chore: update papers, fixed them.

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

4 files changed, 53 insertions, 59 deletions

diff --git a/source/dn001.07/paper.tex b/source/dn001.07/paper.tex
index b759098..270487b 100644
--- a/source/dn001.07/paper.tex
+++ b/source/dn001.07/paper.tex
@@ -14,7 +14,6 @@
 \date{February 2026}
 
 \begin{document}
-\bf
 \maketitle
 
 \begin{center}
@@ -25,25 +24,12 @@
 
 \subsection{Definition}
 
-Let two lemmas for the integrals $\alpha$ and $\beta$. $\beta$ and $\alpha$ shall be true given the following:
-
-\subsubsection{Conditions of $\alpha$}
-
-Let $\alpha$ be an integral of $t \in \mathbb{Z}$ which consists of a sum $S(t) = \sum_{k=1}^{n} (t!), \quad \forall k, n \in \mathbb{Z}$.
-Consider the following integral:
+Let us define the following:
 
 \begin{equation}
-	\alpha(u) = \int S(t) du
+	f(u) = \int_{u_{0}}^{u} S(t) dt
 \end{equation}
-For all $\alpha$ in $0 < \alpha < e$, or $\alpha > e$.
-For $\alpha \in \mathbb{R}, \quad u \in \mathbb{R}, \quad u > 0$.
-
-\subsubsection{Conditions of $\beta$}
-
-Let $\beta$ be an integral of $t \in \mathbb{Z}$ which consists of $\alpha \in \mathbb{R}$.
-$\beta > \alpha$. Such that $\lim_{t \to e} (\alpha \to \infty$+).
-
-\subsubsection{Conclusion}
-We see that $\beta, \quad \alpha$ is defined if and only if $t > e$. We can conclude that the lemma holds and points toward $\infty+$.
+where $u > 0, \quad u \in \mathbb{R}$\\
+let $S : (0, e) \to \mathbb{R}$ be continuous.
 
 \end{document}
diff --git a/source/wg04/paper.2.tex b/source/wg04/paper.2.tex
new file mode 100644
index 0000000..c5f2cd4
--- /dev/null
+++ b/source/wg04/paper.2.tex
@@ -0,0 +1,36 @@
+% AUTHOR: Amlal El Mahrouss
+% PURPOSE: WG04.2: Sum Analysis.
+
+\documentclass[11pt, a4paper]{article}
+\usepackage{graphicx}
+\usepackage{listings}
+\usepackage{amsmath,amssymb,amsthm}
+\usepackage{xcolor}
+\usepackage{hyperref}
+\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry}
+
+\title{Sum Analysis.}
+\author{Amlal El Mahrouss\\\texttt{amlal@nekernel.org}}
+\date{February 2026}
+
+\begin{document}
+
+\maketitle
+
+\begin{center}
+	\rule[0.01cm]{17cm}{0.01cm}
+\end{center}
+
+\section{Definitions}
+\subsection{The Sum Integral:}
+\begin{equation}
+    \int_{a}^{b} S(x)dx, \quad \forall a, b \in \mathbb{R}, \quad 0 < a < b
+\end{equation}
+\subsection{The Sum Function:}
+\begin{equation}
+    S(x) := x^{e}, \quad x \in \mathbb{R}, \quad x > 0
+\end{equation}
+
+
+\end{document}
+
diff --git a/source/wg04/paper.tex b/source/wg04/paper.tex
index b61a97f..13b4faf 100644
--- a/source/wg04/paper.tex
+++ b/source/wg04/paper.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{Integrals and Equations for Analysis.}
@@ -15,58 +16,27 @@
 
 \begin{document}
 
-\bf \maketitle
+\maketitle
 
 \begin{center}
 	\rule[0.01cm]{17cm}{0.01cm}
 \end{center}
 
-\section{Abstract}
+\section{Formulas definitions}
 
-We will introduce in this paper useful primitives in order to help in analysis. We will also define and describe properties in a top-to-bottom order.
-
-\section{Definitions}
-
-Let the following definitions be used in mathematical analysis:
-
-\subsection{A sum of an integral from $a_{0}$ to $a_{n}$}
-
-Let an integral $\alpha$ be defined:
+Let us define four formulas using the definitons below:
+We assume $x \in \mathbb{R}, \quad n \in \mathbb{N}, \quad 0 < k < a < b < 1$
+\subsection{Sum Integral:}
 \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}
+    I_{1} \coloneqq n \int_{a}^{b} S(x) dx
 \end{equation}
-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{An equation Ze in $\mathbb{R}$}
-
-Let an equation $Ze$ be defined as:
-
+\subsection{Sum Formula:}
 \begin{equation}
-	\forall t \in \mathbb{R}, \quad t = Ze_{n}(\Delta t_{n})
+	S(x) \coloneqq \cos(x)\times\sin(x)
 \end{equation}
-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 an integral $Ke(k, u, n)$}
-
-Let $Ke(k, u, n)$ be defined as:
-
+\subsection{Sum Equation:}
 \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
+	I_{2} \coloneqq \int_{k}^{a} Z(x)dx, \quad  Z(x) := (S(x) \times x)
 \end{equation}
-Such that $Ke(k, u, n)$ for $z \in \mathbb{R}$. \\
-Such that $\Phi_{k}(u)$ is the analysis function where $\Phi_{k}(u) \in \mathbb{C}$ if and only if $u \in \mathbb{R}, \quad u > 0$.
-
-\section{Properties}
-
-Let the following properties be used in mathematical analysis:
-
-\subsection{Properties of $\Delta t$ in $Ze$}
-
-Let the properties be for $Ze$:
-
-\begin{itemize}
-    \item $\Delta t \in \mathbb{C}, \quad \forall x \in \Delta t, \quad x \in \mathbb{C}$
-\end{itemize}
-
 \end{document}
 
diff --git a/wg04.mk b/wg04.mk
index fdced61..4805f38 100644
--- a/wg04.mk
+++ b/wg04.mk
@@ -6,6 +6,8 @@
 
 .PHONY: wg04
 wg04: clean
+	$(HTMLTEX) source/wg04/sketch-01.tex
+	$(PDFTEX) source/wg04/sketch-01.tex
 	$(HTMLTEX) source/wg04/paper.tex
 	$(PDFTEX) source/wg04/paper.tex