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| -rw-r--r-- | source/dn001.05/paper.tex | 4 | ||||
| -rw-r--r-- | source/wg04/paper.tex | 60 |
2 files changed, 21 insertions, 43 deletions
diff --git a/source/dn001.05/paper.tex b/source/dn001.05/paper.tex index ab7ccc6..cdbbd00 100644 --- a/source/dn001.05/paper.tex +++ b/source/dn001.05/paper.tex @@ -36,10 +36,10 @@ We will define their concepts alongside their properties. We assume the reader h \section{The Execution Product Formula} Let $\Gamma(x, y, z)$ be an execution product of variable arguments $x$, $y$, $z$ defined in compatible and composable execution context $\mathbb{E}$. \\ -Let the index $\alpha$ denote the current execution domain of an execution product. +Let the index $i$ denote the current execution domain of an execution product. \\ \\ Consider the following formula: \begin{equation} -\Gamma(x, y, z) = \prod_{\alpha=1}^{n}(x_{\alpha} \times y_{\alpha} \times z_{\alpha})+\mathbb{U} +\Gamma(x, y, z) = \prod_{i=1}^{n}(x_{i} \times y_{i} \times z_{i})+\mathbb{U} \end{equation} which we define as the execution product, where $\mathbb{U}$ is the Unknown Execution of $\Gamma(x, y, z)$, now defined as $g(\mathbb{E})$. diff --git a/source/wg04/paper.tex b/source/wg04/paper.tex index c192a19..06ca1f4 100644 --- a/source/wg04/paper.tex +++ b/source/wg04/paper.tex @@ -13,34 +13,6 @@ \author{Amlal El Mahrouss\\\texttt{amlal@nekernel.org}} \date{February 2026} -\definecolor{codegray}{gray}{0.95} -\definecolor{codeblue}{rgb}{0.1,0.1,0.8} -\definecolor{codegreen}{rgb}{0,0.6,0} -\definecolor{codepurple}{rgb}{0.58,0,0.82} - -\lstset{ - language=C++, - backgroundcolor=\color{codegray}, - basicstyle=\footnotesize\ttfamily, - keywordstyle=\color{codeblue}\bfseries, - commentstyle=\color{codegreen}, - stringstyle=\color{codepurple}, - numbers=left, - numberstyle=\tiny\color{gray}, - stepnumber=1, - numbersep=5pt, - breaklines=true, - breakatwhitespace=false, - frame=single, - rulecolor=\color{black}, - captionpos=b, - keepspaces=true, - showspaces=false, - showstringspaces=false, - showtabs=false, - tabsize=2 -} - \begin{document} \bf \maketitle @@ -51,7 +23,7 @@ \abstract { - This technical note defines useful primitives to be used in the case of calculus. The formulas have been kept abstract for the sole purpose of flexibility. + This paper defines, and documents useful primitives to be used in the case of calculus. Them shall been kept abstract for the sole purpose of flexibility. } \begin{center} @@ -60,7 +32,8 @@ \section{Introduction} -We will define alongside this paper useful primitives in order to complete analysis in a mathematical manner. +We will define in this paper useful primitives in order to complete analysis in a mathematical manner.\\ +It will contain definitions and properties in a top-to-bottom order. \section{Definitions} @@ -70,24 +43,24 @@ Let the following definitions be used in mathematical analysis: Let an integral $\alpha$ be defined: \begin{equation} - \int_{a_{0}}^{a_{n}}\sum_{b_{a_{0}},k=a_{0}} f_{b_{a_{0}}}(S(k+b_{a_{0}})) + C + \sum_{k=1}^{a_{n}} \int_{a_{0}}^{a_{n}} f_{k}(S(b_{a_{0}})) + C, \quad b_{a_{0}} > 0, \quad b_{a_{0}} \in \mathbb{R} \end{equation} Such that: \begin{equation} - a \geq 0, \quad a \in \mathbb{N}, \quad k \neq 0, \quad k \in \mathbb{R} + a > 0, \quad a \in \mathbb{R}, \quad k \neq 0, \quad k \in \mathbb{R} \end{equation} -Such that $S$ is a user-defined computing function of a sum $k+b_{a_{0}}$. +Such that $S$ is a user-defined computing function of a sum $b_{a_{0}}$. \subsection{Definition of Ze in $\mathbb{R}$} Let an equation $Ze$ be defined: \begin{equation} - \forall t \in \mathbb{R}, \quad t_{n} = Ze_{n}(\Delta t) + \forall t \in \mathbb{R}, \quad t = Ze_{n}(\Delta t_{n}) \end{equation} Such that: \begin{equation} - n \geq 0, \quad n \in \mathbb{N} + n \geq 0, \quad n \in \mathbb{R} \end{equation} Such that: \begin{equation} @@ -95,6 +68,16 @@ Such that: \end{equation} Such that $Ze_{n}$ be the current yield result of $t_{n} \in \Delta t$. +\subsection{Definition of $Ke(k, u, n)$} + +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 +\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: @@ -104,13 +87,8 @@ Let the following properties be used in mathematical analysis: Let the properties be: \begin{itemize} - \item $\Delta t \in \mathbb{C}, \quad \forall x \in \Delta t$ + \item $\Delta t \in \mathbb{C}, \quad \forall x \in \Delta t, \quad x \in \mathbb{C}$ \end{itemize} - -\section{Conclusion} - -We have defined two primitives and one property in order to help us analyse mathematical domains within $\mathbb{R}$ or $\mathbb{C}$. - \end{document} |