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| -rw-r--r-- | source/dn001.07/paper.tex | 47 | ||||
| -rw-r--r-- | source/wg04/paper.tex | 10 |
2 files changed, 52 insertions, 5 deletions
diff --git a/source/dn001.07/paper.tex b/source/dn001.07/paper.tex new file mode 100644 index 0000000..209c34d --- /dev/null +++ b/source/dn001.07/paper.tex @@ -0,0 +1,47 @@ + +\documentclass[11pt, a4paper]{article} +\usepackage{graphicx} +\usepackage{listings} +\usepackage{xcolor} +\usepackage{hyperref} +\usepackage{amsfonts} +\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry} + +\title{Lemmas for Integrals.} +\author{Amlal El Mahrouss\\amlal@nekernel.org} +\date{February 2026} + +\begin{document} +\bf +\maketitle + +\begin{center} + \rule[0.01cm]{17cm}{0.01cm} +\end{center} + +\section{Lemmas} + +\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 $S(t) = \sum_{k=1}^{t} (t!), \quad \forall k \in \mathbb{Z}$. +Consider the following integral: + +\begin{equation} + \int_{0}^{t} S(t) dt +\end{equation} +For all $\alpha$ in $0 < \alpha < e$, $\alpha > e$ and $\alpha < \pi$. +For $\alpha \in \mathbb{R}$. + +\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 $\infty$. + +\end{document} diff --git a/source/wg04/paper.tex b/source/wg04/paper.tex index 97dc261..9eb3dd3 100644 --- a/source/wg04/paper.tex +++ b/source/wg04/paper.tex @@ -45,7 +45,7 @@ Let an integral $\alpha$ be defined: \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} \end{equation} -Such that: +Such that \begin{equation} a > 0, \quad a \in \mathbb{R}, \quad k \neq 0, \quad k \in \mathbb{Z} \end{equation} @@ -58,11 +58,11 @@ Let an equation $Ze$ be defined: \begin{equation} \forall t \in \mathbb{R}, \quad t = Ze_{n}(\Delta t_{n}) \end{equation} -Such that: +Such that \begin{equation} - n \geq 0, \quad n \in \mathbb{R} + n > 0, \quad n \in \mathbb{R} \end{equation} -Such that: +Such that \begin{equation} \lvert \Delta t \rvert \geq n, \quad \Delta t \in \mathbb{C} \end{equation} @@ -73,7 +73,7 @@ Such that $Ze_{n}$ be the yield result of $t_{n} \in \Delta t$. Let $Ke(k, u, n)$: \begin{equation} - Ke(k, u, n)=\int_{k}^{n} \Phi_{k}(u) + C, \quad \forall k, n \in \mathbb{R}, \quad k \geq 0 + 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 \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 $u \in \mathbb{R}, \quad u > 0$. |