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| -rw-r--r-- | source/dn001.07/paper.tex | 22 | ||||
| -rw-r--r-- | source/wg04/paper.2.tex | 36 | ||||
| -rw-r--r-- | source/wg04/paper.tex | 52 | ||||
| -rw-r--r-- | wg04.mk | 2 |
4 files changed, 53 insertions, 59 deletions
diff --git a/source/dn001.07/paper.tex b/source/dn001.07/paper.tex index b759098..270487b 100644 --- a/source/dn001.07/paper.tex +++ b/source/dn001.07/paper.tex @@ -14,7 +14,6 @@ \date{February 2026} \begin{document} -\bf \maketitle \begin{center} @@ -25,25 +24,12 @@ \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 $S(t) = \sum_{k=1}^{n} (t!), \quad \forall k, n \in \mathbb{Z}$. -Consider the following integral: +Let us define the following: \begin{equation} - \alpha(u) = \int S(t) du + f(u) = \int_{u_{0}}^{u} S(t) dt \end{equation} -For all $\alpha$ in $0 < \alpha < e$, or $\alpha > e$. -For $\alpha \in \mathbb{R}, \quad u \in \mathbb{R}, \quad u > 0$. - -\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 $\infty+$. +where $u > 0, \quad u \in \mathbb{R}$\\ +let $S : (0, e) \to \mathbb{R}$ be continuous. \end{document} diff --git a/source/wg04/paper.2.tex b/source/wg04/paper.2.tex new file mode 100644 index 0000000..c5f2cd4 --- /dev/null +++ b/source/wg04/paper.2.tex @@ -0,0 +1,36 @@ +% AUTHOR: Amlal El Mahrouss +% PURPOSE: WG04.2: Sum Analysis. + +\documentclass[11pt, a4paper]{article} +\usepackage{graphicx} +\usepackage{listings} +\usepackage{amsmath,amssymb,amsthm} +\usepackage{xcolor} +\usepackage{hyperref} +\usepackage[margin=0.5in,top=1in,bottom=1in]{geometry} + +\title{Sum Analysis.} +\author{Amlal El Mahrouss\\\texttt{amlal@nekernel.org}} +\date{February 2026} + +\begin{document} + +\maketitle + +\begin{center} + \rule[0.01cm]{17cm}{0.01cm} +\end{center} + +\section{Definitions} +\subsection{The Sum Integral:} +\begin{equation} + \int_{a}^{b} S(x)dx, \quad \forall a, b \in \mathbb{R}, \quad 0 < a < b +\end{equation} +\subsection{The Sum Function:} +\begin{equation} + S(x) := x^{e}, \quad x \in \mathbb{R}, \quad x > 0 +\end{equation} + + +\end{document} + diff --git a/source/wg04/paper.tex b/source/wg04/paper.tex index b61a97f..13b4faf 100644 --- a/source/wg04/paper.tex +++ b/source/wg04/paper.tex @@ -7,6 +7,7 @@ \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.} @@ -15,58 +16,27 @@ \begin{document} -\bf \maketitle +\maketitle \begin{center} \rule[0.01cm]{17cm}{0.01cm} \end{center} -\section{Abstract} +\section{Formulas definitions} -We will introduce in this paper useful primitives in order to help in analysis. We will also define and describe properties in a top-to-bottom order. - -\section{Definitions} - -Let the following definitions be used in mathematical analysis: - -\subsection{A sum of an integral from $a_{0}$ to $a_{n}$} - -Let an integral $\alpha$ be defined: +Let us define four formulas using the definitons below: +We assume $x \in \mathbb{R}, \quad n \in \mathbb{N}, \quad 0 < k < a < b < 1$ +\subsection{Sum Integral:} \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} + I_{1} \coloneqq n \int_{a}^{b} S(x) dx \end{equation} -Such that $a > 0, \quad a \in \mathbb{R}, \quad k \neq 0, \quad k \in \mathbb{Z}$ Such that $S$ is a user-defined computing function of a sum $b_{a_{0}}$ - -\subsection{An equation Ze in $\mathbb{R}$} - -Let an equation $Ze$ be defined as: - +\subsection{Sum Formula:} \begin{equation} - \forall t \in \mathbb{R}, \quad t = Ze_{n}(\Delta t_{n}) + S(x) \coloneqq \cos(x)\times\sin(x) \end{equation} -Such that $n > 0, \quad n \in \mathbb{R}$ Such that $\lvert \Delta t \rvert \geq n, \quad \Delta t \in \mathbb{C}$ Such that $Ze_{n}$ be the yield result of $t_{n} \in \Delta t$. - -\subsection{Definition of an integral $Ke(k, u, n)$} - -Let $Ke(k, u, n)$ be defined as: - +\subsection{Sum Equation:} \begin{equation} - 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 + I_{2} \coloneqq \int_{k}^{a} Z(x)dx, \quad Z(x) := (S(x) \times x) \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 $u \in \mathbb{R}, \quad u > 0$. - -\section{Properties} - -Let the following properties be used in mathematical analysis: - -\subsection{Properties of $\Delta t$ in $Ze$} - -Let the properties be for $Ze$: - -\begin{itemize} - \item $\Delta t \in \mathbb{C}, \quad \forall x \in \Delta t, \quad x \in \mathbb{C}$ -\end{itemize} - \end{document} @@ -6,6 +6,8 @@ .PHONY: wg04 wg04: clean + $(HTMLTEX) source/wg04/sketch-01.tex + $(PDFTEX) source/wg04/sketch-01.tex $(HTMLTEX) source/wg04/paper.tex $(PDFTEX) source/wg04/paper.tex |