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| Line 331: |
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| | <br> | | <br> |
| | | | |
| − | ==Differential Logic and Dynamic Systems==
| |
| | | | |
| − | ===Table 1. Syntax & Semantics of a Calculus for Propositional Logic===
| |
| − |
| |
| − | <pre>
| |
| − | Table 1. Syntax & Semantics of a Calculus for Propositional Logic
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | Expression | Interpretation | Other Notations |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | " " | True. | 1 |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | () | False. | 0 |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | A | A. | A |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | (A) | Not A. | A' |
| |
| − | | | | ~A |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | A B C | A and B and C. | A & B & C |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | ((A)(B)(C)) | A or B or C. | A v B v C |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | (A (B)) | A implies B. | A => B |
| |
| − | | | If A then B. | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | (A, B) | A not equal to B. | A =/= B |
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| − | | | A exclusive-or B. | A + B |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | ((A, B)) | A is equal to B. | A = B |
| |
| − | | | A if & only if B. | A <=> B |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | (A, B, C) | Just one of | A'B C v |
| |
| − | | | A, B, C | A B'C v |
| |
| − | | | is false. | A B C' |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | ((A),(B),(C)) | Just one of | A B'C' v |
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| − | | | A, B, C | A'B C' v |
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| − | | | is true. | A'B'C |
| |
| − | | | | |
| |
| − | | | Partition all | |
| |
| − | | | into A, B, C. | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | ((A, B), C) | Oddly many of | A + B + C |
| |
| − | | (A, (B, C)) | A, B, C | |
| |
| − | | | are true. | A B C v |
| |
| − | | | | A B'C' v |
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| − | | | | A'B C' v |
| |
| − | | | | A'B'C |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | | (Q, (A),(B),(C)) | Partition Q | Q'A'B'C' v |
| |
| − | | | into A, B, C. | Q A B'C' v |
| |
| − | | | | Q A'B C' v |
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| − | | | Genus Q comprises | Q A'B'C |
| |
| − | | | species A, B, C. | |
| |
| − | o-------------------o-------------------o-------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| |
| − | |+ '''Table 1. Syntax and Semantics of a Calculus for Propositional Logic'''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! Expression
| |
| − | ! Interpretation
| |
| − | ! Other Notations
| |
| − | |-
| |
| − | | " "
| |
| − | | True.
| |
| − | | 1
| |
| − | |-
| |
| − | | ( )
| |
| − | | False.
| |
| − | | 0
| |
| − | |-
| |
| − | | A
| |
| − | | A.
| |
| − | | A
| |
| − | |-
| |
| − | | (A)
| |
| − | | Not A.
| |
| − | | A’ <br> ~A <br> ¬A
| |
| − | |-
| |
| − | | A B C
| |
| − | | A and B and C.
| |
| − | | A ∧ B ∧ C
| |
| − | |-
| |
| − | | ((A)(B)(C))
| |
| − | | A or B or C.
| |
| − | | A ∨ B ∨ C
| |
| − | |-
| |
| − | | (A (B))
| |
| − | | A implies B. <br> If A then B.
| |
| − | | A ⇒ B
| |
| − | |-
| |
| − | | (A, B)
| |
| − | | A not equal to B. <br> A exclusive-or B.
| |
| − | | A ≠ B <br> A + B
| |
| − | |-
| |
| − | | ((A, B))
| |
| − | | A is equal to B. <br> A if & only if B.
| |
| − | | A = B <br> A ⇔ B
| |
| − | |-
| |
| − | | (A, B, C)
| |
| − | | Just one of <br> A, B, C <br> is false.
| |
| − | |
| |
| − | A’B C ∨<br>
| |
| − | A B’C ∨<br>
| |
| − | A B C’
| |
| − | |-
| |
| − | | ((A),(B),(C))
| |
| − | | Just one of <br> A, B, C <br> is true. <br><br>
| |
| − | Partition all <br> into A, B, C.
| |
| − | |
| |
| − | A B’C’ ∨<br>
| |
| − | A’B C’ ∨<br>
| |
| − | A’B’C
| |
| − | |-
| |
| − | | ((A, B), C) <br> <br> (A, (B, C))
| |
| − | | Oddly many of <br> A, B, C <br> are true.
| |
| − | |
| |
| − | A + B + C<br> <br>
| |
| − | A B C ∨<br>
| |
| − | A B’C’ ∨<br>
| |
| − | A’B C’ ∨<br>
| |
| − | A’B’C
| |
| − | |-
| |
| − | | (Q, (A),(B),(C))
| |
| − | | Partition Q <br> into A, B, C.<br>
| |
| − | Genus Q comprises <br> species A, B, C.
| |
| − | |
| |
| − | Q’A’B’C’ ∨<br>
| |
| − | Q A B’C’ ∨<br>
| |
| − | Q A’B C’ ∨<br>
| |
| − | Q A’B’C
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Table 2. Fundamental Notations for Propositional Calculus===
| |
| − |
| |
| − | <pre>
| |
| − | Table 2. Fundamental Notations for Propositional Calculus
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | Symbol | Notation | Description | Type |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | !A! | {a_1, ..., a_n} | Alphabet | [n] = #n# |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | A_i | {(a_i), a_i} | Dimension i | B |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | A | <|!A!|> | Set of cells, | B^n |
| |
| − | | | <|a_i, ..., a_n|> | coordinate tuples,| |
| |
| − | | | {<a_i, ..., a_n>} | interpretations, | |
| |
| − | | | A_1 x ... x A_n | points, or vectors| |
| |
| − | | | Prod_i A_i | in the universe | |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | A* | (hom : A -> B) | Linear functions | (B^n)* = B^n |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | A^ | (A -> B) | Boolean functions | B^n -> B |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | A% | [!A!] | Universe of Disc. | (B^n, (B^n -> B)) |
| |
| − | | | (A, A^) | based on features | (B^n +-> B) |
| |
| − | | | (A +-> B) | {a_1, ..., a_n} | [B^n] |
| |
| − | | | (A, (A -> B)) | | |
| |
| − | | | [a_1, ..., a_n] | | |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
| |
| − | |+ '''Table 2. Fundamental Notations for Propositional Calculus'''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! Symbol
| |
| − | ! Notation
| |
| − | ! Description
| |
| − | ! Type
| |
| − | |-
| |
| − | | <font face="lucida calligraphy">A<font>
| |
| − | | {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}
| |
| − | | Alphabet
| |
| − | | [''n''] = '''n'''
| |
| − | |-
| |
| − | | ''A''<sub>''i''</sub>
| |
| − | | {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}
| |
| − | | Dimension ''i''
| |
| − | | '''B'''
| |
| − | |-
| |
| − | | ''A''
| |
| − | |
| |
| − | 〈<font face="lucida calligraphy">A</font>〉<br>
| |
| − | 〈''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>〉<br>
| |
| − | {‹''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>›}<br>
| |
| − | ''A''<sub>1</sub> × … × ''A''<sub>''n''</sub><br>
| |
| − | ∏<sub>''i''</sub> ''A''<sub>''i''</sub>
| |
| − | |
| |
| − | Set of cells,<br>
| |
| − | coordinate tuples,<br>
| |
| − | points, or vectors<br>
| |
| − | in the universe<br>
| |
| − | of discourse
| |
| − | | '''B'''<sup>''n''</sup>
| |
| − | |-
| |
| − | | ''A''*
| |
| − | | (hom : ''A'' → '''B''')
| |
| − | | Linear functions
| |
| − | | ('''B'''<sup>''n''</sup>)* = '''B'''<sup>''n''</sup>
| |
| − | |-
| |
| − | | ''A''^
| |
| − | | (''A'' → '''B''')
| |
| − | | Boolean functions
| |
| − | | '''B'''<sup>''n''</sup> → '''B'''
| |
| − | |-
| |
| − | | ''A''<sup>•</sup>
| |
| − | |
| |
| − | [<font face="lucida calligraphy">A</font>]<br>
| |
| − | (''A'', ''A''^)<br>
| |
| − | (''A'' +→ '''B''')<br>
| |
| − | (''A'', (''A'' → '''B'''))<br>
| |
| − | [''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>]
| |
| − | |
| |
| − | Universe of discourse<br>
| |
| − | based on the features<br>
| |
| − | {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}
| |
| − | |
| |
| − | ('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
| |
| − | ('''B'''<sup>''n''</sup> +→ '''B''')<br>
| |
| − | ['''B'''<sup>''n''</sup>]
| |
| − | |}</font><br>
| |
| − |
| |
| − | ===Table 3. Analogy of Real and Boolean Types===
| |
| − |
| |
| − | <pre>
| |
| − | Table 3. Analogy of Real and Boolean Types
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | Real Domain R | <-> | Boolean Domain B |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | R^n | Basic Space | B^n |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | R^n -> R | Function Space | B^n -> B |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | (R^n -> R) -> R | Tangent Vector | (B^n -> B) -> B |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | R^n -> ((R^n -> R) -> R)| Vector Field | B^n -> ((B^n -> B) -> B)|
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | (R^n x (R^n -> R)) -> R | ditto | (B^n x (B^n -> B)) -> B |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | ((R^n -> R) x R^n) -> R | ditto | ((B^n -> B) x B^n) -> B |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | (R^n -> R) -> (R^n -> R)| Derivation | (B^n -> B) -> (B^n -> B)|
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | R^n -> R^m | Basic Transformation | B^n -> B^m |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | (R^n -> R) -> (R^m -> R)| Function Transformation | (B^n -> B) -> (B^m -> B)|
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | ... | ... | ... |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 3. Analogy of Real and Boolean Types'''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! Real Domain '''R'''
| |
| − | ! ←→
| |
| − | ! Boolean Domain '''B'''
| |
| − | |-
| |
| − | | '''R'''<sup>''n''</sup>
| |
| − | | Basic Space
| |
| − | | '''B'''<sup>''n''</sup>
| |
| − | |-
| |
| − | | '''R'''<sup>''n''</sup> → '''R'''
| |
| − | | Function Space
| |
| − | | '''B'''<sup>''n''</sup> → '''B'''
| |
| − | |-
| |
| − | | ('''R'''<sup>''n''</sup>→'''R''') → '''R'''
| |
| − | | Tangent Vector
| |
| − | | ('''B'''<sup>''n''</sup>→'''B''') → '''B'''
| |
| − | |-
| |
| − | | '''R'''<sup>''n''</sup> → (('''R'''<sup>''n''</sup>→'''R''')→'''R''')
| |
| − | | Vector Field
| |
| − | | '''B'''<sup>''n''</sup> → (('''B'''<sup>''n''</sup>→'''B''')→'''B''')
| |
| − | |-
| |
| − | | ('''R'''<sup>''n''</sup> × ('''R'''<sup>''n''</sup>→ '''R''')) → '''R'''
| |
| − | | ditto
| |
| − | | ('''B'''<sup>''n''</sup> × ('''B'''<sup>''n''</sup>→ '''B''')) → '''B'''
| |
| − | |-
| |
| − | | (('''R'''<sup>''n''</sup>→'''R''') × '''R'''<sup>''n''</sup>) → '''R'''
| |
| − | | ditto
| |
| − | | (('''B'''<sup>''n''</sup>→'''B''') × '''B'''<sup>''n''</sup>) → '''B'''
| |
| − | |-
| |
| − | | ('''R'''<sup>''n''</sup>→'''R''') → ('''R'''<sup>''n''</sup>→'''R''')
| |
| − | | Derivation
| |
| − | | ('''B'''<sup>''n''</sup>→'''B''') → ('''B'''<sup>''n''</sup>→'''B''')
| |
| − | |-
| |
| − | | '''R'''<sup>''n''</sup> → '''R'''<sup>''m''</sup>
| |
| − | | Basic Transformation
| |
| − | | '''B'''<sup>''n''</sup> → '''B'''<sup>''m''</sup>
| |
| − | |-
| |
| − | | ('''R'''<sup>''n''</sup>→'''R''') → ('''R'''<sup>''m''</sup>→'''R''')
| |
| − | | Function Transformation
| |
| − | | ('''B'''<sup>''n''</sup>→'''B''') → ('''B'''<sup>''m''</sup>→'''B''')
| |
| − | |-
| |
| − | | ...
| |
| − | | ...
| |
| − | | ...
