% AUTHOR: Amlal El Mahrouss
% PURPOSE: WG04: Integrals and Equations for Analysis.

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\title{Integrals and Equations for Analysis.}
\author{Amlal El Mahrouss\\\texttt{amlal@nekernel.org}}
\date{February 2026}

\begin{document}

\maketitle

\begin{abstract}
\begin{center}
    The paper covers several definitions that are made to be used in analysis; their purpose has been kept abstract for flexibility reasons.
\end{center}
\end{abstract}

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	\rule[0.01cm]{17cm}{0.01cm}
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\section{DEFINITION}

We define the following formulas to establish our proofs:

\subsection{DEFINITION OF THE A-INTEGRAL:}
\begin{equation}
    I_{1}(t, a, b, n) \coloneqq \sum_{k=t}^{n} \int_{a}^{b} Z(u) du
\end{equation}
\subsection{DEFINITION OF THE A-FUNCTION:}
Let a function which returns a value in $\mathbb{R}$ be:
\begin{equation}
	Z(x) : (0, x) \to \mathbb{R}
\end{equation}
Named $Z(x) \in \mathbb{R}$. Such that $Z(x)$ is a compute-function of argument $x$.
\subsection{DEFINITION OF THE A-EQUATION AND I-INTEGRAL:}
\begin{equation}
	I_{2}(y, z, t, a, b) \coloneqq \int_{a}^{b} E_{1}(x, y, z, t) dx, \quad E_{1}(x, y, z, t) := \tilde{Z}(x \cdot t, y \cdot z)
\end{equation}
Where $t \in \mathbb{R}$ if and only if $0 < t$.
\subsection{DEFINITION OF THE K-EQUATION:}
\begin{equation}
	K_{n}(u_{n}, a, b) \coloneqq \int_{a}^{b} Z_{n}(u_{n} \cdot t) dt, \quad \forall u_{n}, Z_{n}(u_{n}) \in \mathbb{R}
\end{equation}
Where $K_{n} > 0$ and $Z_{n}(u_{n})$ is an A-Function of index $n$ at $u$.

\section{INTRODUCTION}

\section{LEMMAS}

\subsection{LEMMA ONE: THE A-LEMMA:}

\subsection{LEMMA TWO: THE K-LEMMA:}

\section{CONCLUSION}

\section{REFERENCES}


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