chore: update papers, add assumptions in 'Sum Analysis' paper.

Signed-off-by: Amlal El Mahrouss <amlal@nekernel.org>

1 files changed, 7 insertions, 4 deletions

diff --git a/source/wg04/paper.2.tex b/source/wg04/paper.2.tex
index c5f2cd4..52ddb66 100644
--- a/source/wg04/paper.2.tex
+++ b/source/wg04/paper.2.tex
@@ -7,6 +7,7 @@
 \usepackage{amsmath,amssymb,amsthm}
 \usepackage{xcolor}
 \usepackage{hyperref}
+\usepackage{mathtools}
 \usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
 
 \title{Sum Analysis.}
@@ -22,15 +23,17 @@
 \end{center}
 
 \section{Definitions}
-\subsection{The Sum Integral:}
+
+We assume $x \in \mathbb{R}, \quad a, b \in \mathbb{N}, \quad 0 < a < b < \cos(0)$ for:
+
+\subsection{The S-Integral:}
 \begin{equation}
-    \int_{a}^{b} S(x)dx, \quad \forall a, b \in \mathbb{R}, \quad 0 < a < b
+    I_{1} \coloneqq \int_{a}^{b} S(x)dx, \quad \forall a, b \in \mathbb{R}, \quad 0 < a < b
 \end{equation}
-\subsection{The Sum Function:}
+\subsection{The S-Function:}
 \begin{equation}
     S(x) := x^{e}, \quad x \in \mathbb{R}, \quad x > 0
 \end{equation}
 
-
 \end{document}