Correct bgmlv2 'lowerbounds' to 'upperbounds'

main.tex

@@ -360,45 +360,45 @@ bgmlv2_d_ineq = (
 ).expand()
 
 # Keep hold of lower bound for d
-bgmlv2_d_lowerbound = bgmlv2_d_ineq.rhs()
+bgmlv2_d_upperbound = bgmlv2_d_ineq.rhs()
 \end{sagesilent}
 
 \begin{equation}
-	\label{eqn-bgmlv2_d_lowerbound}
+	\label{eqn-bgmlv2_d_upperbound}
 	\sage{bgmlv2_d_ineq}
 \end{equation}
 
 \begin{sagesilent}
 # Seperate out the terms of the lower bound for d
 
-bgmlv2_d_lowerbound_without_hyp = (
-	bgmlv2_d_lowerbound
+bgmlv2_d_upperbound_without_hyp = (
+	bgmlv2_d_upperbound
 	.subs(1/r == 0)
 )
 
-bgmlv2_d_lowerbound_const_term = (
-	bgmlv2_d_lowerbound_without_hyp
+bgmlv2_d_upperbound_const_term = (
+	bgmlv2_d_upperbound_without_hyp
 	.subs(r==0)
 )
 
-bgmlv2_d_lowerbound_linear_term = (
-	bgmlv2_d_lowerbound_without_hyp
-	- bgmlv2_d_lowerbound_const_term
+bgmlv2_d_upperbound_linear_term = (
+	bgmlv2_d_upperbound_without_hyp
+	- bgmlv2_d_upperbound_const_term
 ).expand()
 
-bgmlv2_d_lowerbound_exp_term = (
-	bgmlv2_d_lowerbound
-	- bgmlv2_d_lowerbound_without_hyp
+bgmlv2_d_upperbound_exp_term = (
+	bgmlv2_d_upperbound
+	- bgmlv2_d_upperbound_without_hyp
 ).expand()
 \end{sagesilent}
 
-Viewing equation \ref{eqn-bgmlv2_d_lowerbound} as a lower bound for $d$ in term
+Viewing equation \ref{eqn-bgmlv2_d_upperbound} as a lower bound for $d$ in term
 of $r$ again, there's a constant term
-$\sage{bgmlv2_d_lowerbound_const_term}$,
+$\sage{bgmlv2_d_upperbound_const_term}$,
 a linear term
-$\sage{bgmlv2_d_lowerbound_linear_term}$,
+$\sage{bgmlv2_d_upperbound_linear_term}$,
 and a hyperbolic term
-$\sage{bgmlv2_d_lowerbound_exp_term}$.
+$\sage{bgmlv2_d_upperbound_exp_term}$.
 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}.
@@ -554,10 +554,22 @@ 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).
 
 \begin{equation*}
-	\sage{ bgmlv3_d_upperbound_const_term }
+	d \geq
+	\sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
+	+ \sage{bgmlv1_d_lowerbound_linear_term}
+	+ \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)}
 \end{equation*}
 \begin{equation*}
-	\sage{bgmlv3_d_upperbound_const_term_alt1.subs(chbv == 0)}
+	d \geq
+	\sage{bgmlv2_d_lowerbound_const_term}
+	+ \sage{bgmlv2_d_lowerbound_linear_term}
+	+ \sage{bgmlv2_d_lowerbound_exp_term}
+\end{equation*}
+\begin{equation*}
+	d \leq
+	\sage{bgmlv3_d_upperbound_const_term_alt.subs(chbv == 0)}
+	+ \sage{bgmlv3_d_upperbound_linear_term}
+	+ \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)}
 \end{equation*}