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| | ===The Extended Universe of Discourse=== | | ===The Extended Universe of Discourse=== |
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| − | <font face="courier new">
| + | ====Table 8==== |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="text-align:left; width:96%"
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| − | |+ '''Table 8. Notation for the Differential Extension of Propositional Calculus'''
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| − | |- style="background:ghostwhite"
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| − | ! Symbol
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| − | ! Notation
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| − | ! Description
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| − | ! Type
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| − | |-
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| − | | d<font face="lucida calligraphy">A<font>
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| − | | {d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
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| − | |
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| − | Alphabet of<br>
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| − | differential<br>
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| − | features
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| − | | [''n''] = '''n'''
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| − | |-
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| − | | d''A''<sub>''i''</sub>
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| − | | {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>}
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| − | |
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| − | Differential<br>
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| − | dimension ''i''
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| − | | '''D'''
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| − | |-
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| − | | d''A''
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| − | |
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| − | 〈d<font face="lucida calligraphy">A</font>〉<br>
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| − | 〈d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>〉<br>
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| − | {‹d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>›}<br>
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| − | d''A''<sub>1</sub> × … × d''A''<sub>''n''</sub><br>
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| − | ∏<sub>''i''</sub> d''A''<sub>''i''</sub>
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| − | |
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| − | Tangent space<br>
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| − | at a point:<br>
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| − | Set of changes,<br>
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| − | motions, steps,<br>
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| − | tangent vectors<br>
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| − | at a point
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| − | | '''D'''<sup>''n''</sup>
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| − | |-
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| − | | d''A''*
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| − | | (hom : d''A'' → '''B''')
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| − | |
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| − | Linear functions<br>
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| − | on d''A''
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| − | | ('''D'''<sup>''n''</sup>)* = '''D'''<sup>''n''</sup>
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| − | |-
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| − | | d''A''^
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| − | | (d''A'' → '''B''')
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| − | |
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| − | Boolean functions<br>
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| − | on d''A''
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| − | | '''D'''<sup>''n''</sup> → '''B'''
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| − | |-
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| − | | d''A''<sup>•</sup>
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| − | |
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| − | [d<font face="lucida calligraphy">A</font>]<br>
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| − | (d''A'', d''A''^)<br>
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| − | (d''A'' +→ '''B''')<br>
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| − | (d''A'', (d''A'' → '''B'''))<br>
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| − | [d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>]
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| − | |
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| − | Tangent universe<br>
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| − | at a point of ''A''<sup>•</sup>,<br>
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| − | based on the<br>
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| − | tangent features<br>
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| − | {d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
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| − | |
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| − | ('''D'''<sup>''n''</sup>, ('''D'''<sup>''n''</sup> → '''B'''))<br>
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| − | ('''D'''<sup>''n''</sup> +→ '''B''')<br>
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| − | ['''D'''<sup>''n''</sup>]
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| − | |}
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| − | </font><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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| | |- | | |- |
| | | <math>\operatorname{d}A_i</math> | | | <math>\operatorname{d}A_i</math> |
| − | | <math>\{ \overline{\operatorname{d}a_i}, \operatorname{d}a_i \}</math> | + | | <math>\{ (\operatorname{d}a_i), \operatorname{d}a_i \}</math> |