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Table 4. An Equivalence Based on the Propositions as Types Analogy===
| |
| − |
| |
| − | <pre>
| |
| − | Table 4. An Equivalence Based on the Propositions as Types Analogy
| |
| − | o-------------------------o------------------------o--------------------------o
| |
| − | | Pattern | Construction | Instance |
| |
| − | o-------------------------o------------------------o--------------------------o
| |
| − | | X -> (Y -> Z) | Vector Field | K^n -> ((K^n -> K) -> K) |
| |
| − | o-------------------------o------------------------o--------------------------o
| |
| − | | (X x Y) -> Z | | (K^n x (K^n -> K)) -> K |
| |
| − | o-------------------------o------------------------o--------------------------o
| |
| − | | (Y x X) -> Z | | ((K^n -> K) x K^n) -> K |
| |
| − | o-------------------------o------------------------o--------------------------o
| |
| − | | Y -> (X -> Z) | Derivation | (K^n -> K) -> (K^n -> K) |
| |
| − | o-------------------------o------------------------o--------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |+ '''Table 4. An Equivalence Based on the Propositions as Types Analogy
| |
| − | '''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! Pattern
| |
| − | ! Construction
| |
| − | ! Instance
| |
| − | |-
| |
| − | | ''X'' → (''Y'' → ''Z'')
| |
| − | | Vector Field
| |
| − | | '''K'''<sup>''n''</sup> → (('''K'''<sup>''n''</sup> → '''K''') → '''K''')
| |
| − | |-
| |
| − | |(''X'' × ''Y'') → ''Z''
| |
| − | |
| |
| − | | ('''K'''<sup>''n''</sup> × ('''K'''<sup>''n''</sup> → '''K''')) → '''K'''
| |
| − | |-
| |
| − | | (''Y'' × ''X'') → ''Z''
| |
| − | |
| |
| − | | (('''K'''<sup>''n''</sup> → '''K''') × '''K'''<sup>''n''</sup>) → '''K'''
| |
| − | |-
| |
| − | | ''Y'' → (''X'' → ''Z'')
| |
| − | | Derivation
| |
| − | | ('''K'''<sup>''n''</sup> → '''K''') → ('''K'''<sup>''n''</sup> → '''K''')
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Table 5. A Bridge Over Troubled Waters===
| |
| − |
| |
| − | <pre>
| |
| − | Table 5. A Bridge Over Troubled Waters
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | Linear Space | Liminal Space | Logical Space |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | | | |
| |
| − | | !X! | !`X`! | !A! |
| |
| − | | | | |
| |
| − | | {x_1, ..., x_n} | {`x`_1, ..., `x`_n} | {a_1, ..., a_n} |
| |
| − | | | | |
| |
| − | | cardinality n | cardinality n | cardinality n |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | | | |
| |
| − | | X_i | `X`_i | A_i |
| |
| − | | | | |
| |
| − | | <|x_i|> | {(`x`_i), `x`_i} | {(a_i), a_i} |
| |
| − | | | | |
| |
| − | | isomorphic to K | isomorphic to B | isomorphic to B |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | | | |
| |
| − | | X | `X` | A |
| |
| − | | | | |
| |
| − | | <|!X!|> | <|!`X`!|> | <|!A!|> |
| |
| − | | | | |
| |
| − | | <|x_1, ..., x_n|> | <|`x`_1, ..., `x`_n|> | <|a_1, ..., a_n|> |
| |
| − | | | | |
| |
| − | | {<x_1, ..., x_n>} | {<`x`_1, ..., `x`_n>} | {<a_1, ..., a_n>} |
| |
| − | | | | |
| |
| − | | X_1 x ... x X_n | `X`_1 x ... x `X`_n | A_1 x ... x A_n |
| |
| − | | | | |
| |
| − | | Prod_i X_i | Prod_i `X`_i | Prod_i A_i |
| |
| − | | | | |
| |
| − | | isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | | | |
| |
| − | | X* | `X`* | A* |
| |
| − | | | | |
| |
| − | | (hom : X -> K) | (hom : `X` -> B) | (hom : A -> B) |
| |
| − | | | | |
| |
| − | | isomorphic to K^n | isomorphic to B^n | isomorphic to B^n |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | | | |
| |
| − | | X^ | `X`^ | A^ |
| |
| − | | | | |
| |
| − | | (X -> K) | (`X` -> B) | (A -> B) |
| |
| − | | | | |
| |
| − | | isomorphic to (K^n -> K)| isomorphic to (B^n -> B)| isomorphic to (B^n -> B)|
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | | | | |
| |
| − | | X% | `X`% | A% |
| |
| − | | | | |
| |
| − | | [!X!] | [!`X`!] | [!A!] |
| |
| − | | | | |
| |
| − | | [x_1, ..., x_n] | [`x`_1, ..., `x`_n] | [a_1, ..., a_n] |
| |
| − | | | | |
| |
| − | | (X, X^) | (`X`, `X`^) | (A, A^) |
| |
| − | | | | |
| |
| − | | (X +-> K) | (`X` +-> B) | (A +-> B) |
| |
| − | | | | |
| |
| − | | (X, (X -> K)) | (`X`, (`X` -> B)) | (A, (A -> B)) |
| |
| − | | | | |
| |
| − | | isomorphic to: | isomorphic to: | isomorphic to: |
| |
| − | | | | |
| |
| − | | (K^n, (K^n -> K)) | (B^n, (B^n -> B)) | (B^n, (B^n -> K)) |
| |
| − | | | | |
| |
| − | | (K^n +-> K) | (B^n +-> B) | (B^n +-> B) |
| |
| − | | | | |
| |
| − | | [K^n] | [B^n] | [B^n] |
| |
| − | o-------------------------o-------------------------o-------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
| |
| − | |+ '''Table 5. A Bridge Over Troubled Waters'''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! Linear Space
| |
| − | ! Liminal Space
| |
| − | ! Logical Space
| |
| − | |-
| |
| − | |
| |
| − | <font face="lucida calligraphy">X</font><br>
| |
| − | {''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>}<br>
| |
| − | cardinality ''n''
| |
| − | |
| |
| − | <font face="lucida calligraphy"><u>X</u></font><br>
| |
| − | {<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>}<br>
| |
| − | cardinality ''n''
| |
| − | |
| |
| − | <font face="lucida calligraphy">A</font><br>
| |
| − | {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}<br>
| |
| − | cardinality ''n''
| |
| − | |-
| |
| − | |
| |
| − | ''X''<sub>''i''</sub><br>
| |
| − | 〈''x''<sub>''i''</sub>〉<br>
| |
| − | isomorphic to '''K'''
| |
| − | |
| |
| − | <u>''X''</u><sub>''i''</sub><br>
| |
| − | {(<u>''x''</u><sub>''i''</sub>), <u>''x''</u><sub>''i''</sub>}<br>
| |
| − | isomorphic to '''B'''
| |
| − | |
| |
| − | ''A''<sub>''i''</sub><br>
| |
| − | {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}<br>
| |
| − | isomorphic to '''B'''
| |
| − | |-
| |
| − | |
| |
| − | ''X''<br>
| |
| − | 〈<font face="lucida calligraphy">X</font>〉<br>
| |
| − | 〈''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>〉<br>
| |
| − | {‹''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>›}<br>
| |
| − | ''X''<sub>1</sub> × … × ''X''<sub>''n''</sub><br>
| |
| − | ∏<sub>''i''</sub> ''X''<sub>''i''</sub><br>
| |
| − | isomorphic to '''K'''<sup>''n''</sup>
| |
| − | |
| |
| − | <u>''X''</u><br>
| |
| − | 〈<font face="lucida calligraphy"><u>X</u></font>〉<br>
| |
| − | 〈<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>〉<br>
| |
| − | {‹<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>›}<br>
| |
| − | <u>''X''</u><sub>1</sub> × … × <u>''X''</u><sub>''n''</sub><br>
| |
| − | ∏<sub>''i''</sub> <u>''X''</u><sub>''i''</sub><br>
| |
| − | isomorphic to '''B'''<sup>''n''</sup>
| |
| − | |
| |
| − | ''A''<br>
| |
| − | 〈<font face="lucida calligraphy">A</font>〉<br>
| |
| − | 〈''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>〉<br>
| |
| − | {‹''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>›}<br>
| |
| − | ''A''<sub>1</sub> × … × ''A''<sub>''n''</sub><br>
| |
| − | ∏<sub>''i''</sub> ''A''<sub>''i''</sub><br>
| |
| − | isomorphic to '''B'''<sup>''n''</sup>
| |
| − | |-
| |
| − | |
| |
| − | ''X''*<br>
| |
| − | (hom : ''X'' → '''K''')<br>
| |
| − | isomorphic to '''K'''<sup>''n''</sup>
| |
| − | |
| |
| − | <u>''X''</u>*<br>
| |
| − | (hom : <u>''X''</u> → '''B''')<br>
| |
| − | isomorphic to '''B'''<sup>''n''</sup>
| |
| − | |
| |
| − | ''A''*<br>
| |
| − | (hom : ''A'' → '''B''')<br>
| |
| − | isomorphic to '''B'''<sup>''n''</sup>
| |
| − | |-
| |
| − | |
| |
| − | ''X''^<br>
| |
| − | (''X'' → '''K''')<br>
| |
| − | isomorphic to:<br>
| |
| − | ('''K'''<sup>''n''</sup> → '''K''')
| |
| − | |
| |
| − | <u>''X''</u>^<br>
| |
| − | (<u>''X''</u> → '''B''')<br>
| |
| − | isomorphic to:<br>
| |
| − | ('''B'''<sup>''n''</sup> → '''B''')
| |
| − | |
| |
| − | ''A''^<br>
| |
| − | (''A'' → '''B''')<br>
| |
| − | isomorphic to:<br>
| |
| − | ('''B'''<sup>''n''</sup> → '''B''')
| |
| − | |-
| |
| − | |
| |
| − | ''X''<sup>•</sup><br>
| |
| − | [<font face="lucida calligraphy">X</font>]<br>
| |
| − | [''x''<sub>1</sub>, …, ''x''<sub>''n''</sub>]<br>
| |
| − | (''X'', ''X''^)<br>
| |
| − | (''X'' +→ '''K''')<br>
| |
| − | (''X'', (''X'' → '''K'''))<br>
| |
| − | isomorphic to:<br>
| |
| − | ('''K'''<sup>''n''</sup>, ('''K'''<sup>''n''</sup> → '''K'''))<br>
| |
| − | ('''K'''<sup>''n''</sup> +→ '''K''')<br>
| |
| − | ['''K'''<sup>''n''</sup>]
| |
| − | |
| |
| − | <u>''X''</u><sup>•</sup><br>
| |
| − | [<font face="lucida calligraphy"><u>X</u></font>]<br>
| |
| − | [<u>''x''</u><sub>1</sub>, …, <u>''x''</u><sub>''n''</sub>]<br>
| |
| − | (<u>''X''</u>, <u>''X''</u>^)<br>
| |
| − | (<u>''X''</u> +→ '''B''')<br>
| |
| − | (<u>''X''</u>, (<u>''X''</u> → '''B'''))<br>
| |
| − | isomorphic to:<br>
| |
| − | ('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
| |
| − | ('''B'''<sup>''n''</sup> +→ '''B''')<br>
| |
| − | ['''B'''<sup>''n''</sup>]
| |
| − | |
| |
| − | ''A''<sup>•</sup><br>
| |
| − | [<font face="lucida calligraphy">A</font>]<br>
| |
| − | [''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>]<br>
| |
| − | (''A'', ''A''^)<br>
| |
| − | (''A'' +→ '''B''')<br>
| |
| − | (''A'', (''A'' → '''B'''))<br>
| |
| − | isomorphic to:<br>
| |
| − | ('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> → '''B'''))<br>
| |
| − | ('''B'''<sup>''n''</sup> +→ '''B''')<br>
| |
| − | ['''B'''<sup>''n''</sup>]
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Table 6. Propositional Forms on One Variable===
| |
| − |
| |
| − | <pre>
| |
| − | Table 6. Propositional Forms on One Variable
| |
| − | o---------o---------o---------o----------o------------------o----------o
| |
| − | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| |
| − | | | | | | | |
| |
| − | | Decimal | Binary | Vector | Cactus | English | Ordinary |
| |
| − | o---------o---------o---------o----------o------------------o----------o
| |
| − | | | x : 1 0 | | | |
| |
| − | o---------o---------o---------o----------o------------------o----------o
| |
| − | | | | | | | |
| |
| − | | f_0 | f_00 | 0 0 | ( ) | false | 0 |
| |
| − | | | | | | | |
| |
| − | | f_1 | f_01 | 0 1 | (x) | not x | ~x |
| |
| − | | | | | | | |
| |
| − | | f_2 | f_10 | 1 0 | x | x | x |
| |
| − | | | | | | | |
| |
| − | | f_3 | f_11 | 1 1 | (( )) | true | 1 |
| |
| − | | | | | | | |
| |
| − | o---------o---------o---------o----------o------------------o----------o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 6. Propositional Forms on One Variable'''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! style="width:16%" | L<sub>1</sub><br>Decimal
| |
| − | ! style="width:16%" | L<sub>2</sub><br>Binary
| |
| − | ! style="width:16%" | L<sub>3</sub><br>Vector
| |
| − | ! style="width:16%" | L<sub>4</sub><br>Cactus
| |
| − | ! style="width:16%" | L<sub>5</sub><br>English
| |
| − | ! style="width:16%" | L<sub>6</sub><br>Ordinary
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | x :
| |
| − | | 1 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |-
| |
| − | | f<sub>0</sub>
| |
| − | | f<sub>00</sub>
| |
| − | | 0 0
| |
| − | | ( )
| |
| − | | false
| |
| − | | 0
| |
| − | |-
| |
| − | | f<sub>1</sub>
| |
| − | | f<sub>01</sub>
| |
| − | | 0 1
| |
| − | | (x)
| |
| − | | not x
| |
| − | | ~x
| |
| − | |-
| |
| − | | f<sub>2</sub>
| |
| − | | f<sub>10</sub>
| |
| − | | 1 0
| |
| − | | x
| |
| − | | x
| |
| − | | x
| |
| − | |-
| |
| − | | f<sub>3</sub>
| |
| − | | f<sub>11</sub>
| |
| − | | 1 1
| |
| − | | (( ))
| |
| − | | true
| |
| − | | 1
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Table 7. Propositional Forms on Two Variables===
| |
| − |
| |
| − | <pre>
| |
| − | Table 7. Propositional Forms on Two Variables
| |
| − | o---------o---------o---------o----------o------------------o----------o
| |
| − | | L_1 | L_2 | L_3 | L_4 | L_5 | L_6 |
| |
| − | | | | | | | |
| |
| − | | Decimal | Binary | Vector | Cactus | English | Ordinary |
| |
| − | o---------o---------o---------o----------o------------------o----------o
| |
| − | | | x : 1 1 0 0 | | | |
| |
| − | | | y : 1 0 1 0 | | | |
| |
| − | o---------o---------o---------o----------o------------------o----------o
| |
| − | | | | | | | |
| |
| − | | f_0 | f_0000 | 0 0 0 0 | () | false | 0 |
| |
| − | | | | | | | |
| |
| − | | f_1 | f_0001 | 0 0 0 1 | (x)(y) | neither x nor y | ~x & ~y |
| |
| − | | | | | | | |
| |
| − | | f_2 | f_0010 | 0 0 1 0 | (x) y | y and not x | ~x & y |
| |
| − | | | | | | | |
| |
| − | | f_3 | f_0011 | 0 0 1 1 | (x) | not x | ~x |
| |
| − | | | | | | | |
| |
| − | | f_4 | f_0100 | 0 1 0 0 | x (y) | x and not y | x & ~y |
| |
| − | | | | | | | |
| |
| − | | f_5 | f_0101 | 0 1 0 1 | (y) | not y | ~y |
| |
| − | | | | | | | |
| |
| − | | f_6 | f_0110 | 0 1 1 0 | (x, y) | x not equal to y | x + y |
| |
| − | | | | | | | |
| |
| − | | f_7 | f_0111 | 0 1 1 1 | (x y) | not both x and y | ~x v ~y |
| |
| − | | | | | | | |
| |
