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Directory talk:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 20:25, 9 July 2008
, 20:25, 9 July 2008→Original Format: HTML → TeX
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<math>\begin{matrix}
<math>\begin{matrix}
\mathcal{X} & = & \{x_1, \ldots, x_n\} \\
\mathcal{X}
& = & \{x_1, \ldots, x_n\} \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\underline\mathcal{X} & = & \{\underline{x}_1, \ldots, \underline{x}_n\} \\
\underline\mathcal{X}
& = & \{\underline{x}_1, \ldots, \underline{x}_n\} \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\mathcal{A} & = & \{a_1, \ldots, a_n\} \\
\mathcal{A}
& = & \{a_1, \ldots, a_n\} \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
X_i & = & \langle x_i \rangle \\
X_i
& = & \langle x_i \rangle \\
& \cong & \mathbb{K} \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\underline{X}_i & = & \{(\underline{x}_i), \underline{x}_i \} \\
\underline{X}_i
& = & \{(\underline{x}_i), \underline{x}_i \} \\
& \cong & \mathbb{B} \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
A_i & = & \{(a_i), a_i \} \\
A_i
& = & \{(a_i), a_i \} \\
& \cong & \mathbb{B} \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
X & = & \langle \mathcal{X} \rangle \\
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>
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<math>\begin{matrix}
<math>\begin{matrix}
\underline{X} & = & \langle \underline\mathcal{X} \rangle \\
\underline{X}
& = & \langle \underline\mathcal{X} \rangle \\
& = & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\
& = & \langle \underline{x}_1, \ldots, \underline{x}_n \rangle \\
& = & \underline{X}_1 \times \ldots \times \underline{X}_n \\
& = & \underline{X}_1 \times \ldots \times \underline{X}_n \\
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<math>\begin{matrix}
<math>\begin{matrix}
A & = & \langle \mathcal{A} \rangle \\
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>
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<math>\begin{matrix}
<math>\begin{matrix}
X^* & = & (\ell : X \to \mathbb{K}) \\
X^*
& = & (\ell : X \to \mathbb{K}) \\
& \cong & \mathbb{K}^n \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
\underline{X}^* & = & (\ell : \underline{X} \to \mathbb{B}) \\
\underline{X}^*
& = & (\ell : \underline{X} \to \mathbb{B}) \\
& \cong & \mathbb{B}^n \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
<math>\begin{matrix}
A^* & = & (\ell : A \to \mathbb{B}) \\
A^*
& = & (\ell : A \to \mathbb{B}) \\
& \cong & \mathbb{B}^n \\
\end{matrix}</math>
\end{matrix}</math>
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<math>\begin{matrix}
(''X'' → '''K''')<br>
X^\uparrow
& = & (X \to \mathbb{K}) \\
('''K'''<sup>''n''</sup> → '''K''')
& \cong & (\mathbb{K}^n \to \mathbb{K}) \\
\end{matrix}</math>
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<u>''X''</u>^<br>
<math>\begin{matrix}
(<u>''X''</u> → '''B''')<br>
\underline{X}^\uparrow
& = & (\underline{X} \to \mathbb{B}) \\
('''B'''<sup>''n''</sup> → '''B''')
& \cong & (\mathbb{B}^n \to \mathbb{B}) \\
\end{matrix}</math>
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<math>\begin{matrix}
(''A'' → '''B''')<br>
A^\uparrow
& = & (A \to \mathbb{B}) \\
('''B'''<sup>''n''</sup> → '''B''')
& \cong & (\mathbb{B}^n \to \mathbb{B}) \\
\end{matrix}</math>
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