feat: paper improvements for WG04. (papers)

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

1 files changed, 9 insertions, 21 deletions

diff --git a/source/wg04/paper.tex b/source/wg04/paper.tex
index 9eb3dd3..09cee31 100644
--- a/source/wg04/paper.tex
+++ b/source/wg04/paper.tex
@@ -39,38 +39,26 @@ It will contain definitions and properties in a top-to-bottom order.
 
 Let the following definitions be used in mathematical analysis:
 
-\subsection{Definition of a sum of an integral from $a_{0}$ to $a_{n}$}
+\subsection{A sum of an integral from $a_{0}$ to $a_{n}$}
 
 Let an integral $\alpha$ be defined:
 \begin{equation}
     \sum_{k=1} \int_{a_{0}}^{a_{n}} f_{k}(S(b_{a_{0}})) \quad db_{a_{0}}, \quad b_{a_{0}} > 0, \quad b_{a_{0}} \in \mathbb{R}
 \end{equation}
-Such that
-\begin{equation}
-    a > 0, \quad a \in \mathbb{R}, \quad k \neq 0, \quad k \in \mathbb{Z}
-\end{equation}
-Such that $S$ is a user-defined computing function of a sum $b_{a_{0}}$
+Such that $a > 0, \quad a \in \mathbb{R}, \quad k \neq 0, \quad k \in \mathbb{Z}$ Such that $S$ is a user-defined computing function of a sum $b_{a_{0}}$
 
-\subsection{Definition of Ze in $\mathbb{R}$}
+\subsection{An equation Ze in $\mathbb{R}$}
 
-Let an equation $Ze$ be defined:
+Let an equation $Ze$ be defined as:
 
 \begin{equation}
 	\forall t \in \mathbb{R}, \quad t = Ze_{n}(\Delta t_{n})
 \end{equation}
-Such that
-\begin{equation}
-	n > 0, \quad n \in \mathbb{R}
-\end{equation}
-Such that
-\begin{equation}
-	\lvert \Delta t \rvert \geq n, \quad \Delta t \in \mathbb{C}
-\end{equation}
-Such that $Ze_{n}$ be the yield result of $t_{n} \in \Delta t$.
+Such that $n > 0, \quad n \in \mathbb{R}$ Such that $\lvert \Delta t \rvert \geq n, \quad \Delta t \in \mathbb{C}$ Such that $Ze_{n}$ be the yield result of $t_{n} \in \Delta t$.
 
-\subsection{Definition of $Ke(k, u, n)$}
+\subsection{Definition of an integral $Ke(k, u, n)$}
 
-Let $Ke(k, u, n)$:
+Let $Ke(k, u, n)$ be defined as:
 
 \begin{equation}
 	Ke(k, u, n)=\int_{k}^{n} \Phi_{k}(u) \quad du, \quad \forall u \in \mathbb{R}, \quad \forall k, n \in \mathbb{Z}, \quad k \geq 0
@@ -82,9 +70,9 @@ Such that $\Phi_{k}(u)$ is the analysis function where $\Phi_{k}(u) \in \mathbb{
 
 Let the following properties be used in mathematical analysis:
 
-\subsection{Properties of $\Delta t$}
+\subsection{Properties of $\Delta t$ in $Ze$}
 
-Let the properties be:
+Let the properties be for $Ze$:
 
 \begin{itemize}
     \item $\Delta t \in \mathbb{C}, \quad \forall x \in \Delta t, \quad x \in \mathbb{C}$