| − | | f_8 | f_1000 | 1 0 0 0 | x y | x and y | x & y |
| |
| − | | | | | | | |
| |
| − | | f_9 | f_1001 | 1 0 0 1 | ((x, y)) | x equal to y | x = y |
| |
| − | | | | | | | |
| |
| − | | f_10 | f_1010 | 1 0 1 0 | y | y | y |
| |
| − | | | | | | | |
| |
| − | | f_11 | f_1011 | 1 0 1 1 | (x (y)) | not x without y | x => y |
| |
| − | | | | | | | |
| |
| − | | f_12 | f_1100 | 1 1 0 0 | x | x | x |
| |
| − | | | | | | | |
| |
| − | | f_13 | f_1101 | 1 1 0 1 | ((x) y) | not y without x | x <= y |
| |
| − | | | | | | | |
| |
| − | | f_14 | f_1110 | 1 1 1 0 | ((x)(y)) | x or y | x v y |
| |
| − | | | | | | | |
| |
| − | | f_15 | f_1111 | 1 1 1 1 | (()) | true | 1 |
| |
| − | | | | | | | |
| |
| − | o---------o---------o---------o----------o------------------o----------o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 7. Propositional Forms on Two Variables'''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! style="width:16%" | L<sub>1</sub><br>Decimal
| |
| − | ! style="width:16%" | L<sub>2</sub><br>Binary
| |
| − | ! style="width:16%" | L<sub>3</sub><br>Vector
| |
| − | ! style="width:16%" | L<sub>4</sub><br>Cactus
| |
| − | ! style="width:16%" | L<sub>5</sub><br>English
| |
| − | ! style="width:16%" | L<sub>6</sub><br>Ordinary
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | x :
| |
| − | | 1 1 0 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | y :
| |
| − | | 1 0 1 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |-
| |
| − | | f<sub>0</sub> || f<sub>0000</sub> || 0 0 0 0 || ( ) || false || 0
| |
| − | |-
| |
| − | | f<sub>1</sub> || f<sub>0001</sub> || 0 0 0 1 || (x)(y) || neither x nor y || ¬x ∧ ¬y
| |
| − | |-
| |
| − | | f<sub>2</sub> || f<sub>0010</sub> || 0 0 1 0 || (x) y || y and not x || ¬x ∧ y
| |
| − | |-
| |
| − | | f<sub>3</sub> || f<sub>0011</sub> || 0 0 1 1 || (x) || not x || ¬x
| |
| − | |-
| |
| − | | f<sub>4</sub> || f<sub>0100</sub> || 0 1 0 0 || x (y) || x and not y || x ∧ ¬y
| |
| − | |-
| |
| − | | f<sub>5</sub> || f<sub>0101</sub> || 0 1 0 1 || (y) || not y || ¬y
| |
| − | |-
| |
| − | | f<sub>6</sub> || f<sub>0110</sub> || 0 1 1 0 || (x, y) || x not equal to y || x ≠ y
| |
| − | |-
| |
| − | | f<sub>7</sub> || f<sub>0111</sub> || 0 1 1 1 || (x y) || not both x and y || ¬x ∨ ¬y
| |
| − | |-
| |
| − | | f<sub>8</sub> || f<sub>1000</sub> || 1 0 0 0 || x y || x and y || x ∧ y
| |
| − | |-
| |
| − | | f<sub>9</sub> || f<sub>1001</sub> || 1 0 0 1 || ((x, y)) || x equal to y || x = y
| |
| − | |-
| |
| − | | f<sub>10</sub> || f<sub>1010</sub> || 1 0 1 0 || y || y || y
| |
| − | |-
| |
| − | | f<sub>11</sub> || f<sub>1011</sub> || 1 0 1 1 || (x (y)) || not x without y || x → y
| |
| − | |-
| |
| − | | f<sub>12</sub> || f<sub>1100</sub> || 1 1 0 0 || x || x || x
| |
| − | |-
| |
| − | | f<sub>13</sub> || f<sub>1101</sub> || 1 1 0 1 || ((x) y) || not y without x || x ← y
| |
| − | |-
| |
| − | | f<sub>14</sub> || f<sub>1110</sub> || 1 1 1 0 || ((x)(y)) || x or y || x ∨ y
| |
| − | |-
| |
| − | | f<sub>15</sub> || f<sub>1111</sub> || 1 1 1 1 || (( )) || true || 1
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Table 8. Notation for the Differential Extension of Propositional Calculus===
| |
| − |
| |
| − | <pre>
| |
| − | Table 8. Notation for the Differential Extension of Propositional Calculus
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | Symbol | Notation | Description | Type |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | d!A! | {da_1, ..., da_n} | Alphabet of | [n] = #n# |
| |
| − | | | | differential | |
| |
| − | | | | features | |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | dA_i | {(da_i), da_i} | Differential | D |
| |
| − | | | | dimension i | |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | dA | <|d!A!|> | Tangent space | D^n |
| |
| − | | | <|da_i,...,da_n|> | at a point: | |
| |
| − | | | {<da_i,...,da_n>} | Set of changes, | |
| |
| − | | | dA_1 x ... x dA_n | motions, steps, | |
| |
| − | | | Prod_i dA_i | tangent vectors | |
| |
| − | | | | at a point | |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | dA* | (hom : dA -> B) | Linear functions | (D^n)* ~=~ D^n |
| |
| − | | | | on dA | |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | dA^ | (dA -> B) | Boolean functions | D^n -> B |
| |
| − | | | | on dA | |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | | dA% | [d!A!] | Tangent universe | (D^n, (D^n -> B)) |
| |
| − | | | (dA, dA^) | at a point of A%, | (D^n +-> B) |
| |
| − | | | (dA +-> B) | based on the | [D^n] |
| |
| − | | | (dA, (dA -> B)) | tangent features | |
| |
| − | | | [da_1, ..., da_n] | {da_1, ..., da_n} | |
| |
| − | o---------o-------------------o-------------------o-------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
| |
| − | |+ '''Table 8. Notation for the Differential Extension of Propositional Calculus'''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! Symbol
| |
| − | ! Notation
| |
| − | ! Description
| |
| − | ! Type
| |
| − | |-
| |
| − | | d<font face="lucida calligraphy">A<font>
| |
| − | | {d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
| |
| − | |
| |
| − | Alphabet of<br>
| |
| − | differential<br>
| |
| − | features
| |
| − | | [''n''] = '''n'''
| |
| − | |-
| |
| − | | d''A''<sub>''i''</sub>
| |
| − | | {(d''a''<sub>''i''</sub>), d''a''<sub>''i''</sub>}
| |
| − | |
| |
| − | Differential<br>
| |
| − | dimension ''i''
| |
| − | | '''D'''
| |
| − | |-
| |
| − | | d''A''
| |
| − | |
| |
| − | 〈d<font face="lucida calligraphy">A</font>〉<br>
| |
| − | 〈d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>〉<br>
| |
| − | {‹d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>›}<br>
| |
| − | d''A''<sub>1</sub> × … × d''A''<sub>''n''</sub><br>
| |
| − | ∏<sub>''i''</sub> d''A''<sub>''i''</sub>
| |
| − | |
| |
| − | Tangent space<br>
| |
| − | at a point:<br>
| |
| − | Set of changes,<br>
| |
| − | motions, steps,<br>
| |
| − | tangent vectors<br>
| |
| − | at a point
| |
| − | | '''D'''<sup>''n''</sup>
| |
| − | |-
| |
| − | | d''A''*
| |
| − | | (hom : d''A'' → '''B''')
| |
| − | |
| |
| − | Linear functions<br>
| |
| − | on d''A''
| |
| − | | ('''D'''<sup>''n''</sup>)* = '''D'''<sup>''n''</sup>
| |
| − | |-
| |
| − | | d''A''^
| |
| − | | (d''A'' → '''B''')
| |
| − | |
| |
| − | Boolean functions<br>
| |
| − | on d''A''
| |
| − | | '''D'''<sup>''n''</sup> → '''B'''
| |
| − | |-
| |
| − | | d''A''<sup>•</sup>
| |
| − | |
| |
| − | [d<font face="lucida calligraphy">A</font>]<br>
| |
| − | (d''A'', d''A''^)<br>
| |
| − | (d''A'' +→ '''B''')<br>
| |
| − | (d''A'', (d''A'' → '''B'''))<br>
| |
| − | [d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>]
| |
| − | |
| |
| − | Tangent universe<br>
| |
| − | at a point of ''A''<sup>•</sup>,<br>
| |
| − | based on the<br>
| |
| − | tangent features<br>
| |
| − | {d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}
| |
| − | |
| |
| − | ('''D'''<sup>''n''</sup>, ('''D'''<sup>''n''</sup> → '''B'''))<br>
| |
| − | ('''D'''<sup>''n''</sup> +→ '''B''')<br>
| |
| − | ['''D'''<sup>''n''</sup>]
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Table 9. Higher Order Differential Features===
| |
| − |
| |
| − | <pre>
| |
| − | Table 9. Higher Order Differential Features
| |
| − | o----------------------------------------o----------------------------------------o
| |
| − | | | |
| |
| − | | !A! = d^0.!A! = {a_1, ..., a_n} | E^0.!A! = d^0.!A! |
| |
| − | | | |
| |
| − | | d!A! = d^1.!A! = {da_1, ..., da_n} | E^1.!A! = d^0.!A! |_| d^1.!A! |
| |
| − | | | |
| |
| − | | d^k.!A! = {d^k.a_1,...,d^k.a_n}| E^k.!A! = d^0.!A! |_| ... |_| d^k.!A! |
| |
| − | | | |
| |
| − | | d*!A! = {d^0.!A!, ..., d^k.!A!, ...} | E^oo.!A! = |_| d*!A! |
| |
| − | | | |
| |
| − | o----------------------------------------o----------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="10" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
| |
| − | |+ '''Table 9. Higher Order Differential Features'''
| |
| − | | width=50% |
| |
| − | <font face="lucida calligraphy">A</font> = d<sup>0</sup><font face="lucida calligraphy">A</font> = {''a''<sub>1</sub>, …, ''a''<sub>''n''</sub>}<br><br>
| |
| − | d<font face="lucida calligraphy">A</font> = d<sup>1</sup><font face="lucida calligraphy">A</font> = {d''a''<sub>1</sub>, …, d''a''<sub>''n''</sub>}<br><br>
| |
| − | d<sup>''k''</sup><font face="lucida calligraphy">A</font> = {d<sup>''k''</sup>''a''<sub>''1''</sub>, …, d<sup>''k''</sup>''a''<sub>''n''</sub>}<br><br>
| |
| − | d<sup>*</sup><font face="lucida calligraphy">A</font> = {d<sup>0</sup><font face="lucida calligraphy">A</font>, …, d<sup>''k''</sup><font face="lucida calligraphy">A</font>, …}
| |
| − | | width=50% |
| |
| − | E<sup>0</sup><font face="lucida calligraphy">A</font> = d<sup>0</sup><font face="lucida calligraphy">A</font><br><br>
| |
| − | E<sup>1</sup><font face="lucida calligraphy">A</font> = d<sup>0</sup><font face="lucida calligraphy">A</font> ∪ d<sup>1</sup><font face="lucida calligraphy">A</font><br><br>
| |
| − | E<sup>''k''</sup><font face="lucida calligraphy">A</font> = d<sup>0</sup><font face="lucida calligraphy">A</font> ∪ … ∪ d<sup>''k''</sup><font face="lucida calligraphy">A</font><br><br>
| |
| − | E<sup>∞</sup><font face="lucida calligraphy">A</font> = ∪ d<sup>*</sup><font face="lucida calligraphy">A</font>
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
| |
| − | |+ '''Table 9. Higher Order Differential Features'''
| |
| − | | width=50% |
| |
| − | {| cellpadding="4" style="background:lightcyan"
| |
| − | | <font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | d<sup>0</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | {''a''<sub>1</sub>,
| |
| − | | …,
| |
| − | | ''a''<sub>''n''</sub>}
| |
| − | |-
| |
| − | | d<font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | d<sup>1</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | {d''a''<sub>1</sub>,
| |
| − | | …,
| |
| − | | d''a''<sub>''n''</sub>}
| |
| − | |-
| |
| − | |
| |
| − | |
| |
| − | | d<sup>''k''</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | {d<sup>''k''</sup>''a''<sub>''1''</sub>,
| |
| − | | …,
| |
| − | | d<sup>''k''</sup>''a''<sub>''n''</sub>}
| |
| − | |-
| |
| − | | d<sup>*</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | {d<sup>0</sup><font face="lucida calligraphy">A</font>,
| |
| − | | …,
| |
| − | | d<sup>''k''</sup><font face="lucida calligraphy">A</font>,
| |
| − | | …}
| |
| − | |}
| |
| − | | width=50% |
| |
| − | {| cellpadding="4" style="background:lightcyan"
| |
| − | | E<sup>0</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | d<sup>0</sup><font face="lucida calligraphy">A</font>
| |
| − | |-
| |
| − | | E<sup>1</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | d<sup>0</sup><font face="lucida calligraphy">A</font> ∪ d<sup>1</sup><font face="lucida calligraphy">A</font>
| |
| − | |-
| |
| − | | E<sup>''k''</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | d<sup>0</sup><font face="lucida calligraphy">A</font> ∪ … ∪ d<sup>''k''</sup><font face="lucida calligraphy">A</font>
| |
| − | |-
| |
| − | | E<sup>∞</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | ∪ d<sup>*</sup><font face="lucida calligraphy">A</font>
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Table 10. A Realm of Intentional Features===
| |
| − |
| |
| − | <pre>
| |
| − | Table 10. A Realm of Intentional Features
| |
| − | o---------------------------------------o----------------------------------------o
| |
| − | | | |
| |
| − | | p^0.!A! = !A! = {a_1, ..., a_n} | Q^0.!A! = !A! |
| |
| − | | | |
| |
| − | | p^1.!A! = !A!' = {a_1', ..., a_n'} | Q^1.!A! = !A! |_| !A!' |
| |
| − | | | |
| |
| − | | p^2.!A! = !A!" = {a_1", ..., a_n"} | Q^2.!A! = !A! |_| !A!' |_| !A!" |
| |
| − | | | |
| |
| − | | ... ... ... | ... ... |
| |
| − | | | |
| |
| − | | p^k.!A! = {p^k.a_1, ..., p^k.a_n} | Q^k.!A! = !A! |_| ... |_| p^k.!A! |
| |
| − | | | |
| |
| − | o---------------------------------------o----------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
| |
| − | |+ '''Table 10. A Realm of Intentional Features'''
| |
| − | | width=50% |
| |
| − | {| cellpadding="4" style="background:lightcyan"
| |
| − | | p<sup>0</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | <font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | {''a''<sub>1</sub> ,
| |
| − | | …,
| |
| − | | ''a''<sub>''n''</sub> }
| |
| − | |-
| |
| − | | p<sup>1</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | <font face="lucida calligraphy">A</font>′
| |
| − | | =
| |
| − | | {''a''<sub>1</sub>′,
| |
| − | | …,
| |
| − | | ''a''<sub>''n''</sub>′}
| |
| − | |-
| |
| − | | p<sup>2</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | <font face="lucida calligraphy">A</font>″
| |
| − | | =
| |
| − | | {''a''<sub>1</sub>″,
| |
| − | | …,
| |
| − | | ''a''<sub>''n''</sub>″}
| |
| − | |-
| |
| − | | ...
