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Directory talk:Jon Awbrey/Papers/Differential Logic and Dynamic Systems 2.0 (view source)
Revision as of 01:32, 14 July 2008
, 01:32, 14 July 2008→Table 9: cleanup
====Table 9====
====Table 9====
{| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
{| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%"
|+ '''Table 9. Higher Order Differential Features'''
|+ '''Table 9. Higher Order Differential Features'''
|
<p><math>\begin{array}{lllll}
<math>\begin{smallmatrix}
\operatorname{d}^0 \mathcal{A}
\operatorname{d}^0 \mathcal{A}
& = &
& = & \{a_1, \ldots, a_n\}
\{a_1, & \ldots, & a_n\} \\
& = & \mathcal{A} \\
\operatorname{d}^1 \mathcal{A}
\operatorname{d}^1 \mathcal{A}
& = &
& = & \{\operatorname{d}a_1, \ldots, \operatorname{d}a_n\}
\{\operatorname{d}a_1, & \ldots, & \operatorname{d}a_n\} \\
& = & \operatorname{d}\mathcal{A} \\
\end{array}</math></p>
<p><math>\begin{array}{lll}
\operatorname{d}^k \mathcal{A}
\operatorname{d}^k \mathcal{A}
& = &
& = & \{\operatorname{d}^k a_1, \ldots, \operatorname{d}^k a_n\} \\
\{\operatorname{d}^k a_1,
\operatorname{d}^k a_n\} \\
\operatorname{d}^* \mathcal{A}
\operatorname{d}^* \mathcal{A}
& = &
& = & \{\operatorname{d}^0 \mathcal{A}, \ldots, \operatorname{d}^k \mathcal{A}, \ldots \} \\
\{\operatorname{d}^0 \mathcal{A},
\end{array}</math></p>
|
\operatorname{d}^k \mathcal{A},
<p><math>\begin{array}{lll}
\operatorname{E}^0 \mathcal{A}
\end{smallmatrix}</math>
& = & \operatorname{d}^0 \mathcal{A} \\
\operatorname{E}^1 \mathcal{A}
<math>\begin{array}{lll}
& = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \mathcal{A} \\
\operatorname{E}^0 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A} \\
\operatorname{E}^k \mathcal{A}
\operatorname{E}^1 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \mathcal{A} \\
& = & \operatorname{d}^0 \mathcal{A}\ \cup\ \ldots\ \cup\ \operatorname{d}^k \mathcal{A} \\
\operatorname{E}^k \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \ldots\ \cup\ \operatorname{d}^k \mathcal{A} \\
\operatorname{E}^\infty \mathcal{A}
\operatorname{E}^\infty \mathcal{A} & = & \bigcup\ \operatorname{d}^* \mathcal{A} \\
& = & \bigcup\ \operatorname{d}^* \mathcal{A} \\
\end{array}</math>
\end{array}</math></p>
|}<br>
|}<br>