From 7145b478d6d243c4de8cf2fda89ede79e8b9e9e3 Mon Sep 17 00:00:00 2001
From: Luke Naylor <l.naylor@sms.ed.ac.uk>
Date: Wed, 10 May 2023 15:38:38 +0100
Subject: [PATCH] Add conditions on r for bounds
---
main.tex | 9 ++++++---
1 file changed, 6 insertions(+), 3 deletions(-)
diff --git a/main.tex b/main.tex
index c3f8668..df4844f 100644
--- a/main.tex
+++ b/main.tex
@@ -564,17 +564,20 @@ vertical wall (TODO as discussed in ref).
d &\geq&
\sage{bgmlv1_d_lowerbound_linear_term}
&+ \sage{bgmlv1_d_lowerbound_const_term_alt.subs(chbv == 0)}
- +& \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)}
+ +& \sage{bgmlv1_d_lowerbound_exp_term_alt.subs(chbv == 0)},
+ &\qquad\text{when\:} r > \frac{R}{2}
\\
d &\leq&
\sage{bgmlv2_d_upperbound_linear_term}
&+ \sage{bgmlv2_d_upperbound_const_term}
- +& \sage{bgmlv2_d_upperbound_exp_term}
+ +& \sage{bgmlv2_d_upperbound_exp_term},
+ &\qquad\text{when\:} r > 0
\\
d &\leq&
\sage{bgmlv3_d_upperbound_linear_term}
&+ \sage{bgmlv3_d_upperbound_const_term_alt.subs(chbv == 0)}
- +& \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)}
+ +& \sage{bgmlv3_d_upperbound_exp_term_alt.subs(chbv == 0)},
+ &\qquad\text{when\:} r > R
\end{align}
Furthermore, we get an extra bound for $d$ resulting from the condition that the
--
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