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | | ...
| |
| − | |-
| |
| − | | p<sup>''k''</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | |
| |
| − | |
| |
| − | | {p<sup>''k''</sup>''a''<sub>1</sub>,
| |
| − | | …,
| |
| − | | p<sup>''k''</sup>''a''<sub>''n''</sub>}
| |
| − | |}
| |
| − | | width=50% |
| |
| − | {| cellpadding="4" style="background:lightcyan"
| |
| − | | Q<sup>0</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | <font face="lucida calligraphy">A</font>
| |
| − | |-
| |
| − | | Q<sup>1</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | <font face="lucida calligraphy">A</font> ∪ <font face="lucida calligraphy">A</font>′
| |
| − | |-
| |
| − | | Q<sup>2</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | <font face="lucida calligraphy">A</font> ∪ <font face="lucida calligraphy">A</font>′ ∪ <font face="lucida calligraphy">A</font>″
| |
| − | |-
| |
| − | | ...
| |
| − | |
| |
| − | | ...
| |
| − | |-
| |
| − | | Q<sup>''k''</sup><font face="lucida calligraphy">A</font>
| |
| − | | =
| |
| − | | <font face="lucida calligraphy">A</font> ∪ <font face="lucida calligraphy">A</font>′ ∪ … ∪ p<sup>''k''</sup><font face="lucida calligraphy">A</font>
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Formula Display 1===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | From (A) & (dA) infer (A) next. |
| |
| − | | |
| |
| − | | From (A) & dA infer A next. |
| |
| − | | |
| |
| − | | From A & (dA) infer A next. |
| |
| − | | |
| |
| − | | From A & dA infer (A) next. |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | | || From || (''A'') || and || (d''A'') || infer || (''A'') || next. ||
| |
| − | |-
| |
| − | | || From || (''A'') || and || d''A'' || infer || ''A'' || next. ||
| |
| − | |-
| |
| − | | || From || ''A'' || and || (d''A'') || infer || ''A'' || next. ||
| |
| − | |-
| |
| − | | || From || ''A'' || and || d''A'' || infer || (''A'') || next. ||
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Table 11. A Pair of Commodious Trajectories===
| |
| − |
| |
| − | <pre>
| |
| − | Table 11. A Pair of Commodious Trajectories
| |
| − | o---------o-------------------o-------------------o
| |
| − | | Time | Trajectory 1 | Trajectory 2 |
| |
| − | o---------o-------------------o-------------------o
| |
| − | | | | |
| |
| − | | 0 | A dA (d^2.A) | (A) (dA) d^2.A |
| |
| − | | | | |
| |
| − | | 1 | (A) dA d^2.A | (A) dA d^2.A |
| |
| − | | | | |
| |
| − | | 2 | A (dA) (d^2.A) | A (dA) (d^2.A) |
| |
| − | | | | |
| |
| − | | 3 | A (dA) (d^2.A) | A (dA) (d^2.A) |
| |
| − | | | | |
| |
| − | | 4 | " " " | " " " |
| |
| − | | | | |
| |
| − | o---------o-------------------o-------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |+ '''Table 11. A Pair of Commodious Trajectories'''
| |
| − | |- style="background:paleturquoise"
| |
| − | ! Time
| |
| − | ! Trajectory 1
| |
| − | ! Trajectory 2
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |-
| |
| − | | 2
| |
| − | |-
| |
| − | | 3
| |
| − | |-
| |
| − | | 4
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"
| |
| − | | ''A'' || d''A'' || (d<sup>2</sup>''A'')
| |
| − | |-
| |
| − | | (''A'') || d''A'' || d<sup>2</sup>''A''
| |
| − | |-
| |
| − | | ''A'' || (d''A'') || (d<sup>2</sup>''A'')
| |
| − | |-
| |
| − | | ''A'' || (d''A'') || (d<sup>2</sup>''A'')
| |
| − | |-
| |
| − | | " || " || "
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; text-align:center"
| |
| − | | (''A'') || (d''A'') || d<sup>2</sup>''A''
| |
| − | |-
| |
| − | | (''A'') || d''A'' || d<sup>2</sup>''A''
| |
| − | |-
| |
| − | | ''A'' || (d''A'') || (d<sup>2</sup>''A'')
| |
| − | |-
| |
| − | | ''A'' || (d''A'') || (d<sup>2</sup>''A'')
| |
| − | |-
| |
| − | | " || " || "
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Figure 12. The Anchor===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | E^2.X |
| |
| − | | |
| |
| − | | o-------------o |
| |
| − | | / \ |
| |
| − | | / A \ |
| |
| − | | / \ |
| |
| − | | / ->- \ |
| |
| − | | o / \ o |
| |
| − | | | \ / | |
| |
| − | | | -o- | |
| |
| − | | | ^ | |
| |
| − | | o---o---------o | o---------o---o |
| |
| − | | / \ \|/ / \ |
| |
| − | | / \ o | / \ |
| |
| − | | / \ | /|\ / \ |
| |
| − | | / \ | / | \ / \ |
| |
| − | | o o-|-o--|--o---o o |
| |
| − | | | | | | | | |
| |
| − | | | ---->o<----o | |
| |
| − | | | | | | |
| |
| − | | o dA o o d^2.A o |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o-------------o o-------------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 12. The Anchor
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 13. The Tiller===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | ->- |
| |
| − | | / \ |
| |
| − | | \ / |
| |
| − | | o-------------o -o- |
| |
| − | | / \ ^ |
| |
| − | | / dA \/ A |
| |
| − | | / /\ |
| |
| − | | / / \ |
| |
| − | | o o / o |
| |
| − | | | \ / | |
| |
| − | | | \ / | |
| |
| − | o------------|-------\-------/-------|------------o
| |
| − | | | \ / | |
| |
| − | | | \ / | |
| |
| − | | o v / o |
| |
| − | | \ o / |
| |
| − | | \ ^ / |
| |
| − | | \ | / d^2.A |
| |
| − | | \ | / |
| |
| − | | o------|------o |
| |
| − | | | |
| |
| − | | | |
| |
| − | | o |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 13. The Tiller
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 14. Differential Propositions===
| |
| − |
| |
| − | <pre>
| |
| − | Table 14. Differential Propositions
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | A : 1 1 0 0 | | | |
| |
| − | | | dA : 1 0 1 0 | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | f_0 | g_0 | 0 0 0 0 | () | False | 0 |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | | g_1 | 0 0 0 1 | (A)(dA) | Neither A nor dA | ~A & ~dA |
| |
| − | | | | | | | |
| |
| − | | | g_2 | 0 0 1 0 | (A) dA | Not A but dA | ~A & dA |
| |
| − | | | | | | | |
| |
| − | | | g_4 | 0 1 0 0 | A (dA) | A but not dA | A & ~dA |
| |
| − | | | | | | | |
| |
| − | | | g_8 | 1 0 0 0 | A dA | A and dA | A & dA |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | f_1 | g_3 | 0 0 1 1 | (A) | Not A | ~A |
| |
| − | | | | | | | |
| |
| − | | f_2 | g_12 | 1 1 0 0 | A | A | A |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | | g_6 | 0 1 1 0 | (A, dA) | A not equal to dA | A + dA |
| |
| − | | | | | | | |
| |
| − | | | g_9 | 1 0 0 1 | ((A, dA)) | A equal to dA | A = dA |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | | g_5 | 0 1 0 1 | (dA) | Not dA | ~dA |
| |
| − | | | | | | | |
| |
| − | | | g_10 | 1 0 1 0 | dA | dA | dA |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | | g_7 | 0 1 1 1 | (A dA) | Not both A and dA | ~A v ~dA |
| |
| − | | | | | | | |
| |
| − | | | g_11 | 1 0 1 1 | (A (dA)) | Not A without dA | A => dA |
| |
| − | | | | | | | |
| |
| − | | | g_13 | 1 1 0 1 | ((A) dA) | Not dA without A | A <= dA |
| |
| − | | | | | | | |
| |
| − | | | g_14 | 1 1 1 0 | ((A)(dA)) | A or dA | A v dA |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | | | | | | | |
| |
| − | | f_3 | g_15 | 1 1 1 1 | (()) | True | 1 |
| |
| − | | | | | | | |
| |
| − | o-------o--------o---------o-----------o-------------------o----------o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 14. Differential Propositions'''
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | A :
| |
| − | | 1 1 0 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | dA :
| |
| − | | 1 0 1 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |-
| |
| − | | f<sub>0</sub>
| |
| − | | g<sub>0</sub>
| |
| − | | 0 0 0 0
| |
| − | | ( )
| |
| − | | False
| |
| − | | 0
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>1</sub>
| |
| − | | 0 0 0 1
| |
| − | | (A)(dA)
| |
| − | | Neither A nor dA
| |
| − | | ¬A ∧ ¬dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>2</sub>
| |
| − | | 0 0 1 0
| |
| − | | (A) dA
| |
| − | | Not A but dA
| |
| − | | ¬A ∧ dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>4</sub>
| |
| − | | 0 1 0 0
| |
| − | | A (dA)
| |
| − | | A but not dA
| |
| − | | A ∧ ¬dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>8</sub>
| |
| − | | 1 0 0 0
| |
| − | | A dA
| |
| − | | A and dA
| |
| − | | A ∧ dA
| |
| − | |-
| |
| − | | f<sub>1</sub>
| |
| − | | g<sub>3</sub>
| |
| − | | 0 0 1 1
| |
| − | | (A)
| |
| − | | Not A
| |
| − | | ¬A
| |
| − | |-
| |
| − | | f<sub>2</sub>
| |
| − | | g<sub>12</sub>
| |
| − | | 1 1 0 0
| |
| − | | A
| |
| − | | A
| |
| − | | A
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>6</sub>
| |
| − | | 0 1 1 0
| |
| − | | (A, dA)
| |
| − | | A not equal to dA
| |
| − | | A ≠ dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>9</sub>
| |
| − | | 1 0 0 1
| |
| − | | ((A, dA))
| |
| − | | A equal to dA
| |
| − | | A = dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>5</sub>
| |
| − | | 0 1 0 1
| |
| − | | (dA)
| |
| − | | Not dA
| |
| − | | ¬dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>10</sub>
| |
| − | | 1 0 1 0
| |
| − | | dA
| |
| − | | dA
| |
| − | | dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>7</sub>
| |
| − | | 0 1 1 1
| |
| − | | (A dA)
| |
| − | | Not both A and dA
| |
| − | | ¬A ∨ ¬dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>11</sub>
| |
| − | | 1 0 1 1
| |
| − | | (A (dA))
| |
| − | | Not A without dA
| |
| − | | A → dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>13</sub>
| |
| − | | 1 1 0 1
| |
| − | | ((A) dA)
| |
| − | | Not dA without A
| |
| − | | A ← dA
| |
| − | |-
| |
| − | |
| |
| − | | g<sub>14</sub>
| |
| − | | 1 1 1 0
| |
| − | | ((A)(dA))
| |
| − | | A or dA
| |
| − | | A ∨ dA
| |
| − | |-
| |
| − | | f<sub>3</sub>
| |
| − | | g<sub>15</sub>
| |
| − | | 1 1 1 1
| |
| − | | (( ))
| |
| − | | True
| |
| − | | 1
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 14. Differential Propositions'''
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | A :
| |
| − | | 1 1 0 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |- style="background:paleturquoise"
| |
| − | |
| |
| − | | align="right" | dA :
| |
| − | | 1 0 1 0
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |-
| |
| − | | f<sub>0</sub>
| |
| − | | g<sub>0</sub>
| |
| − | | 0 0 0 0
| |
| − | | ( )
| |
| − | | False
| |
| − | | 0
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | <br>
| |
| − | <br>
| |
| − | <br>
| |
| − |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>1</sub><br>
| |
| − | g<sub>2</sub><br>
| |
| − | g<sub>4</sub><br>
| |
| − | g<sub>8</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 0 0 1<br>
| |
| − | 0 0 1 0<br>
| |
| − | 0 1 0 0<br>
| |
| − | 1 0 0 0
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (A)(dA)<br>
| |
| − | (A) dA <br>
| |
| − | A (dA)<br>
| |
| − | A dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | Neither A nor dA<br>
| |
| − | Not A but dA<br>
| |
| − | A but not dA<br>
| |
| − | A and dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | ¬A ∧ ¬dA<br>
| |
| − | ¬A ∧ dA<br>
| |
| − | A ∧ ¬dA<br>
| |
| − | A ∧ dA
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | f<sub>1</sub><br>
| |
| − | f<sub>2</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>3</sub><br>
| |
| − | g<sub>12</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 0 1 1<br>
| |
| − | 1 1 0 0
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (A)<br>
| |
| − | A
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | Not A<br>
| |
| − | A
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | ¬A<br>
| |
| − | A
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | <br>
| |
| − |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>6</sub><br>
| |
| − | g<sub>9</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 1 1 0<br>
| |
| − | 1 0 0 1
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (A, dA)<br>
| |
| − | ((A, dA))
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | A not equal to dA<br>
| |
| − | A equal to dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | A ≠ dA<br>
| |
| − | A = dA
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | <br>
| |
| − |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>5</sub><br>
| |
| − | g<sub>10</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 1 0 1<br>
| |
| − | 1 0 1 0
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (dA)<br>
| |
| − | dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | Not dA<br>
| |
| − | dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | ¬dA<br>
| |
| − | dA
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | <br>
| |
| − | <br>
| |
| − | <br>
| |
| − |
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | g<sub>7</sub><br>
| |
| − | g<sub>11</sub><br>
