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% AUTHOR: Amlal El Mahrouss
% PURPOSE: WG04: Integrals and Equations for Analysis.
\documentclass[11pt, a4paper]{article}
\usepackage{graphicx}
\usepackage{listings}
\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.}
\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}
\begin{center}
\rule[0.01cm]{17cm}{0.01cm}
\end{center}
\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) := 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}
\end{document}
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