
{
fact
}{
Fact
}
[section]
\newtheorem
{
example
}{
Example
}
[section]
...
...
@@ -730,7 +730,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\end{proof}
\begin{cor
r
olary}
[Numerical Tests for All `left' Pseudo-walls]
\begin{coro
l
lary}
[Numerical Tests for All `left' Pseudo-walls]
\label
{
cor:num
_
test
_
prob2
}
Given a Chern character
$
v
$
with positive rank and
$
\Delta
(
v
)
\geq
0
$
,
such that
$
\beta
_{
-
}
\coloneqq\beta
_{
-
}
(
v
)
\in
\QQ
$
.
...
...
@@ -750,7 +750,7 @@ problem with the help of lemma \ref{lem:pseudo_wall_numerical_tests}.
\item
$
\chern
_
2
^{
\beta
_{
-
}}
(
u
)
>
0
$
\label
{
item:radiuscond:lem:num
_
test
_
prob2
}
\end{enumerate}
\end{cor
r
olary}
\end{coro
l
lary}
\begin{proof}
This is a specialization of the previous lemma, using
$
P
=(
\beta
_{
-
}
,
0
)
$
.
...
...
@@ -996,7 +996,7 @@ u = Chern_Char(*var("r c d", domain="real"))
Recall from condition
\ref
{
item:chern1bound:lem:num
_
test
_
prob1
}
in
lemma
\ref
{
lem:num
_
test
_
prob1
}
(or cor
r
olary
\ref
{
cor:num
_
test
_
prob2
}
)
(or coro
l
lary
\ref
{
cor:num
_
test
_
prob2
}
)
that
$
\chern
_
1
^{
\beta
}
(
u
)
$
has fixed bounds in terms of
$
\chern
_
1
^{
\beta
}
(
v
)
$
,
and so we can write:
...
...
@@ -1030,7 +1030,7 @@ and we shall be varying $\chern_0(E) = r$ to see when certain inequalities fail.
This section studies the numerical conditions that
$
u
$
must satisfy as per
lemma
\ref
{
lem:num
_
test
_
prob1
}
(or cor
r
olary
\ref
{
cor:num
_
test
_
prob2
}
)
(or coro
l
lary
\ref
{
cor:num
_
test
_
prob2
}
)
\subsubsection
{
Size of pseudo-wall:
$
\chern
_
2
^
P
(
u
)
>
0
$
}
\label
{
subsect-d-bound-radiuscond
}
...
...
@@ -1038,7 +1038,7 @@ lemma \ref{lem:num_test_prob1}
This condition refers to condition
\ref
{
item:radiuscond:lem:num
_
test
_
prob1
}
from lemma
\ref
{
lem:num
_
test
_
prob1
}
(or cor
r
olary
\ref
{
cor:num
_
test
_
prob2
}
).
(or coro
l
lary
\ref
{
cor:num
_
test
_
prob2
}
).
In the case where we are tackling problem
\ref
{
problem:problem-statement-2
}
(with
$
\beta
=
\beta
_{
-
}$
), this condition, when expressed as a bound on
$
d
$
,
...
...
@@ -1061,7 +1061,7 @@ amounts to:
This condition refers to condition
\ref
{
item:bgmlvu:lem:num
_
test
_
prob1
}
from lemma
\ref
{
lem:num
_
test
_
prob1
}
(or cor
r
olary
\ref
{
cor:num
_
test
_
prob2
}
).
(or coro
l
lary
\ref
{
cor:num
_
test
_
prob2
}
).
\begin{sagesilent}
...
...
@@ -1089,7 +1089,7 @@ bgmlv2_with_q = (
\noindent
This can be rearranged to express a bound on
$
d
$
as follows
(recall from condition
\ref
{
item:rankpos:lem:num
_
test
_
prob1
}
in lemma
\ref
{
lem:num
_
test
_
prob1
}
or cor
r
olary
in lemma
\ref
{
lem:num
_
test
_
prob1
}
or coro
l
lary
\ref
{
cor:num
_
test
_
prob2
}
that
$
r>
0
$
):
\begin{sagesilent}
...
...
@@ -1156,7 +1156,7 @@ for the bound found for $d$ in subsubsection \ref{subsect-d-bound-radiuscond}.
This condition refers to condition
\ref
{
item:bgmlvv-u:lem:num
_
test
_
prob1
}
from lemma
\ref
{
lem:num
_
test
_
prob1
}
(or cor
r
olary
\ref
{
cor:num
_
test
_
prob2
}
).
(or coro
l
lary
\ref
{
cor:num
_
test
_
prob2
}
).
Expressing
$
\Delta
(
v
-
u
)
\geq
0
$
in term of
$
q
$
and rearranging as a bound on
$
d
$
yields:
...
...
@@ -1297,7 +1297,7 @@ $d=\chern^{\beta_{-}}_2(u)$ induced by conditions
\ref
{
item:bgmlvu:lem:num
_
test
_
prob2
}
,
\ref
{
item:bgmlvv-u:lem:num
_
test
_
prob2
}
, and
\ref
{
item:radiuscond:lem:num
_
test
_
prob1
}
from cor
r
olary
\ref
{
cor:num
_
test
_
prob2
}
have the same constant and linear
from coro
l
lary
\ref
{
cor:num
_
test
_
prob2
}
have the same constant and linear
terms in
$
r
$
, but different hyperbolic terms.
These give bounds with the same assymptotes when we take
$
r
\to\infty
$
(for any fixed
$
q
=
\chern
_
1
^{
\beta
_{
-
}}
(
u
)
$
).
...
...
@@ -1685,7 +1685,7 @@ r_upper_bound_all_q = (
)
\end{sagesilent}
\begin{cor
r
olary}
[Bound on
$
r
$
\#
2]
\begin{coro
l
lary}
[Bound on
$
r
$
\#
2]
\label
{
cor:direct
_
rmax
_
with
_
uniform
_
eps
}
Let
$
v
$
be a fixed Chern character and
$
R
\coloneqq\chern
_
0
(
v
)
\leq
n
^
2
\Delta
(
v
)
$
.
...
...
@@ -1700,7 +1700,7 @@ r_upper_bound_all_q = (
\sage
{
r
_
upper
_
bound
_
all
_
q.expand()
}
\end{equation*}
\egroup
\end{cor
r
olary}
\end{coro
l
lary}
\begin{proof}
\bgroup
...
...
@@ -1749,7 +1749,7 @@ corrolary_bound = (
.subs(n==recurring.n)
)
\end{sagesilent}
Using the above cor
r
olary
\ref
{
cor:direct
_
rmax
_
with
_
uniform
_
eps
}
, we get that
Using the above coro
l
lary
\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
)
}$
,
which is much closer to real maximum 25 than the original bound 144.
...
...
@@ -1772,7 +1772,7 @@ corrolary_bound = (
.subs(n==extravagant.n)
)
\end{sagesilent}
Using the above cor
r
olary
\ref
{
cor:direct
_
rmax
_
with
_
uniform
_
eps
}
, we get that
Using the above coro
l
lary
\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
)
}$
,
which is much closer to real maximum
$
\sage
{
extravagant.actual
_
rmax
}$
than the
...
...
This diff is collapsed.
Click to expand it.
Preview
0%
Loading
Try again
or
attach a new file
.
Cancel
You are about to add
0
people
to the discussion. Proceed with caution.
Finish editing this message first!
Save comment
Cancel
Please
register
or
sign in
to comment