| |
| − | g<sub>13</sub><br>
| |
| − | g<sub>14</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | 0 1 1 1<br>
| |
| − | 1 0 1 1<br>
| |
| − | 1 1 0 1<br>
| |
| − | 1 1 1 0
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | (A dA)<br>
| |
| − | (A (dA))<br>
| |
| − | ((A) dA)<br>
| |
| − | ((A)(dA))
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | Not both A and dA<br>
| |
| − | Not A without dA<br>
| |
| − | Not dA without A<br>
| |
| − | A or dA
| |
| − | |}
| |
| − | |
| |
| − | {| style="background:lightcyan"
| |
| − | |
| |
| − | ¬A ∨ ¬dA<br>
| |
| − | A → dA<br>
| |
| − | A ← dA<br>
| |
| − | A ∨ dA
| |
| − | |}
| |
| − | |-
| |
| − | | f<sub>3</sub>
| |
| − | | g<sub>15</sub>
| |
| − | | 1 1 1 1
| |
| − | | (( ))
| |
| − | | True
| |
| − | | 1
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Table 15. Tacit Extension of [''A''] to [''A'', d''A'']===
| |
| − |
| |
| − | <pre>
| |
| − | Table 15. Tacit Extension of [A] to [A, dA]
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | 0 = 0 . ((dA), dA) = 0 |
| |
| − | | |
| |
| − | | (A) = (A) . ((dA), dA) = (A)(dA) + (A) dA |
| |
| − | | |
| |
| − | | A = A . ((dA), dA) = A (dA) + A dA |
| |
| − | | |
| |
| − | | 1 = 1 . ((dA), dA) = 1 |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |+ '''Table 15. Tacit Extension of [''A''] to [''A'', d''A'']'''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |
| |
| − | | 0
| |
| − | | =
| |
| − | | 0
| |
| − | | ·
| |
| − | | ((d''A''), d''A'')
| |
| − | | =
| |
| − | | 0
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | | (''A'')
| |
| − | | =
| |
| − | | (''A'')
| |
| − | | ·
| |
| − | | ((d''A''), d''A'')
| |
| − | | =
| |
| − | | (''A'')(d''A'') + (''A'') d''A''
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | | ''A''
| |
| − | | =
| |
| − | | ''A''
| |
| − | | ·
| |
| − | | ((d''A''), d''A'')
| |
| − | | =
| |
| − | | ''A'' (d''A'') + ''A'' d''A''
| |
| − | |
| |
| − | |-
| |
| − | |
| |
| − | | 1
| |
| − | | =
| |
| − | | 1
| |
| − | | ·
| |
| − | | ((d''A''), d''A'')
| |
| − | | =
| |
| − | | 1
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Figure 16-a. A Couple of Fourth Gear Orbits: 1===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | o |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | o o |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | o o o |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | o 5 o 7 o o |
| |
| − | | / \ ^| / \ ^| / \ / \ |
| |
| − | | / \/ | / \/ | / \ / \ |
| |
| − | | / /\ | / /\ | / \ / \ |
| |
| − | | / / \|/ / \|/ \ / \ |
| |
| − | | o 4<---|----/----|----3 o o |
| |
| − | | |\ /|\ / /|\ ^ / \ /| |
| |
| − | | | \ / | \/ / | \/ / \ / | |
| |
| − | | | \ / | /\ / | /\ / \ / | |
| |
| − | | | \ / v/ \ / |/ \ / \ / | |
| |
| − | | | o 6 o | o o | |
| |
| − | | | |\ / \ /| / \ /| | |
| |
| − | | | | \ / \/ | / \ / | | |
| |
| − | | | | \ / /\ | / \ / | | |
| |
| − | | | d^0.A \ / / \|/ \ / d^1.A | |
| |
| − | | o----+----o 2<---|----1 o----+----o |
| |
| − | | | \ /|\ ^ / | |
| |
| − | | | \ / | \/ / | |
| |
| − | | | \ / | /\ / | |
| |
| − | | | d^2.A \ / v/ \ / d^3.A | |
| |
| − | | o---------o 0 o---------o |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | o |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 16-a. A Couple of Fourth Gear Orbits: 1
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 16-b. A Couple of Fourth Gear Orbits: 2===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | o |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | o 0 o |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | o 5 o 2 o |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | o o o 6 o |
| |
| − | | / \ / \ / \ / \ |
| |
| − | | / \ / \ / \ / \ |
| |
| − | | / \ / \ / \ / \ |
| |
| − | | / \ / \ / \ / \ |
| |
| − | | o o 7 o o 4 o |
| |
| − | | |\ / \ / \ / \ /| |
| |
| − | | | \ / \ / \ / \ / | |
| |
| − | | | \ / \ / \ / \ / | |
| |
| − | | | \ / \ / \ / \ / | |
| |
| − | | | o o 3 o 1 o | |
| |
| − | | | |\ / \ / \ /| | |
| |
| − | | | | \ / \ / \ / | | |
| |
| − | | | | \ / \ / \ / | | |
| |
| − | | | d^0.A \ / \ / \ / d^1.A | |
| |
| − | | o----+----o o o----+----o |
| |
| − | | | \ / \ / | |
| |
| − | | | \ / \ / | |
| |
| − | | | \ / \ / | |
| |
| − | | | d^2.A \ / \ / d^3.A | |
| |
| − | | o---------o o---------o |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | o |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | Figure 16-b. A Couple of Fourth Gear Orbits: 2
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 2===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | r(q) = Sum_k d_k . 2^(-k) = Sum_k d^k.A(q) . 2^(-k) |
| |
| − | | |
| |
| − | | = |
| |
| − | | |
| |
| − | | s(q)/t = (Sum_k d_k . 2^(m-k)) / 2^m = (Sum_k d^k.A(q) . 2^(m-k)) / 2^m |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | | ''r''(''q'')
| |
| − | | =
| |
| − | | ∑<sub>''k''</sub> ''d''<sub>''k''</sub> · 2<sup>-''k''</sup>
| |
| − | | =
| |
| − | | ∑<sub>''k''</sub> d<sup>''k''</sup>''A''(''q'') · 2<sup>-''k''</sup>
| |
| − | |-
| |
| − | | =
| |
| − | |-
| |
| − | | ''s''(''q'')/''t''
| |
| − | | =
| |
| − | | (∑<sub>''k''</sub> ''d''<sub>''k''</sub> · 2<sup>(''m''-''k'')</sup>) / 2<sup>''m''</sup>
| |
| − | | =
| |
| − | | (∑<sub>''k''</sub> d<sup>''k''</sup>''A''(''q'') · 2<sup>(''m''-''k'')</sup>) / 2<sup>''m''</sup>
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:center; width:96%"
| |
| − | | <math>r(q)\!</math>
| |
| − | | <math>=</math>
| |
| − | | <math>\sum_k d_k \cdot 2^{-k}</math>
| |
| − | | <math>=</math>
| |
| − | | <math>\sum_k \mbox{d}^k A(q) \cdot 2^{-k}</math>
| |
| − | |-
| |
| − | | <math>=</math>
| |
| − | |-
| |
| − | | <math>\frac{s(q)}{t}</math>
| |
| − | | <math>=</math>
| |
| − | | <math>\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}</math>
| |
| − | | <math>=</math>
| |
| − | | <math>\frac{\sum_k \mbox{d}^k A(q) \cdot 2^{(m-k)}}{2^m}</math>
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1===
| |
| − |
| |
| − | <pre>
| |
| − | Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1
| |
| − | o---------o---------o---------o---------o---------o---------o---------o
| |
| − | | Time | State | A | dA | | | |
| |
| − | | p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A |
| |
| − | o---------o---------o---------o---------o---------o---------o---------o
| |
| − | | | | |
| |
| − | | p_0 | q_01 | 0. 0 0 0 1 |
| |
| − | | | | |
| |
| − | | p_1 | q_03 | 0. 0 0 1 1 |
| |
| − | | | | |
| |
| − | | p_2 | q_05 | 0. 0 1 0 1 |
| |
| − | | | | |
| |
| − | | p_3 | q_15 | 0. 1 1 1 1 |
| |
| − | | | | |
| |
| − | | p_4 | q_17 | 1. 0 0 0 1 |
| |
| − | | | | |
| |
| − | | p_5 | q_19 | 1. 0 0 1 1 |
| |
| − | | | | |
| |
| − | | p_6 | q_21 | 1. 0 1 0 1 |
| |
| − | | | | |
| |
| − | | p_7 | q_31 | 1. 1 1 1 1 |
| |
| − | | | | |
| |
| − | o---------o---------o---------o---------o---------o---------o---------o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1'''
| |
| − | |- style="background:paleturquoise"
| |
| − | | Time
| |
| − | | State
| |
| − | | ''A''
| |
| − | | d''A''
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |- style="background:paleturquoise"
| |
| − | | ''p''<sub>''i''</sub>
| |
| − | | ''q''<sub>''j''</sub>
| |
| − | | d<sup>0</sup>''A''
| |
| − | | d<sup>1</sup>''A''
| |
| − | | d<sup>2</sup>''A''
| |
| − | | d<sup>3</sup>''A''
| |
| − | | d<sup>4</sup>''A''
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"
| |
| − | | ''p''<sub>0</sub>
| |
| − | |-
| |
| − | | ''p''<sub>1</sub>
| |
| − | |-
| |
| − | | ''p''<sub>2</sub>
| |
| − | |-
| |
| − | | ''p''<sub>3</sub>
| |
| − | |-
| |
| − | | ''p''<sub>4</sub>
| |
| − | |-
| |
| − | | ''p''<sub>5</sub>
| |
| − | |-
| |
| − | | ''p''<sub>6</sub>
| |
| − | |-
| |
| − | | ''p''<sub>7</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"
| |
| − | | ''q''<sub>01</sub>
| |
| − | |-
| |
| − | | ''q''<sub>03</sub>
| |
| − | |-
| |
| − | | ''q''<sub>05</sub>
| |
| − | |-
| |
| − | | ''q''<sub>15</sub>
| |
| − | |-
| |
| − | | ''q''<sub>17</sub>
| |
| − | |-
| |
| − | | ''q''<sub>19</sub>
| |
| − | |-
| |
| − | | ''q''<sub>21</sub>
| |
| − | |-
| |
| − | | ''q''<sub>31</sub>
| |
| − | |}
| |
| − | | colspan="5" |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0. || 0 || 0 || 0 || 1
| |
| − | |-
| |
| − | | 0. || 0 || 0 || 1 || 1
| |
| − | |-
| |
| − | | 0. || 0 || 1 || 0 || 1
| |
| − | |-
| |
| − | | 0. || 1 || 1 || 1 || 1
| |
| − | |-
| |
| − | | 1. || 0 || 0 || 0 || 1
| |
| − | |-
| |
| − | | 1. || 0 || 0 || 1 || 1
| |
| − | |-
| |
| − | | 1. || 0 || 1 || 0 || 1
| |
| − | |-
| |
| − | | 1. || 1 || 1 || 1 || 1
| |
| − | |}
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2===
| |
| − |
| |
| − | <pre>
| |
| − | Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2
| |
| − | o---------o---------o---------o---------o---------o---------o---------o
| |
| − | | Time | State | A | dA | | | |
| |
| − | | p_i | q_j | d^0.A | d^1.A | d^2.A | d^3.A | d^4.A |
| |
| − | o---------o---------o---------o---------o---------o---------o---------o
| |
| − | | | | |
| |
| − | | p_0 | q_25 | 1. 1 0 0 1 |
| |
| − | | | | |
| |
| − | | p_1 | q_11 | 0. 1 0 1 1 |
| |
| − | | | | |
| |
| − | | p_2 | q_29 | 1. 1 1 0 1 |
| |
| − | | | | |
| |
| − | | p_3 | q_07 | 0. 0 1 1 1 |
| |
| − | | | | |
| |
| − | | p_4 | q_09 | 0. 1 0 0 1 |
| |
| − | | | | |
| |
| − | | p_5 | q_27 | 1. 1 0 1 1 |
| |
| − | | | | |
| |
| − | | p_6 | q_13 | 0. 1 1 0 1 |
| |
| − | | | | |
| |
| − | | p_7 | q_23 | 1. 0 1 1 1 |
| |
| − | | | | |
| |
| − | o---------o---------o---------o---------o---------o---------o---------o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" border="1" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2'''
| |
| − | |- style="background:paleturquoise"
| |
| − | | Time
| |
| − | | State
| |
| − | | ''A''
| |
| − | | d''A''
| |
| − | |
| |
| − | |
| |
| − | |
| |
| − | |- style="background:paleturquoise"
| |
| − | | ''p''<sub>''i''</sub>
| |
| − | | ''q''<sub>''j''</sub>
| |
| − | | d<sup>0</sup>''A''
| |
| − | | d<sup>1</sup>''A''
| |
| − | | d<sup>2</sup>''A''
| |
| − | | d<sup>3</sup>''A''
| |
| − | | d<sup>4</sup>''A''
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"
| |
| − | | ''p''<sub>0</sub>
| |
| − | |-
| |
| − | | ''p''<sub>1</sub>
| |
| − | |-
| |
| − | | ''p''<sub>2</sub>
| |
| − | |-
| |
| − | | ''p''<sub>3</sub>
| |
| − | |-
| |
| − | | ''p''<sub>4</sub>
| |
| − | |-
| |
| − | | ''p''<sub>5</sub>
| |
| − | |-
| |
| − | | ''p''<sub>6</sub>
| |
| − | |-
| |
| − | | ''p''<sub>7</sub>
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center"
| |
| − | | ''q''<sub>25</sub>
| |
| − | |-
| |
| − | | ''q''<sub>11</sub>
| |
| − | |-
| |
| − | | ''q''<sub>29</sub>
| |
| − | |-
| |
| − | | ''q''<sub>07</sub>
| |
| − | |-
| |
| − | | ''q''<sub>09</sub>
| |
| − | |-
| |
| − | | ''q''<sub>27</sub>
| |
| − | |-
| |
| − | | ''q''<sub>13</sub>
| |
| − | |-
| |
| − | | ''q''<sub>23</sub>
| |
| − | |}
| |
| − | | colspan="5" |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1. || 1 || 0 || 0 || 1
| |
| − | |-
| |
| − | | 0. || 1 || 0 || 1 || 1
| |
| − | |-
| |
| − | | 1. || 1 || 1 || 0 || 1
| |
| − | |-
| |
| − | | 0. || 0 || 1 || 1 || 1
| |
| − | |-
| |
| − | | 0. || 1 || 0 || 0 || 1
| |
| − | |-
| |
| − | | 1. || 1 || 0 || 1 || 1
| |
| − | |-
| |
| − | | 0. || 1 || 1 || 0 || 1
| |
| − | |-
| |
| − | | 1. || 0 || 1 || 1 || 1
| |
| − | |}
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Figure 18-a. Extension from 1 to 2 Dimensions: Areal===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | o o |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / o o 1 1 o |
| |
| − | | / / \ / \ / \ |
| |
| − | | / / \ / \ / \ |
| |
| − | | / 1 / \ / \ / \ |
| |
| − | | / / \ !e! / \ / \ |
| |
| − | | o / o ----> o 1 0 o 0 1 o |
| |
| − | | |\ / / |\ / \ /| |
| |
| − | | | \ / 0 / | \ / \ / | |
| |
| − | | | \ / / | \ / \ / | |
| |
| − | | |x_1\ / / |x_1\ / \ /x_2| |
| |
| − | | o----o / o----o 0 0 o----o |
| |
| − | | \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | o o |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 18-a. Extension from 1 to 2 Dimensions: Areal
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 18-b. Extension from 1 to 2 Dimensions: Bundle===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------o o-------------------o
| |
| − | | | | |
| |
| − | | | | o-------o |
| |
| − | | o---------o | | / \ |
| |
| − | | / \ | | o o |
| |
| − | | / o------------------------| | dx | |
| |
| − | | / \ | | o o |
| |
| − | | / \ | | \ / |
| |
| − | | o o | | o-------o |
| |
