From be46e57ab98e358e0fce982094bf3a7503de4d80 Mon Sep 17 00:00:00 2001 From: Amlal El Mahrouss Date: Mon, 16 Feb 2026 20:17:14 +0100 Subject: chore: update papers, add assumptions in 'Integrals for Equations and Analysis' paper. Signed-off-by: Amlal El Mahrouss --- source/wg04/paper.tex | 14 +++++++++++--- 1 file changed, 11 insertions(+), 3 deletions(-) (limited to 'source') diff --git a/source/wg04/paper.tex b/source/wg04/paper.tex index f36939b..50b0e45 100644 --- a/source/wg04/paper.tex +++ b/source/wg04/paper.tex @@ -24,10 +24,10 @@ \section{Definitions} -We assume $x, y, z \in \mathbb{R}, \quad n \in \mathbb{N}, \quad \cos(\pi) < a < b < \cos(0), \quad t = a$ +We assume $x, y, z \in \mathbb{R}, \quad n \in \mathbb{N}, \quad \cos(\pi) < a < b, \quad t = a$ \subsection{The A-Integral:} \begin{equation} - I_{1} \coloneqq \sum_{n}^{t} \int_{a}^{b} S(x) dx + I_{1} \coloneqq \sum_{n}^{t} \int_{a}^{b} S(x) d \cdot x \end{equation} \subsection{The A-Function:} \begin{equation} @@ -35,7 +35,15 @@ We assume $x, y, z \in \mathbb{R}, \quad n \in \mathbb{N}, \quad \cos(\pi) < a < \end{equation} \subsection{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} \coloneqq \int_{a}^{b} E_{1}d \cdot x, \quad E_{1} = Z(x) := \frac{(S(x) \cdot y)}{\Delta t \cdot z} \end{equation} +Where $\delta t \in \mathbb{R}$ such that $0 < \delta t$. +\subsection{The K-Equation:} +\begin{equation} + K_{n}=\int_{a}^{b} S_{k}{t}(u_{n}) \quad d \cdot u_{n}, \quad \forall u_{n} \in \mathbb{R} +\end{equation} +Where $K_{n} > 0$ for $z \in \mathbb{R}$. \\ +Where $S_{k}(u)$ is the A-Function where $S_{k}(u) \in \mathbb{C}$ if and only if $u \in \mathbb{R}, \quad u > 0$. + \end{document} -- cgit v1.2.3