feat: update and add papers.

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

2 files changed, 8 insertions, 6 deletions

diff --git a/source/wg04/paper.2.tex b/source/wg04/paper.2.tex
index 93c3a59..0de2bb0 100644
--- a/source/wg04/paper.2.tex
+++ b/source/wg04/paper.2.tex
@@ -32,11 +32,11 @@ We assume $x \in \mathbb{R}, \quad a, b \in \mathbb{N}, \quad 0 < a < b < \cos(0
 
 \subsection{DEFINITION OF THE S-INTEGRAL:}
 \begin{equation}
-    I_{1} \coloneqq \int_{a}^{b} S(x)dx
+    I_{1}(a, b, x) \coloneqq \int_{a}^{b} S(x)dx
 \end{equation}
 \subsection{DEFINITION OF THE S-FUNCTION:}
 \begin{equation}
-    S(x) = \sum_{k=1}^{t} (\frac{dx}{du}) \cdot x, \quad x > 0, \quad t > k
+    S(x, u) = \sum_{k=1}^{t} (\frac{dx}{du}\cdot k) \cdot x, \quad x > 0, \quad t > k
 \end{equation}
 
 \end{document}
diff --git a/source/wg04/paper.tex b/source/wg04/paper.tex
index e780a6a..3c64e6a 100644
--- a/source/wg04/paper.tex
+++ b/source/wg04/paper.tex
@@ -34,23 +34,25 @@ This section will cover the definitions and properties of the formulas we will u
 We assume $x, y, z \in \mathbb{R}, \quad n \in \mathbb{N}, \quad \cos(\pi) < a < b, \quad t = a$
 \subsection{DEFINITION OF THE A-INTEGRAL:}
 \begin{equation}
-    I_{1} \coloneqq \sum_{n}^{t} \int_{a}^{b} S(x) dx
+    I_{1}(x, t, a, b, n) \coloneqq \sum_{n}^{t} \int_{a}^{b} S(x) dx
 \end{equation}
 \subsection{DEFINITION OF THE A-FUNCTION:}
 \begin{equation}
-	S(x) : (x, y) \to \mathbb{R}
+	S(x, y) : (x, y) \to \mathbb{R}
 \end{equation}
 \subsection{DEFINITION OF THE A-EQUATION:}
 \begin{equation}
-	I_{2} \coloneqq \int_{a}^{b} (E_{1})dx, \quad E_{1} = Z(x) := \frac{(S(x) \cdot y)}{\Delta t \cdot z}
+	I_{2}(x, y, z, a, b, \Delta t) \coloneqq \int_{a}^{b} (E_{1}(x, y, z, \Delta t))dx, \quad E_{1}(x, y, z, \Delta t) := \frac{(S(x) \cdot y)}{\Delta t \cdot z}
 \end{equation}
 Where $\Delta t \in \mathbb{R}$ if and only if $0 < \Delta t$.
 \subsection{DEFINITION OF THE K-EQUATION:}
 \begin{equation}
-	K_{n} \coloneqq \int_{a}^{b} S_{n}(u_{n}) \quad d \cdot u_{n}, \quad \forall u_{n} \in \mathbb{R}
+	K_{n}(u_{n}, a, b) \coloneqq \int_{a}^{b} S_{n}(u_{n}) \quad d \cdot u_{n}, \quad \forall u_{n} \in \mathbb{R}
 \end{equation}
 Where $K_{n} > 0$ such that $S_{n}(u_{n})$ is the A-Function at index $n$ of $u$.
 
 \section{INTRODUCTION}
 
+We will in this paper introduce several equations and their usages in analysis.
+
 \end{document}