| − | | | | | | |
| |
| − | | | | | o-------------------o
| |
| − | | | x | |
| |
| − | | | | | o-------------------o
| |
| − | | | | | | |
| |
| − | | o o | | o-------o |
| |
| − | | \ / | | / \ |
| |
| − | | \ / | | o o |
| |
| − | | \ / o------------| | dx | |
| |
| − | | \ / | | o o |
| |
| − | | o---------o | | \ / |
| |
| − | | | | o-------o |
| |
| − | | | | |
| |
| − | o-----------------------------o o-------------------o
| |
| − | Figure 18-b. Extension from 1 to 2 Dimensions: Bundle
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 18-c. Extension from 1 to 2 Dimensions: Compact===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | o-----------------o |
| |
| − | | / o \ |
| |
| − | | / (dx) / \ \ dx |
| |
| − | | / v o--------------------->o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | o o |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | x | (x) |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | o o |
| |
| − | | \ / o |
| |
| − | | \ / / \ |
| |
| − | | \ o<---------------------o v |
| |
| − | | \ / dx \ / (dx) |
| |
| − | | \ / o |
| |
| − | | o-----------------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 18-c. Extension from 1 to 2 Dimensions: Compact
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 18-d. Extension from 1 to 2 Dimensions: Digraph===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | dx |
| |
| − | | .--. .---------->----------. .--. |
| |
| − | | | \ / \ / | |
| |
| − | | (dx) ^ @ x (x) @ v (dx) |
| |
| − | | | / \ / \ | |
| |
| − | | *--* *----------<----------* *--* |
| |
| − | | dx |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 19-a. Extension from 2 to 4 Dimensions: Areal===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | o o |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ o 1100 o |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ !e! / \ / \ |
| |
| − | | o 1 1 o ----> o 1101 o 1110 o |
| |
| − | | / \ / \ / \ / \ / \ |
| |
| − | | / \ / \ / \ / \ / \ |
| |
| − | | / \ / \ / \ / \ / \ |
| |
| − | | / \ / \ o 1001 o 1111 o 0110 o |
| |
| − | | / \ / \ / \ / \ / \ / \ |
| |
| − | | / \ / \ / \ / \ / \ / \ |
| |
| − | | / \ / \ / \ / \ / \ / \ |
| |
| − | | o 1 0 o 0 1 o o 1000 o 1011 o 0111 o 0100 o |
| |
| − | | |\ / \ /| |\ / \ / \ / \ /| |
| |
| − | | | \ / \ / | | \ / \ / \ / \ / | |
| |
| − | | | \ / \ / | | \ / \ / \ / \ / | |
| |
| − | | | \ / \ / | | o 1010 o 0011 o 0101 o | |
| |
| − | | | \ / \ / | | |\ / \ / \ /| | |
| |
| − | | | \ / \ / | | | \ / \ / \ / | | |
| |
| − | | | x_1 \ / \ / x_2 | |x_1| \ / \ / \ / |x_2| |
| |
| − | | o-------o 0 0 o-------o o---+---o 0010 o 0001 o---+---o |
| |
| − | | \ / | \ / \ / | |
| |
| − | | \ / | \ / \ / | |
| |
| − | | \ / | x_3 \ / \ / x_4 | |
| |
| − | | \ / o-------o 0000 o-------o |
| |
| − | | \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | o o |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | Figure 19-a. Extension from 2 to 4 Dimensions: Areal
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 19-b. Extension from 2 to 4 Dimensions: Bundle===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------o
| |
| − | | o-----o o-----o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | o o o o |
| |
| − | @ | du | | dv | |
| |
| − | /| o o o o |
| |
| − | / | \ \ / / |
| |
| − | / | \ o / |
| |
| − | / | \ / \ / |
| |
| − | / | o-----o o-----o |
| |
| − | / o-----------------------------o
| |
| − | /
| |
| − | o-----------------------------------------/---o o-----------------------------o
| |
| − | | / | | o-----o o-----o |
| |
| − | | @ | | / \ / \ |
| |
| − | | o---------o o---------o | | / o \ |
| |
| − | | / \ / \ | | / / \ \ |
| |
| − | | / o \ | | o o o o |
| |
| − | | / / \ @-------\-----------@ | du | | dv | |
| |
| − | | / / @ \ \ | | o o o o |
| |
| − | | / / \ \ \ | | \ \ / / |
| |
| − | | / / \ \ \ | | \ o / |
| |
| − | | o o \ o o | | \ / \ / |
| |
| − | | | | \| | | | o-----o o-----o |
| |
| − | | | | | | | o-----------------------------o
| |
| − | | | u | |\ v | |
| |
| − | | | | | \ | | o-----------------------------o
| |
| − | | | | | \ | | | o-----o o-----o |
| |
| − | | o o o \ o | | / \ / \ |
| |
| − | | \ \ / \ / | | / o \ |
| |
| − | | \ \ / \ / | | / / \ \ |
| |
| − | | \ \ / \ / | | o o o o |
| |
| − | | \ @-----\-/-----------\-------------@ | du | | dv | |
| |
| − | | \ o / | | o o o o |
| |
| − | | \ / \ / \ | | \ \ / / |
| |
| − | | o---------o o---------o \ | | \ o / |
| |
| − | | \ | | \ / \ / |
| |
| − | | \ | | o-----o o-----o |
| |
| − | o-----------------------------------------\---o o-----------------------------o
| |
| − | \
| |
| − | \ o-----------------------------o
| |
| − | \ | o-----o o-----o |
| |
| − | \ | / \ / \ |
| |
| − | \ | / o \ |
| |
| − | \ | / / \ \ |
| |
| − | \| o o o o |
| |
| − | @ | du | | dv | |
| |
| − | | o o o o |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o-----o o-----o |
| |
| − | o-----------------------------o
| |
| − | Figure 19-b. Extension from 2 to 4 Dimensions: Bundle
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 19-c. Extension from 2 to 4 Dimensions: Compact===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | o-------------------o o-------------------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | o o (du).(dv) o o |
| |
| − | | | | -->-- | | |
| |
| − | | | | \ / | | |
| |
| − | | | dv .(du) | \ / | du .(dv) | |
| |
| − | | | u <---------------@---------------> v | |
| |
| − | | | | | | | |
| |
| − | | | | | | | |
| |
| − | | | | | | | |
| |
| − | | o o | o o |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \|/ / |
| |
| − | | \ | / |
| |
| − | | \ /|\ / |
| |
| − | | o-------------------o | o-------------------o |
| |
| − | | | |
| |
| − | | du . dv |
| |
| − | | | |
| |
| − | | V |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 19-c. Extension from 2 to 4 Dimensions: Compact
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 19-d. Extension from 2 to 4 Dimensions: Digraph===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | .->-. |
| |
| − | | | | |
| |
| − | | * * |
| |
| − | | \ / |
| |
| − | | .-->--@--<--. |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / . . \ |
| |
| − | | / | | \ |
| |
| − | | / | | \ |
| |
| − | | / | | \ |
| |
| − | | . | | . |
| |
| − | | | | | | |
| |
| − | | v | | v |
| |
| − | | .--. | .---------->----------. | .--. |
| |
| − | | | \|/ | | \|/ | |
| |
| − | | ^ @ ^ v @ v |
| |
| − | | | /|\ | | /|\ | |
| |
| − | | *--* | *----------<----------* | *--* |
| |
| − | | ^ | | ^ |
| |
| − | | | | | | |
| |
| − | | * | | * |
| |
| − | | \ | | / |
| |
| − | | \ | | / |
| |
| − | | \ | | / |
| |
| − | | \ . . / |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | *-->--@--<--* |
| |
| − | | / \ |
| |
| − | | . . |
| |
| − | | | | |
| |
| − | | *-<-* |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 20-i. Thematization of Conjunction (Stage 1)===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | | | | |
| |
| − | | o-----o o-----o | | o-----o o-----o |
| |
| − | | / \ / \ | | / \ / \ |
| |
| − | | / o \ | | / o \ |
| |
| − | | / /`\ \ | | / /`\ \ |
| |
| − | | o o```o o | | o o```o o |
| |
| − | | | u |```| v | | | | u |```| v | |
| |
| − | | o o```o o | | o o```o o |
| |
| − | | \ \`/ / | | \ \`/ / |
| |
| − | | \ o / | | \ o / |
| |
| − | | \ / \ / | | \ / \ / |
| |
| − | | o-----o o-----o | | o-----o o-----o |
| |
| − | | | | |
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | u v \ J /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | o
| |
| − | Figure 20-i. Thematization of Conjunction (Stage 1)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 20-ii. Thematization of Conjunction (Stage 2)===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | | | | |
| |
| − | | o-----o o-----o | | o-----o o-----o |
| |
| − | | / \ / \ | | / \ / \ |
| |
| − | | / o \ | | / o \ |
| |
| − | | / /`\ \ | | / /`\ \ |
| |
| − | | o o```o o | | o o```o o |
| |
| − | | | u |```| v | | | | u |```| v | |
| |
| − | | o o```o o | | o o```o o |
| |
| − | | \ \`/ / | | \ \`/ / |
| |
| − | | \ o / | | \ o / |
| |
| − | | \ / \ / | | \ / \ / |
| |
| − | | o-----o o-----o | | o-----o o-----o |
| |
| − | | | | |
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ J /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o----------\---------/----------o o----------\---------/----------o
| |
| − | | \ / | | \ / |
| |
| − | | \ / | | \ / |
| |
| − | | o-----@-----o | | o-----@-----o |
| |
| − | | /`````````````\ | | /`````````````\ |
| |
| − | | /```````````````\ | | /```````````````\ |
| |
| − | | /`````````````````\ | | /`````````````````\ |
| |
| − | | o```````````````````o | | o```````````````````o |
| |
| − | | |```````````````````| | | |```````````````````| |
| |
| − | | |```````` J ````````| | | |```````` x ````````| |
| |
| − | | |```````````````````| | | |```````````````````| |
| |
| − | | o```````````````````o | | o```````````````````o |
| |
| − | | \`````````````````/ | | \`````````````````/ |
| |
| − | | \```````````````/ | | \```````````````/ |
| |
| − | | \`````````````/ | | \`````````````/ |
| |
| − | | o-----------o | | o-----------o |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | J = u v x = J<u, v>
| |
| − |
| |
| − | Figure 20-ii. Thematization of Conjunction (Stage 2)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 20-iii. Thematization of Conjunction (Stage 3)===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | | | |```````````````````````````````|
| |
| − | | | |````````````o-----o````````````|
| |
| − | | | |```````````/ \```````````|
| |
| − | | | |``````````/ \``````````|
| |
| − | | | |`````````/ \`````````|
| |
| − | | | |````````/ \````````|
| |
| − | | J | |```````o x o```````|
| |
| − | | | |```````| |```````|
| |
| − | | | |```````| |```````|
| |
| − | | | |```````| |```````|
| |
| − | | o-----o o-----o | |```````o-----o o-----o```````|
| |
| − | | / \ / \ | |``````/`\ \ / /`\``````|
| |
| − | | / o \ | |`````/```\ o /```\`````|
| |
| − | | / /`\ \ | |````/`````\ /`\ /`````\````|
| |
| − | | / /```\ \ | |```/```````\ /```\ /```````\```|
| |
| − | | o o`````o o | |``o`````````o-----o`````````o``|
| |
| − | | | u |`````| v | | |``|`````````| |`````````|``|
| |
| − | o--o---------o-----o---------o--o |``|``` u ```| |``` v ```|``|
| |
| − | |``|`````````| |`````````|``| |``|`````````| |`````````|``|
| |
| − | |``o`````````o o`````````o``| |``o`````````o o`````````o``|
| |
| − | |```\`````````\ /`````````/```| |```\`````````\ /`````````/```|
| |
| − | |````\`````````\ /`````````/````| |````\`````````\ /`````````/````|
| |
| − | |`````\`````````o`````````/`````| |`````\`````````o`````````/`````|
| |
| − | |``````\```````/`\```````/``````| |``````\```````/`\```````/``````|
| |
| − | |```````o-----o```o-----o```````| |```````o-----o```o-----o```````|
| |
| − | |```````````````````````````````| |```````````````````````````````|
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | \ /
| |
| − | \ /
| |
| − | J = u v \ /
| |
| − | \ !j! /
| |
| − | \ /
| |
| − | !j! = (( x , u v )) \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | @
| |
| − | Figure 20-iii. Thematization of Conjunction (Stage 3)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 21. Thematization of Disjunction and Equality===
| |
| − |
| |
| − | <pre>
| |
| − | f g
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | | | |```````````````````````````````|
| |
| − | | o-----o o-----o | |```````o-----o```o-----o```````|
| |
| − | | /```````\ /```````\ | |``````/ \`/ \``````|
| |
| − | | /`````````o`````````\ | |`````/ o \`````|
| |
| − | | /`````````/`\`````````\ | |````/ /`\ \````|
| |
| − | | /`````````/```\`````````\ | |```/ /```\ \```|
| |
| − | | o`````````o`````o```````` o | |``o o`````o o``|
| |
| − | | |`````````|`````|`````````| | |``| |`````| |``|
| |
| − | | |``` u ```|`````|``` v ```| | |``| u |`````| v |``|
| |
| − | | |`````````|`````|`````````| | |``| |`````| |``|
| |
| − | | o`````````o`````o`````````o | |``o o`````o o``|
| |
| − | | \`````````\```/`````````/ | |```\ \```/ /```|
| |
| − | | \`````````\`/`````````/ | |````\ \`/ /````|
| |
| − | | \`````````o`````````/ | |`````\ o /`````|
| |
| − | | \```````/ \```````/ | |``````\ /`\ /``````|
| |
| − | | o-----o o-----o | |```````o-----o```o-----o```````|
| |
| − | | | |```````````````````````````````|
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | ((u)(v)) ((u , v))
| |
| − |
| |
| − | | |
| |
| − | | |
| |
| − | theta theta
| |
| − | | |
| |
| − | | |
| |
| − | v v
| |
| − |
| |
| − | !f! !g!