| | | Differential<br> | | | Differential<br> |
| | dimension <math>i\!</math> | | dimension <math>i\!</math> |
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| | <math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br> | | <math>(\mathbb{D}^n\ +\!\to \mathbb{B})</math><br> |
| | <math>[\mathbb{D}^n]</math> | | <math>[\mathbb{D}^n]</math> |
| − | |} | + | |}<br> |
| − | <br> | + | |
| | + | ====Table 9==== |
| | | | |
| | {| 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%" |
| − | |+ '''Table 2. Propositional Calculus : Basic Notation''' | + | |+ '''Table 9. Higher Order Differential Features''' |
| − | |- style="background:ghostwhite"
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| − | ! Symbol
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| − | ! Notation
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| − | ! Description
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| − | ! Type
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| − | |-
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| − | | <math>\mathfrak{A}</math>
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| − | | <math>\lbrace\!</math> “<math>a_1\!</math>” <math>, \ldots,\!</math> “<math>a_n\!</math>” <math>\rbrace\!</math>
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| − | | <math>\operatorname{Alphabet}</math>
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| − | | <math>[n] = \mathbf{n}</math>
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| − | |-
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| − | | <math>\mathcal{A}</math>
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| − | | <math>\{a_1, \ldots, a_n\}</math>
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| − | | <math>\operatorname{Basis}</math>
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| − | | <math>[n] = \mathbf{n}</math>
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| − | |-
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| − | | <math>A_i\!</math>
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| − | | <math>\{(a_i), a_i\}\!</math>
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| − | | <math>\operatorname{Dimension}\ i</math>
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| − | | <math>\mathbb{B}</math>
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| − | |-
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| − | | <math>A\!</math>
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| − | |
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| − | <p><math>\langle \mathcal{A} \rangle</math></p>
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| − | <p><math>\langle a_1, \ldots, a_n \rangle</math></p>
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| − | <p><math>\{(a_1, \ldots, a_n)\}</math></p>
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| − | <p><math>A_1 \times \ldots \times A_n</math></p>
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| − | <p><math>\textstyle \prod_{i=1}^n A_i</math></p>
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| − | |
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| − | <p><math>\operatorname{Set~of~cells},</math></p>
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| − | <p><math>\operatorname{coordinate~tuples},</math></p>
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| − | <p><math>\operatorname{points,~or~vectors}</math></p>
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| − | <p><math>\operatorname{in~the~universe}</math></p>
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| − | <p><math>\operatorname{of~discourse}</math></p>
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| − | | <math>\mathbb{B}^n</math>
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| − | |-
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| − | | <math>A^*\!</math>
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| − | | <math>(\operatorname{hom} : A \to \mathbb{B})</math>
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| − | | <math>\operatorname{Linear~functions}</math>
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| − | | <math>(\mathbb{B}^n)^* \cong \mathbb{B}^n</math>
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| − | |-
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| − | | <math>A^\uparrow</math>
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| − | | <math>(A \to \mathbb{B})</math>
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| − | | <math>\operatorname{Boolean~functions}</math>
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| − | | <math>\mathbb{B}^n \to \mathbb{B}</math>
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| − | |-
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| − | | <math>A^\circ</math>
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| − | |
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| − | <p><math>[\mathcal{A}]</math></p>
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| − | <p><math>(A, A^\uparrow)</math></p>
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| − | <p><math>(A\ +\!\to \mathbb{B})</math></p>
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| − | <p><math>(A, (A \to \mathbb{B}))</math></p>
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| − | <p><math>[a_1, \ldots, a_n]</math></p>
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| | | | | | |
| − | <p><math>\operatorname{Universe~of~discourse}</math></p> | + | <p><math>\begin{array}{lllll} |
| − | <p><math>\operatorname{based~on~the~features}</math></p> | + | \operatorname{d}^0 \mathcal{A} |
| − | <p><math>\{a_1, \ldots, a_n\}</math></p>
| + | & = & \{a_1, \ldots, a_n\} |
| | + | & = & \mathcal{A} \\ |
| | + | \operatorname{d}^1 \mathcal{A} |
| | + | & = & \{\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 a_1, \ldots, \operatorname{d}^k a_n\} \\ |
| | + | \operatorname{d}^* \mathcal{A} |
| | + | & = & \{\operatorname{d}^0 \mathcal{A}, \ldots, \operatorname{d}^k \mathcal{A}, \ldots \} \\ |
| | + | \end{array}</math></p> |
| | | | | | |
| − | <p><math>(\mathbb{B}^n, (\mathbb{B}^n \to \mathbb{B}))</math></p> | + | <p><math>\begin{array}{lll} |
| − | <p><math>(\mathbb{B}^n\ +\!\to \mathbb{B})</math></p>
| + | \operatorname{E}^0 \mathcal{A} |
| − | <p><math>[\mathbb{B}^n]</math></p>
| + | & = & \operatorname{d}^0 \mathcal{A} \\ |
| | + | \operatorname{E}^1 \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> |
| | |}<br> | | |}<br> |