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\fancyhead[L]{\small\itshape Ne.org, Vol.~2, No.~1, March 2026}
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%% ============================================================
%% FRONT MATTER
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\title{%
\vspace{-1.5em}%
\large\textbf{All you need is Integrals for Debugging.}%
}
\author{%
\textbf{Amlal El Mahrouss}$^{1}$\thanks{Corresponding author. Email: amlal@nekernel.org}
\\amlal@nekernel.org, amlalelmahrouss@icloud.com\\[0.4em]
\small $^{1}$Ne.org Journal
}
\date{%
\small 3 March 2026
}
%% ============================================================
\begin{document}
\twocolumn[{%
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\rule{\linewidth}{0.4pt}
%% ---- Abstract -------------------------------------------
\begin{center}
\begin{minipage}{0.92\linewidth}
\small
\textbf{Abstract.}\enspace
This development presents an approach of software debugging[1] in which we'll name 'Integrals of Debuggging' (ID).
We propose a mathematical of which integrals are used to analyze, modify, and apply, software patches to programs.
\medskip
\textbf{Keywords:}\enspace
Computer Science; Mathematics;
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\section{Introduction}
Consider an open interval from $n$ to $p$ over a field $\mathbb{K}$ which describes a program and its properties. What are the techniques in which we can analyze and audit its properties and behavior? We will try to impose a framework in this report.
\section{Definition of the LongReturn(x, t) function}
\label{sec:cl-example-integrals}
Let an integral of $\operatorname{LongReturn(x, t)}$[2] be defined where the value $n > 0$ and $p > 0, \quad n < p$:
\begin{equation}
\operatorname{LongReturn(x, t)} \coloneqq \int_{n}^{p} \operatorname{FastReturn(x, t)} \cdot d \operatorname{t}
\end{equation}
We assume that $x$ and $t$ are always greater or equal than one, $\operatorname{LongReturn(x, t)}$ shall always be greater than $\operatorname{FastReturn(x, t)}$ for all such value in $\mathbb{K}$.
This introduction of the LongReturn, will let us jump to our next section about its applications in Computer Analysis.
\section{Constraints}
However before starting, we shall provide the following conditions from [2]:
\begin{equation}
\operatorname{LongReturn(x, t)} \geq \operatorname{FastReturn(x, t)}, \quad \operatorname{x} \geq 1
\end{equation}
The variable x shall also be defined in $\mathbb{K}$ throughout the integral. We shall now proceed into the next section.
\section{Definition of the DerRet(x, t) function}
The following is the derivative of $\operatorname{LongReturn(x, t)}$, which equals to:
\begin{equation}
\operatorname{DerRet(State, Time\_State)} \coloneqq \frac{\delta \operatorname{Travel(State)}}{\delta Time\_State} + Rem
\end{equation}
Such that the following is assumed:
\begin{equation}
\quad State, \operatorname{Travel(State)} \in \mathbb{K}
\end{equation}
Indeed, the function $\operatorname{Travel(State)}$ is equal to the program state at the variable $\operatorname{State}$ which represents the program's state.
\section{Applications}
One application of the $\operatorname{LongReturn(x, t)}$ is software debugging, more so, the break-point system of a debugger. The following section includes graphs to measure the efficiency of our approach in such process.
\end{document}
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