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | |```````````````````````````````| | |
| |
| − | |````````````o-----o````````````| | o-----o |
| |
| − | |```````````/ \```````````| | /```````\ |
| |
| − | |``````````/ \``````````| | /`````````\ |
| |
| − | |`````````/ \`````````| | /```````````\ |
| |
| − | |````````/ \````````| | /`````````````\ |
| |
| − | |```````o f o```````| | o`````` g ``````o |
| |
| − | |```````| |```````| | |```````````````| |
| |
| − | |```````| |```````| | |```````````````| |
| |
| − | |```````| |```````| | |```````````````| |
| |
| − | |```````o-----o o-----o```````| | o-----o```o-----o |
| |
| − | |``````/ \`````\ /`````/ \``````| | /`\ \`/ /`\ |
| |
| − | |`````/ \`````o`````/ \`````| | /```\ o /```\ |
| |
| − | |````/ \```/`\```/ \````| | /`````\ /`\ /`````\ |
| |
| − | |```/ \`/```\`/ \```| | /```````\ /```\ /```````\ |
| |
| − | |``o o-----o o``| | o`````````o-----o`````````o |
| |
| − | |``| | | |``| | |`````````| |`````````| |
| |
| − | |``| u | | v |``| | |``` u ```| |``` v ```| |
| |
| − | |``| | | |``| | |`````````| |`````````| |
| |
| − | |``o o o o``| | o`````````o o`````````o |
| |
| − | |```\ \ / /```| | \`````````\ /`````````/ |
| |
| − | |````\ \ / /````| | \`````````\ /`````````/ |
| |
| − | |`````\ o /`````| | \`````````o`````````/ |
| |
| − | |``````\ /`\ /``````| | \```````/ \```````/ |
| |
| − | |```````o-----o```o-----o```````| | o-----o o-----o |
| |
| − | |```````````````````````````````| | |
| |
| − | o-------------------------------o o-------------------------------o
| |
| − | ((f , ((u)(v)) )) ((g , ((u , v)) ))
| |
| − |
| |
| − | Figure 21. Thematization of Disjunction and Equality
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 22. Disjunction ''f'' and Equality ''g''===
| |
| − |
| |
| − | <pre>
| |
| − | Table 22. Disjunction f and Equality g
| |
| − | o-------------------o-------------------o
| |
| − | | u v | f g |
| |
| − | o-------------------o-------------------o
| |
| − | | | |
| |
| − | | 0 0 | 0 1 |
| |
| − | | | |
| |
| − | | 0 1 | 1 0 |
| |
| − | | | |
| |
| − | | 1 0 | 1 0 |
| |
| − | | | |
| |
| − | | 1 1 | 1 1 |
| |
| − | | | |
| |
| − | o-------------------o-------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
| |
| − | |+ '''Table 22. Disjunction ''f'' and Equality ''g'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''f'' || ''g''
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0
| |
| − | |-
| |
| − | | 0 || 1
| |
| − | |-
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 1
| |
| − | |-
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)===
| |
| − |
| |
| − | <pre>
| |
| − | Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)
| |
| − | o-----------------o-----------o o-----------------o-----------o
| |
| − | | u v f | x !f! | | u v g | y !g! |
| |
| − | o-----------------o-----------o o-----------------o-----------o
| |
| − | | | | | | |
| |
| − | | 0 0 --> | 0 1 | | 0 0 --> | 1 1 |
| |
| − | | | | | | |
| |
| − | | 0 1 --> | 1 1 | | 0 1 --> | 0 1 |
| |
| − | | | | | | |
| |
| − | | 1 0 --> | 1 1 | | 1 0 --> | 0 1 |
| |
| − | | | | | | |
| |
| − | | 1 1 --> | 1 1 | | 1 1 --> | 1 1 |
| |
| − | | | | | | |
| |
| − | o-----------------o-----------o o-----------------o-----------o
| |
| − | | | | | | |
| |
| − | | 0 0 | 1 0 | | 0 0 | 0 0 |
| |
| − | | | | | | |
| |
| − | | 0 1 | 0 0 | | 0 1 | 1 0 |
| |
| − | | | | | | |
| |
| − | | 1 0 | 0 0 | | 1 0 | 1 0 |
| |
| − | | | | | | |
| |
| − | | 1 1 | 0 0 | | 1 1 | 0 0 |
| |
| − | | | | | | |
| |
| − | o-----------------o-----------o o-----------------o-----------o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" style="width:96%"
| |
| − | |+ '''Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)'''
| |
| − | |
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
| − | |+ '''Table 23-i. Disjunction ''f'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v'' || ''f''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''x'' || φ
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || →
| |
| − | |-
| |
| − | | 0 || 1 || →
| |
| − | |-
| |
| − | | 1 || 0 || →
| |
| − | |-
| |
| − | | 1 || 1 || →
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 1
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 ||
| |
| − | |-
| |
| − | | 0 || 1 ||
| |
| − | |-
| |
| − | | 1 || 0 ||
| |
| − | |-
| |
| − | | 1 || 1 ||
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 0 || 0
| |
| − | |-
| |
| − | | 0 || 0
| |
| − | |-
| |
| − | | 0 || 0
| |
| − | |}
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
| − | |+ '''Table 23-ii. Equality ''g'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v'' || ''g''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''y'' || γ
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || →
| |
| − | |-
| |
| − | | 0 || 1 || →
| |
| − | |-
| |
| − | | 1 || 0 || →
| |
| − | |-
| |
| − | | 1 || 1 || →
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1 || 1
| |
| − | |-
| |
| − | | 0 || 1
| |
| − | |-
| |
| − | | 0 || 1
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 ||
| |
| − | |-
| |
| − | | 0 || 1 ||
| |
| − | |-
| |
| − | | 1 || 0 ||
| |
| − | |-
| |
| − | | 1 || 1 ||
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0
| |
| − | |-
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 0 || 0
| |
| − | |}
| |
| − | |}
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)===
| |
| − |
| |
| − | <pre>
| |
| − | Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)
| |
| − | o-----------------------o-----o o-----------------------o-----o
| |
| − | | u v f x | !f! | | u v g y | !g! |
| |
| − | o-----------------------o-----o o-----------------------o-----o
| |
| − | | | | | | |
| |
| − | | 0 0 --> 0 | 1 | | 0 0 0 | 0 |
| |
| − | | | | | | |
| |
| − | | 0 0 1 | 0 | | 0 0 --> 1 | 1 |
| |
| − | | | | | | |
| |
| − | | 0 1 0 | 0 | | 0 1 --> 0 | 1 |
| |
| − | | | | | | |
| |
| − | | 0 1 --> 1 | 1 | | 0 1 1 | 0 |
| |
| − | | | | | | |
| |
| − | o-----------------------o-----o o-----------------------o-----o
| |
| − | | | | | | |
| |
| − | | 1 0 0 | 0 | | 1 0 --> 0 | 1 |
| |
| − | | | | | | |
| |
| − | | 1 0 --> 1 | 1 | | 1 0 1 | 0 |
| |
| − | | | | | | |
| |
| − | | 1 1 0 | 0 | | 1 1 0 | 0 |
| |
| − | | | | | | |
| |
| − | | 1 1 --> 1 | 1 | | 1 1 --> 1 | 1 |
| |
| − | | | | | | |
| |
| − | o-----------------------o-----o o-----------------------o-----o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" style="width:96%"
| |
| − | |+ '''Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)'''
| |
| − | |
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
| − | |+ '''Table 24-i. Disjunction ''f'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v'' || ''f'' || ''x''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | φ
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || → || 0
| |
| − | |-
| |
| − | | 0 || 0 || || 1
| |
| − | |-
| |
| − | | 0 || 1 || || 0
| |
| − | |-
| |
| − | | 0 || 1 || → || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1 || 0 || || 0
| |
| − | |-
| |
| − | | 1 || 0 || → || 1
| |
| − | |-
| |
| − | | 1 || 1 || || 0
| |
| − | |-
| |
| − | | 1 || 1 || → || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |}
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
| − | |+ '''Table 24-ii. Equality ''g'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v'' || ''g'' || ''y''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | γ
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || || 0
| |
| − | |-
| |
| − | | 0 || 0 || → || 1
| |
| − | |-
| |
| − | | 0 || 1 || → || 0
| |
| − | |-
| |
| − | | 0 || 1 || || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |-
| |
| − | | 1
| |
| − | |-
| |
| − | | 0
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1 || 0 || → || 0
| |
| − | |-
| |
| − | | 1 || 0 || || 1
| |
| − | |-
| |
| − | | 1 || 1 || || 0
| |
| − | |-
| |
| − | | 1 || 1 || → || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |}
| |
| − | |}
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)===
| |
| − |
| |
| − | <pre>
| |
| − | Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)
| |
| − | o-----------------------o-----o o-----------------------o-----o
| |
| − | | u v f x | !f! | | u v g y | !g! |
| |
| − | o-----------------------o-----o o-----------------------o-----o
| |
| − | | | | | | |
| |
| − | | 0 0 --> 0 | 1 | | 0 0 0 | 0 |
| |
| − | | | | | | |
| |
| − | | 0 1 0 | 0 | | 0 1 --> 0 | 1 |
| |
| − | | | | | | |
| |
| − | | 1 0 0 | 0 | | 1 0 --> 0 | 1 |
| |
| − | | | | | | |
| |
| − | | 1 1 0 | 0 | | 1 1 0 | 0 |
| |
| − | | | | | | |
| |
| − | o-----------------------o-----o o-----------------------o-----o
| |
| − | | | | | | |
| |
| − | | 0 0 1 | 0 | | 0 0 --> 1 | 1 |
| |
| − | | | | | | |
| |
| − | | 0 1 --> 1 | 1 | | 0 1 1 | 0 |
| |
| − | | | | | | |
| |
| − | | 1 0 --> 1 | 1 | | 1 0 1 | 0 |
| |
| − | | | | | | |
| |
| − | | 1 1 --> 1 | 1 | | 1 1 --> 1 | 1 |
| |
| − | | | | | | |
| |
| − | o-----------------------o-----o o-----------------------o-----o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" style="width:96%"
| |
| − | |+ '''Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)'''
| |
| − | |
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
| − | |+ '''Table 25-i. Disjunction ''f'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v'' || ''f'' || ''x''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | φ
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || → || 0
| |
| − | |-
| |
| − | | 0 || 1 || || 0
| |
| − | |-
| |
| − | | 1 || 0 || || 0
| |
| − | |-
| |
| − | | 1 || 1 || || 0
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 0
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || || 1
| |
| − | |-
| |
| − | | 0 || 1 || → || 1
| |
| − | |-
| |
| − | | 1 || 0 || → || 1
| |
| − | |-
| |
| − | | 1 || 1 || → || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |-
| |
| − | | 1
| |
| − | |-
| |
| − | | 1
| |
| − | |}
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
| − | |+ '''Table 25-ii. Equality ''g'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v'' || ''g'' || ''y''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | γ
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || || 0
| |
| − | |-
| |
| − | | 0 || 1 || → || 0
| |
| − | |-
| |
| − | | 1 || 0 || → || 0
| |
| − | |-
| |
| − | | 1 || 1 || || 0
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |-
| |
| − | | 1
| |
| − | |-
| |
| − | | 0
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || → || 1
| |
| − | |-
| |
| − | | 0 || 1 || || 1
| |
| − | |-
| |
| − | | 1 || 0 || || 1
| |
| − | |-
| |
| − | | 1 || 1 || → || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 0
| |
| − | |-
| |
| − | | 1
| |
| − | |}
| |
| − | |}
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Tables 26-i and 26-ii. Tacit Extension and Thematization===
| |
| − |
| |
| − | <pre>
| |
| − | Tables 26-i and 26-ii. Tacit Extension and Thematization
| |
| − | o-----------------o-----------o o-----------------o-----------o
| |
| − | | u v x | !e!f !f! | | u v y | !e!g !g! |
| |
| − | o-----------------o-----------o o-----------------o-----------o
| |
| − | | | | | | |
| |
| − | | 0 0 0 | 0 1 | | 0 0 0 | 1 0 |
| |
| − | | | | | | |
| |
| − | | 0 0 1 | 0 0 | | 0 0 1 | 1 1 |
| |
| − | | | | | | |
| |
| − | | 0 1 0 | 1 0 | | 0 1 0 | 0 1 |
| |
| − | | | | | | |
| |
| − | | 0 1 1 | 1 1 | | 0 1 1 | 0 0 |
| |
| − | | | | | | |
| |
| − | o-----------------o-----------o o-----------------o-----------o
| |
| − | | | | | | |
| |
| − | | 1 0 0 | 1 0 | | 1 0 0 | 0 1 |
| |
| − | | | | | | |
| |
| − | | 1 0 1 | 1 1 | | 1 0 1 | 0 0 |
| |
| − | | | | | | |
| |
| − | | 1 1 0 | 1 0 | | 1 1 0 | 1 0 |
| |
| − | | | | | | |
| |
| − | | 1 1 1 | 1 1 | | 1 1 1 | 1 1 |
| |
| − | | | | | | |
| |
| − | o-----------------o-----------o o-----------------o-----------o
| |
| − | </pre>
| |
| − |
| |
| − | {| align="center" style="width:96%"
| |
| − | |+ '''Tables 26-i and 26-ii. Tacit Extension and Thematization'''
| |
| − | |
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
| − | |+ '''Table 26-i. Disjunction ''f'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v'' || ''x''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ε''f'' || θ''f''
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || 0
| |
| − | |-
| |
| − | | 0 || 0 || 1
| |
| − | |-
| |
| − | | 0 || 1 || 0
| |
| − | |-
| |
| − | | 0 || 1 || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 1
| |
| − | |-
| |
| − | | 0 || 0
| |
| − | |-
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1 || 0 || 0
| |
| − | |-
| |
| − | | 1 || 0 || 1
| |
| − | |-
| |
| − | | 1 || 1 || 0
| |
| − | |-
| |
| − | | 1 || 1 || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |-
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |}
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="1" cellpadding="0" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:90%"
| |
| − | |+ '''Table 26-ii. Equality ''g'' '''
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ''u'' || ''v'' || ''y''
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:paleturquoise; font-weight:bold; text-align:center; width:100%"
| |
| − | | ε''g'' || θ''g''
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 0 || 0
| |
| − | |-
| |
| − | | 0 || 0 || 1
| |
| − | |-
| |
| − | | 0 || 1 || 0
| |
| − | |-
| |
| − | | 0 || 1 || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |-
| |
| − | | 0 || 1
| |
| − | |-
| |
| − | | 0 || 0
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 1 || 0 || 0
| |
| − | |-
| |
| − | | 1 || 0 || 1
| |
| − | |-
| |
| − | | 1 || 1 || 0
| |
| − | |-
| |
| − | | 1 || 1 || 1
| |
| − | |}
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="6" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| − | | 0 || 1
| |
| − | |-
| |
| − | | 0 || 0
| |
| − | |-
| |
| − | | 1 || 0
| |
| − | |-
| |
| − | | 1 || 1
| |
| − | |}
| |
| − | |}
| |
| − | |}
| |
| − | <br>
| |
| − |
| |
| − | ===Table 27. Thematization of Bivariate Propositions===
| |
| − |
| |
| − | <pre>
| |
| − | Table 27. Thematization of Bivariate Propositions
| |
| − | o---------o---------o----------o--------------------o--------------------o
| |
| − | | u : 1 1 0 0 | f | theta (f) | theta (f) |
| |
| − | | v : 1 0 1 0 | | | |
| |
| − | o---------o---------o----------o--------------------o--------------------o
| |
| − | | | | | | |
| |
| − | | f_0 | 0 0 0 0 | () | (( f , () )) | f + 1 |
| |
| − | | | | | | |
| |
| − | | f_1 | 0 0 0 1 | (u)(v) | (( f , (u)(v) )) | f + u + v + uv |
| |
| − | | | | | | |
| |
| − | | f_2 | 0 0 1 0 | (u) v | (( f , (u) v )) | f + v + uv + 1 |
| |
| − | | | | | | |
| |
| − | | f_3 | 0 0 1 1 | (u) | (( f , (u) )) | f + u |
| |
| − | | | | | | |
| |
| − | | f_4 | 0 1 0 0 | u (v) | (( f , u (v) )) | f + u + uv + 1 |
| |
| − | | | | | | |
| |
| − | | f_5 | 0 1 0 1 | (v) | (( f , (v) )) | f + v |
| |
| − | | | | | | |
| |
| − | | f_6 | 0 1 1 0 | (u, v) | (( f , (u, v) )) | f + u + v + 1 |
| |
| − | | | | | | |
| |
| − | | f_7 | 0 1 1 1 | (u v) | (( f , (u v) )) | f + uv |
| |
| − | | | | | | |
| |
| − | o---------o---------o----------o--------------------o--------------------o
| |
| − | | | | | | |
| |
| − | | f_8 | 1 0 0 0 | u v | (( f , u v )) | f + uv + 1 |
| |
| − | | | | | | |
| |
| − | | f_9 | 1 0 0 1 | ((u, v)) | (( f , ((u, v)) )) | f + u + v |
| |
| − | | | | | | |
| |
| − | | f_10 | 1 0 1 0 | v | (( f , v )) | f + v + 1 |
| |
| − | | | | | | |
| |
| − | | f_11 | 1 0 1 1 | (u (v)) | (( f , (u (v)) )) | f + u + uv |
| |
| − | | | | | | |
| |
| − | | f_12 | 1 1 0 0 | u | (( f , u )) | f + u + 1 |
| |
| − | | | | | | |
| |
| − | | f_13 | 1 1 0 1 | ((u) v) | (( f , ((u) v) )) | f + v + uv |
| |
| − | | | | | | |
| |
| − | | f_14 | 1 1 1 0 | ((u)(v)) | (( f , ((u)(v)) )) | f + u + v + uv + 1 |
| |
| − | | | | | | |
| |
| − | | f_15 | 1 1 1 1 | (()) | (( f , (()) )) | f |
| |
| − | | | | | | |
