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Revision as of 05:16, 19 May 2008
, 05:16, 19 May 2008→Current Version @ PlanetMath : TeX Format: update
\section{Formal development}
\section{Formal development}
Table 4
\begin{center}\begin{tabular}{|l|l|l|l|}
\begin{center}\begin{tabular}{|l|l|l|l|}
$\mathcal{A}$ & $\{ a_1, \ldots, a_n \}$ & Alphabet & $[n] = \mathbf{n}$ \\
$\mathcal{A}$ & $\{ a_1, \ldots, a_n \}$ & Alphabet & $[n] = \mathbf{n}$ \\
\hline
\hline
$A_i$ & $\{ (a_i), a_i \}$ & Dimension $i$ & $\mathbb{B}$ \\
$A_i$ & $\{ \overline{a_i}, a_i \}$ & Dimension $i$ & $\mathbb{B}$ \\
\hline
\hline
$A$ & $\langle \mathcal{A} \rangle$ & Set of cells, & $\mathbb{B}^n$ \\
$A$ & $\langle \mathcal{A} \rangle$ & Set of cells, & $\mathbb{B}^n$ \\
& $(A, (A \to \mathbb{B}))$ & & \\
& $(A, (A \to \mathbb{B}))$ & & \\
& $[ a_1, \ldots, a_n ]$ & & \\
& $[ a_1, \ldots, a_n ]$ & & \\
\hline
\end{tabular}\end{center}
Table 5
\begin{center}\begin{tabular}{|l|l|l|l|}
\multicolumn{4}{c}{\textbf{Table 5. Differential Extension : Basic Notation}} \\
\hline
\textbf{Symbol} &
\textbf{Notation} &
\textbf{Description} &
\textbf{Type} \\
\hline
$\operatorname{d}\mathcal{A}$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
Alphabet of differential features &
$[n] = \mathbf{n}$ \\
\hline
$\operatorname{d}A_i$ &
$\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}$ &
Differential dimension $i$ &
$\mathbb{D}$ \\
\hline
$\operatorname{d}A$ &
$\langle \operatorname{d}\mathcal{A} \rangle$ &
Tangent space at a point: &
$\mathbb{D}^n$
\\
&
$\langle \operatorname{d}a_1, \ldots, \operatorname{d}a_n \rangle$ &
Set of changes, &
\\
&
$\{ (\operatorname{d}a_1, \ldots, \operatorname{d}a_n) \}$ &
motions, steps, &
\\
&
$\operatorname{d}A_1 \times \ldots \times \operatorname{d}A_n$ &
tangent vectors &
\\
&
$\textstyle \prod_{i=1}^n \operatorname{d}A_i$ &
at a point &
\\
\hline
$\operatorname{d}A^*$ &
$(\operatorname{hom} : \operatorname{d}A \to \mathbb{B})$ &
Linear functions on $\operatorname{d}A$ &
$(\mathbb{D}^n)^* \cong \mathbb{D}^n$ \\
\hline
$\operatorname{d}A^\uparrow$ &
$(\operatorname{d}A \to \mathbb{B})$ &
Boolean functions on $\operatorname{d}A$ &
$\mathbb{D}^n \to \mathbb{B}$ \\
\hline
$\operatorname{d}A^\circ$ &
$[ \operatorname{d}\mathcal{A} ]$ &
Tangent universe &
$(\mathbb{D}^n, (\mathbb{D}^n \to \mathbb{B}))$
\\
&
$(\operatorname{d}A, \operatorname{d}A^\uparrow)$ &
at a point of $A^\circ,$ &
$(\mathbb{D}^n\ +\!\to \mathbb{B})$
\\
&
$(\operatorname{d}A\ +\!\to \mathbb{B})$ &
based on the &
$[\mathbb{D}^n]$
\\
&
$(\operatorname{d}A, (\operatorname{d}A \to \mathbb{B}))$ &
tangent features &
\\
&
$[ \operatorname{d}a_1, \ldots, \operatorname{d}a_n ]$ &
$\{ \operatorname{d}a_1, \ldots, \operatorname{d}a_n \}$ &
\\
\hline
\hline
\end{tabular}\end{center}
\end{tabular}\end{center}
