From 12ab81033cb05e505ddd2c96264d7cc5f76a338b Mon Sep 17 00:00:00 2001
From: Douglas Rumbaugh <dbr4@psu.edu>
Date: Thu, 12 Jun 2025 22:07:38 -0400
Subject: presentation updates

---
 paper.tex | 27 +++++++++++++++++++++++++++
 1 file changed, 27 insertions(+)

(limited to 'paper.tex')

diff --git a/paper.tex b/paper.tex
index 1e9bfd7..76d07fd 100644
--- a/paper.tex
+++ b/paper.tex
@@ -438,6 +438,33 @@ of Engineering Science and Mechanics
 
 \backmatter
 
+\newpage
+\begin{align*}
+	B(n) &\in \Theta(n \log n) \\
+	\mathscr{Q}_S(n) &\in \Theta(\log n + k)
+\end{align*}
+
+\begin{align}
+I(n) \in \Theta\left(B\left(\frac{n}{2}\right)\right) \\
+I(n) \in \Theta(n \log n)
+\end{align}
+
+\begin{equation}
+\mathscr{Q}(n) = \mathscr{Q}_S\left(\frac{n}{2}\right) + \mathscr{Q}_S\left(\frac{n}{2}\right)
+\end{equation}
+
+\begin{equation}
+\mathscr{Q}(n) \in \Theta\left(\log^2n + k\log n\right)
+\end{equation}
+
+
+
+
+
+\begin{align*}
+I_A(n) \in \Theta \left(\frac{B(n)}{n}\cdot \log n\right) \\
+\mathscr{Q}(n) \in O \left(\mathscr{Q}_S(n) \cdot \log n\right)
+\end{align*}
 %Vita
 \vita{chapters/vita}
 
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