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| | ====Table 9==== | | ====Table 9==== |
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| − | <font face="courier new">
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| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="text-align:left; width:96%"
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| − | |+ '''Table 9. Higher Order Differential Features'''
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| − | | width=50% |
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| − | {| cellpadding="4"
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| − | | <font face="lucida calligraphy">A</font>
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| − | | =
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| − | | d<sup>0</sup><font face="lucida calligraphy">A</font>
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| − | | =
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| − | | {''a''<sub>1</sub>,
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| − | | …,
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| − | | ''a''<sub>''n''</sub>}
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| − | |-
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| − | | d<font face="lucida calligraphy">A</font>
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| − | | =
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| − | | d<sup>1</sup><font face="lucida calligraphy">A</font>
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| − | | =
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| − | | {d''a''<sub>1</sub>,
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| − | | …,
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| − | | d''a''<sub>''n''</sub>}
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| − | |-
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| − | |
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| − | |
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| − | | d<sup>''k''</sup><font face="lucida calligraphy">A</font>
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| − | | =
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| − | | {d<sup>''k''</sup>''a''<sub>''1''</sub>,
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| − | | …,
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| − | | d<sup>''k''</sup>''a''<sub>''n''</sub>}
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| − | |-
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| − | | d<sup>*</sup><font face="lucida calligraphy">A</font>
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| − | | =
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| − | | {d<sup>0</sup><font face="lucida calligraphy">A</font>,
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| − | | …,
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| − | | d<sup>''k''</sup><font face="lucida calligraphy">A</font>,
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| − | | …}
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| − | |}
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| − | | width=50% |
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| − | {| cellpadding="4"
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| − | | E<sup>0</sup><font face="lucida calligraphy">A</font>
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| − | | =
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| − | | d<sup>0</sup><font face="lucida calligraphy">A</font>
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| − | |-
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| − | | E<sup>1</sup><font face="lucida calligraphy">A</font>
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| − | | =
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| − | | d<sup>0</sup><font face="lucida calligraphy">A</font> ∪ d<sup>1</sup><font face="lucida calligraphy">A</font>
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| − | |-
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| − | | E<sup>''k''</sup><font face="lucida calligraphy">A</font>
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| − | | =
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| − | | d<sup>0</sup><font face="lucida calligraphy">A</font> ∪ … ∪ d<sup>''k''</sup><font face="lucida calligraphy">A</font>
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| − | |-
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| − | | E<sup>∞</sup><font face="lucida calligraphy">A</font>
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| − | | =
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| − | | ∪ d<sup>*</sup><font face="lucida calligraphy">A</font>
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| − | |}
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| − | |}
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| − | </font><br>
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| − |
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| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:center; width:96%"
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| − | |+ '''Table 9. Higher Order Differential Features'''
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| − | | width=50% |
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| − | <math>\begin{smallmatrix}
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| − | \mathcal{A}
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| − | & = &
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| − | \operatorname{d}^0 \mathcal{A}
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| − | & = &
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| − | \{a_1, & \ldots, & a_n\} \\
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| − | \operatorname{d}\mathcal{A}
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| − | & = &
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| − | \operatorname{d}^1 \mathcal{A}
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| − | & = &
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| − | \{\operatorname{d}a_1, & \ldots, & \operatorname{d}a_n\} \\
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| − | & &
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| − | \operatorname{d}^k \mathcal{A}
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| − | & = &
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| − | \{\operatorname{d}^k a_1,
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| − | & \ldots, &
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| − | \operatorname{d}^k a_n\} \\
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| − | \operatorname{d}^* \mathcal{A}
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| − | & = &
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| − | \{\operatorname{d}^0 \mathcal{A},
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| − | & \ldots, &
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| − | \operatorname{d}^k \mathcal{A},
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| − | & \ldots \}
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| − | \end{smallmatrix}</math>
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| − | | width=50% |
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| − | <math>\begin{array}{lll}
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| − | \operatorname{E}^0 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A} \\
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| − | \operatorname{E}^1 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \mathcal{A} \\
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| − | \operatorname{E}^k \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \ldots\ \cup\ \operatorname{d}^k \mathcal{A} \\
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| − | \operatorname{E}^\infty \mathcal{A} & = & \bigcup\ \operatorname{d}^* \mathcal{A} \\
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| − | \end{array}</math>
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| − | |}<br>
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| | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%" | | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%" |
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| | <p><math>\begin{array}{lll} | | <p><math>\begin{array}{lll} |
| − | \operatorname{E}^0 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A} \\ | + | \operatorname{E}^0 \mathcal{A} |
| − | \operatorname{E}^1 \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \mathcal{A} \\ | + | & = & \operatorname{d}^0 \mathcal{A} \\ |
| − | \operatorname{E}^k \mathcal{A} & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \ldots\ \cup\ \operatorname{d}^k \mathcal{A} \\ | + | \operatorname{E}^1 \mathcal{A} |
| − | \operatorname{E}^\infty \mathcal{A} & = & \bigcup\ \operatorname{d}^* \mathcal{A} \\ | + | & = & \operatorname{d}^0 \mathcal{A}\ \cup\ \operatorname{d}^1 \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} |
| | + | & = & \bigcup\ \operatorname{d}^* \mathcal{A} \\ |
| | \end{array}</math></p> | | \end{array}</math></p> |
| | |}<br> | | |}<br> |