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Directory talk:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 23:52, 9 July 2008
, 23:52, 9 July 2008→Original Format: restructure table
====Original Format====
====Original Format====
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%"
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 5. A Bridge Over Troubled Waters'''
|+ '''Table 5. A Bridge Over Troubled Waters'''
|- style="background:ghostwhite"
|- style="background:ghostwhite"
|
|
<math>\begin{matrix}
<math>\begin{matrix}
X
X \\
= & \langle \mathcal{X} \rangle \\
= & \langle x_1, \ldots, x_n \rangle \\
= & X_1 \times \ldots \times X_n \\
= & \prod_{i=1}^n X_i \\
\cong & \mathbb{K}^n \\
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
\underline{X}
\underline{X} \\
= & \langle \underline\mathcal{X} \rangle \\
= & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\
= & \underline{X}_1 \times \ldots \times \underline{X}_n \\
= & \prod_{i=1}^n \underline{X}_i \\
\cong & \mathbb{B}^n \\
\end{matrix}</math>
\end{matrix}</math>
|
|
<math>\begin{matrix}
<math>\begin{matrix}
A
A \\
= & \langle \mathcal{A} \rangle \\
= & \langle a_1, \ldots, a_n \rangle \\
= & A_1 \times \ldots \times A_n \\
= & \prod_{i=1}^n A_i \\
\cong & \mathbb{B}^n \\
\end{matrix}</math>
\end{matrix}</math>
|-
|-
|
|
<math>\begin{matrix}
<math>\begin{matrix}
X^\circ
X^\circ \\
= & [\mathcal{X}] \\
= & [x_1, \ldots, x_n] \\
= & (X, X^\uparrow) \\
= & (X\ +\!\to \mathbb{K}) \\
= & (X, (X \to \mathbb{K})) \\
\cong & (\mathbb{K}^n, (\mathbb{K}^n \to \mathbb{K})) \\
= & (\mathbb{K}^n\ +\!\to \mathbb{K}) \\
= & [\mathbb{K}^n] \\
\end{matrix}</math>
\end{matrix}</math>
|
|
<u>''X''</u><sup>•</sup><br>
<math>\begin{matrix}
[<font face="lucida calligraphy"><u>X</u></font>]<br>
\underline{X}^\circ \\
[<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>]<br>
= & [\underline\mathcal{X}] \\
(<u>''X''</u>, <u>''X''</u>^)<br>
= & [\underline{x}_1, \ldots, \underline{x}_n] \\
(<u>''X''</u> +→ '''B''')<br>
= & (\underline{X}, \underline{X}^\uparrow) \\
(<u>''X''</u>, (<u>''X''</u> → '''B'''))<br>
= & (\underline{X}\ +\!\to \mathbb{B}) \\
= & (\underline{X}, (\underline{X} \to \mathbb{B})) \\
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\
('''B'''<sup>''n''</sup> +→ '''B''')<br>
= & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\
['''B'''<sup>''n''</sup>]
= & [\mathbb{B}^n] \\
\end{matrix}</math>
|
|
<math>\begin{matrix}
[<font face="lucida calligraphy">A</font>]<br>
A^\circ \\
[''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>]<br>
= & [\mathcal{A}] \\
(''A'', ''A''^)<br>
= & [a_1, \ldots, a_n] \\
(''A'' +→ '''B''')<br>
= & (A, A^\uparrow) \\
(''A'', (''A'' → '''B'''))<br>
= & (A\ +\!\to \mathbb{B}) \\
= & (A, (A \to \mathbb{B})) \\
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
\cong & (\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B})) \\
('''B'''<sup>''n''</sup> +→ '''B''')<br>
= & (\mathbb{B}^n\ +\!\to \mathbb{B}) \\
['''B'''<sup>''n''</sup>]
= & [\mathbb{B}^n] \\
\end{matrix}</math>
|}<br>
|}<br>