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luke naylor latex documents
research
Max Destabilizer Rank
Commits
2c9071bc
Commit
2c9071bc
authored
1 year ago
by
Luke Naylor
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Add alternative pdf strings for mathmode in section titles
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229662ee
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#26958
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1 year ago
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main.tex
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2c9071bc
...
...
@@ -190,7 +190,13 @@ For the next subsections, we consider $q$ to be fixed with one of these values,
and we shall be varying
$
\chern
_
0
(
E
)
=
r
$
to see when certain inequalities fail.
\subsection
{$
\Delta
(
E
)
+
\Delta
(
G
)
\leq
\Delta
(
F
)
$}
\subsection
{
\texorpdfstring
{
$
\Delta
(
E
)
+
\Delta
(
G
)
\leq
\Delta
(
F
)
$
}{
Δ(E) + Δ(G) ≤ Δ(F)
}
}
\label
{
subsect-d-bound-bgmlv1
}
This condition expressed in terms of
$
R,C,D,r,c,d
$
looks as follows:
...
...
@@ -300,7 +306,13 @@ In the case $\beta = \beta_{-}$ (or $\beta_{+}$) we have
$
\chern
^{
\beta
}_
2
(
F
)
=
0
$
,
so some of these expressions simplify.
\subsection
{$
\Delta
(
E
)
\geq
0
$}
\subsection
{
\texorpdfstring
{
$
\Delta
(
E
)
\geq
0
$
}{
Δ(E) ≥ 0
}
}
This condition expressed in terms of
$
R,C,D,r,c,d
$
looks as follows:
...
...
@@ -384,7 +396,13 @@ Notice that for $\beta = \beta_{-}$ (or $\beta_{+}$), that is when
$
\chern
^{
\beta
}_
2
(
F
)=
0
$
, the constant and linear terms match up with the ones
for the bound found for
$
d
$
in subsection
\ref
{
subsect-d-bound-bgmlv1
}
.
\subsection
{$
\Delta
(
G
)
\geq
0
$}
\subsection
{
\texorpdfstring
{
$
\Delta
(
G
)
\geq
0
$
}{
Δ(G) ≥ 0
}
}
\label
{
subsect-d-bound-bgmlv3
}
This condition expressed in terms of
$
R,C,D,r,c,d
$
looks as follows:
...
...
@@ -516,14 +534,14 @@ $\chern^{\beta}_2(F) = 0$,
so some of these expressions simplify, and in particular, the constant and
linear terms match those of the other bounds in the previous subsections.
\subsection
{
Bounds on
$
r
$
}
\subsection
{
Bounds on
\texorpdfstring
{$
r
$}{
r
}
}
Now, the inequalities from the last three subsections will be used to find, for
each given
$
q
=
\chern
^{
\beta
}_
1
(
E
)
$
, how large
$
r
$
needs to be in order to leave
no possible solutions for
$
d
$
. At that point, there are no Chern characters
$
(
r,c,d
)
$
that satisfy all inequalities to give a pseudowall.
\subsubsection
{
All circular walls left of vertical wall
}
\subsubsection
{
All circular
pseudo
walls left of vertical wall
}
Suppose we take
$
\beta
=
\beta
_{
-
}$
in the previous subsections, to find all
circular walls to the left of the vertical wall (TODO as discussed in ref).
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