| |
| − | o---------o---------o----------o--------------------o--------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 28. Propositions on Two Variables===
| |
| − |
| |
| − | <pre>
| |
| − | Table 28. Propositions on Two Variables
| |
| − | o-------o-----o----------------------------------------------------------------o
| |
| − | | u v | | f f f f f f f f f f f f f f f f |
| |
| − | | | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 |
| |
| − | o-------o-----o----------------------------------------------------------------o
| |
| − | | | | |
| |
| − | | 0 0 | | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| |
| − | | | | |
| |
| − | | 0 1 | | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| |
| − | | | | |
| |
| − | | 1 0 | | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| |
| − | | | | |
| |
| − | | 1 1 | | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| |
| − | | | | |
| |
| − | o-------o-----o----------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 29. Thematic Extensions of Bivariate Propositions===
| |
| − |
| |
| − | <pre>
| |
| − | Table 29. Thematic Extensions of Bivariate Propositions
| |
| − | o-------o-----o----------------------------------------------------------------o
| |
| − | | u v | f^¢ |!f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! !f! |
| |
| − | | | | 00 01 02 03 04 05 06 07 08 09 10 11 12 13 14 15 |
| |
| − | o-------o-----o----------------------------------------------------------------o
| |
| − | | | | |
| |
| − | | 0 0 | 0 | 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 |
| |
| − | | | | |
| |
| − | | 0 0 | 1 | 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 |
| |
| − | | | | |
| |
| − | | 0 1 | 0 | 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 |
| |
| − | | | | |
| |
| − | | 0 1 | 1 | 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 |
| |
| − | | | | |
| |
| − | | 1 0 | 0 | 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 |
| |
| − | | | | |
| |
| − | | 1 0 | 1 | 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 |
| |
| − | | | | |
| |
| − | | 1 1 | 0 | 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 |
| |
| − | | | | |
| |
| − | | 1 1 | 1 | 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 |
| |
| − | | | | |
| |
| − | o-------o-----o----------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 30. Generic Frame of a Logical Transformation===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------------o
| |
| − | | U |
| |
| − | | |
| |
| − | | o-----------o o-----------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | o o o o |
| |
| − | | | | | | |
| |
| − | | | u | | v | |
| |
| − | | | | | | |
| |
| − | | o o o o |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o-----------o o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o---------------------------o---------------------------o
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | o-------------------------o o-------------------------o o-------------------------o
| |
| − | | U | | U | | U |
| |
| − | | o---o o---o | | o---o o---o | | o---o o---o |
| |
| − | | / \ / \ | | / \ / \ | | / \ / \ |
| |
| − | | / o \ | | / o \ | | / o \ |
| |
| − | | / / \ \ | | / / \ \ | | / / \ \ |
| |
| − | | o o o o | | o o o o | | o o o o |
| |
| − | | | u | | v | | | | u | | v | | | | u | | v | |
| |
| − | | o o o o | | o o o o | | o o o o |
| |
| − | | \ \ / / | | \ \ / / | | \ \ / / |
| |
| − | | \ o / | | \ o / | | \ o / |
| |
| − | | \ / \ / | | \ / \ / | | \ / \ / |
| |
| − | | o---o o---o | | o---o o---o | | o---o o---o |
| |
| − | | | | | | |
| |
| − | o-------------------------o o-------------------------o o-------------------------o
| |
| − | \ | \ / | /
| |
| − | \ | \ / | /
| |
| − | \ | \ / | /
| |
| − | \ | \ / | /
| |
| − | \ g | \ f / | h /
| |
| − | \ | \ / | /
| |
| − | \ | \ / | /
| |
| − | \ | \ / | /
| |
| − | \ | \ / | /
| |
| − | \ o----------|-----------\-----/-----------|----------o /
| |
| − | \ | X | \ / | | /
| |
| − | \ | | \ / | | /
| |
| − | \ | | o-----o-----o | | /
| |
| − | \| | / \ | |/
| |
| − | \ | / \ | /
| |
| − | |\ | / \ | /|
| |
| − | | \ | / \ | / |
| |
| − | | \ | / \ | / |
| |
| − | | \ | o x o | / |
| |
| − | | \ | | | | / |
| |
| − | | \ | | | | / |
| |
| − | | \ | | | | / |
| |
| − | | \ | | | | / |
| |
| − | | \ | | | | / |
| |
| − | | \| | | |/ |
| |
| − | | o--o--------o o--------o--o |
| |
| − | | / \ \ / / \ |
| |
| − | | / \ \ / / \ |
| |
| − | | / \ o / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | / \ / \ / \ |
| |
| − | | o o--o-----o--o o |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | | y | | z | |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | o o o o |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | o-----------o o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o---------------------------------------------------o
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ p , q /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | o
| |
| − |
| |
| − | Figure 30. Generic Frame of a Logical Transformation
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 3===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | x = f<u, v> |
| |
| − | | |
| |
| − | | y = g<u, v> |
| |
| − | | |
| |
| − | | z = h<u, v> |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| − | | width="20%" |
| |
| − | | width="20%" | ''x''
| |
| − | | width="20%" | =
| |
| − | | width="20%" | ''f''‹''u'', ''v''›
| |
| − | | width="20%" |
| |
| − | |-
| |
| − | | || ''y'' || = || ''g''‹''u'', ''v''› ||
| |
| − | |-
| |
| − | | || ''z'' || = || ''h''‹''u'', ''v''› ||
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Figure 31. Operator Diagram (1)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | U% F X% |
| |
| − | | o------------------>o |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | !W! | | !W! |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | v v |
| |
| − | | o------------------>o |
| |
| − | | !W!U% !W!F !W!X% |
| |
| − | | |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Figure 31. Operator Diagram (1)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 32. Operator Diagram (2)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | U% !W! !W!U% |
| |
| − | | o------------------>o |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | F | | !W!F |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | v v |
| |
| − | | o------------------>o |
| |
| − | | X% !W! !W!X% |
| |
| − | | |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Figure 32. Operator Diagram (2)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 33-i. Analytic Diagram (1)===
| |
| − |
| |
| − | <pre>
| |
| − | U% $E$ $E$U% $E$U% $E$U%
| |
| − | o------------------>o============o============o
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | F | | $E$F = | $d$^0.F + | $r$^0.F
| |
| − | | | | |
| |
| − | | | | |
| |
| − | | | | |
| |
| − | v v v v
| |
| − | o------------------>o============o============o
| |
| − | X% $E$ $E$X% $E$X% $E$X%
| |
| − |
| |
| − | Figure 33-i. Analytic Diagram (1)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 33-ii. Analytic Diagram (2)===
| |
| − |
| |
| − | <pre>
| |
| − | U% $E$ $E$U% $E$U% $E$U% $E$U%
| |
| − | o------------------>o============o============o============o
| |
| − | | | | | |
| |
| − | | | | | |
| |
| − | | | | | |
| |
| − | | | | | |
| |
| − | F | | $E$F = | $d$^0.F + | $d$^1.F + | $r$^1.F
| |
| − | | | | | |
| |
| − | | | | | |
| |
| − | | | | | |
| |
| − | v v v v v
| |
| − | o------------------>o============o============o============o
| |
| − | X% $E$ $E$X% $E$X% $E$X% $E$X%
| |
| − |
| |
| − | Figure 33-ii. Analytic Diagram (2)
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 4===
| |
| − |
| |
| − | <pre>
| |
| − | o--------------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> |
| |
| − | | |
| |
| − | | ... |
| |
| − | | |
| |
| − | | x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> |
| |
| − | | |
| |
| − | | |
| |
| − | | dx_1 = EF_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1 + du_1, ..., u_n + du_n> |
| |
| − | | |
| |
| − | | ... |
| |
| − | | |
| |
| − | | dx_k = EF_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1 + du_1, ..., u_n + du_n> |
| |
| − | | |
| |
| − | o--------------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| − | | width="8%" | ''x''<sub>1</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| |
| − | |-
| |
| − | | ...
| |
| − | |-
| |
| − | | width="8%" | ''x''<sub>''k''</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| − | | width="8%" | d''x''<sub>1</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | E''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, …, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›
| |
| − | |-
| |
| − | | ...
| |
| − | |-
| |
| − | | width="8%" | d''x''<sub>''k''</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | E''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub> + d''u''<sub>1</sub>, …, ''u''<sub>''n''</sub> + d''u''<sub>''n''</sub>›
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Formula Display 5===
| |
| − |
| |
| − | <pre>
| |
| − | o--------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | x_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> |
| |
| − | | |
| |
| − | | ... |
| |
| − | | |
| |
| − | | x_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> |
| |
| − | | |
| |
| − | | |
| |
| − | | dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> |
| |
| − | | |
| |
| − | | ... |
| |
| − | | |
| |
| − | | dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> |
| |
| − | | |
| |
| − | o--------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| − | | width="8%" | ''x''<sub>1</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| |
| − | |-
| |
| − | | ...
| |
| − | |-
| |
| − | | width="8%" | ''x''<sub>''k''</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| |
| − | |}
| |
| − | |-
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| − | | width="8%" | d''x''<sub>1</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| |
| − | |-
| |
| − | | ...
| |
| − | |-
| |
| − | | width="8%" | d''x''<sub>''k''</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Formula Display 6===
| |
| − |
| |
| − | <pre>
| |
| − | o--------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | dx_1 = !e!F_1 <u_1, ..., u_n, du_1, ..., du_n> = F_1 <u_1, ..., u_n> |
| |
| − | | |
| |
| − | | ... |
| |
| − | | |
| |
| − | | dx_k = !e!F_k <u_1, ..., u_n, du_1, ..., du_n> = F_k <u_1, ..., u_n> |
| |
| − | | |
| |
| − | o--------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="4" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| − | | width="8%" | d''x''<sub>1</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | <math>\epsilon</math>''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>1</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| |
| − | |-
| |
| − | | ...
| |
| − | |-
| |
| − | | width="8%" | d''x''<sub>''k''</sub>
| |
| − | | width="4%" | =
| |
| − | | width="44%" | <math>\epsilon</math>''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>, d''u''<sub>1</sub>, …, d''u''<sub>''n''</sub>›
| |
| − | | width="4%" | =
| |
| − | | width="40%" | ''F''<sub>''k''</sub>‹''u''<sub>1</sub>, …, ''u''<sub>''n''</sub>›
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Formula Display 7===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | $D$ = $E$ - $e$ |
| |
| − | | |
| |
| − | | = $r$^0 |
| |
| − | | |
| |
| − | | = $d$^1 + $r$^1 |
| |
| − | | |
| |
| − | | = $d$^1 + ... + $d$^m + $r$^m |
| |
| − | | |
| |
| − | | = Sum_(i = 1 ... m) $d$^i + $r$^m |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | <br><font face="courier new">
| |
| − | {| align="center" border="1" cellpadding="12" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:96%"
| |
| − | |
| |
| − | {| align="center" border="0" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:left; width:100%"
| |
| − | | <font face=georgia>'''D'''</font>
| |
| − | | =
| |
| − | | <font face=georgia>'''E'''</font> – <font face=georgia>'''e'''</font>
| |
| − | |-
| |
| − | |
| |
| − | | =
| |
| − | | <font face=georgia>'''r'''</font><sup>0</sup>
| |
| − | |-
| |
| − | |
| |
| − | | =
| |
| − | | <font face=georgia>'''d'''</font><sup>1</sup> + <font face=georgia>'''r'''</font><sup>1</sup>
| |
| − | |-
| |
| − | |
| |
| − | | =
| |
| − | | <font face=georgia>'''d'''</font><sup>1</sup> + … + <font face=georgia>'''d'''</font><sup>''m''</sup> + <font face=georgia>'''r'''</font><sup>''m''</sup>
| |
| − | |-
| |
| − | |
| |
| − | | =
| |
| − | | <font size="+2">∑</font><sub>(''i'' = 1 … ''m'')</sub> <font face=georgia>'''d'''</font><sup>''i''</sup> + <font face=georgia>'''r'''</font><sup>''m''</sup>
| |
| − | |}
| |
| − | |}
| |
| − | </font><br>
| |
| − |
| |
| − | ===Figure 34. Tangent Functor Diagram===
| |
| − |
| |
| − | <pre>
| |
| − | U% $T$ $T$U% $T$U%
| |
| − | o------------------>o============o
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | F | | $T$F = | <!e!, d> F
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | v v v
| |
| − | o------------------>o============o
| |
| − | X% $T$ $T$X% $T$X%
| |
| − |
| |
| − | Figure 34. Tangent Functor Diagram
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 35. Conjunction as Transformation===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | o---------o o---------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / /`\ \ |
| |
| − | | / /```\ \ |
| |
| − | | o o`````o o |
| |
| − | | | |`````| | |
| |
| − | | | u |`````| v | |
| |
| − | | | |`````| | |
| |
| − | | o o`````o o |
| |
| − | | \ \```/ / |
| |
| − | | \ \`/ / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o---------o o---------o |
| |
| − | | |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ J /
| |
| − | \ /
| |
| − | \ /
| |
| − | \ /
| |
| − | o--------------\---------/--------------o
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | o------@------o |
| |
| − | | /```````````````\ |
| |
| − | | /`````````````````\ |
| |
| − | | /```````````````````\ |
| |
| − | | /`````````````````````\ |
| |
| − | | o```````````````````````o |
| |
| − | | |```````````````````````| |
| |
| − | | |`````````` x ``````````| |
| |
| − | | |```````````````````````| |
| |
| − | | o```````````````````````o |
| |
| − | | \`````````````````````/ |
| |
| − | | \```````````````````/ |
| |
| − | | \`````````````````/ |
| |
| − | | \```````````````/ |
| |
| − | | o-------------o |
| |
| − | | |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Figure 35. Conjunction as Transformation
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 36. Computation of !e!J===
| |
| − |
| |
| − | <pre>
| |
| − | Table 36. Computation of !e!J
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | !e!J = J<u, v> |
| |
| − | | |
| |
| − | | = u v |
| |
| − | | |
| |
| − | | = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | !e!J = u v (du)(dv) + |
| |
| − | | u v (du) dv + |
| |
| − | | u v du (dv) + |
| |
| − | | u v du dv |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 37-a. Tacit Extension of J (Areal)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | o |
| |
| − | | /%\ |
| |
| − | | /%%%\ |
| |
| − | | /%%%%%\ |
| |
| − | | o%%%%%%%o |
| |
| − | | /%\%%%%%/%\ |
| |
| − | | /%%%\%%%/%%%\ |
| |
| − | | /%%%%%\%/%%%%%\ |
| |
| − | | o%%%%%%%o%%%%%%%o |
| |
| − | | / \%%%%%/%\%%%%%/ \ |
| |
| − | | / \%%%/%%%\%%%/ \ |
| |
| − | | / \%/%%%%%\%/ \ |
| |
| − | | o o%%%%%%%o o |
| |
| − | | / \ / \%%%%%/ \ / \ |
| |
| − | | / \ / \%%%/ \ / \ |
| |
| − | | / \ / \%/ \ / \ |
| |
| − | | o o o o o |
| |
| − | | |\ / \ / \ / \ /| |
| |
| − | | | \ / \ / \ / \ / | |
| |
| − | | | \ / \ / \ / \ / | |
| |
| − | | | o o o o | |
| |
| − | | | |\ / \ / \ /| | |
| |
| − | | | | \ / \ / \ / | | |
| |
| − | | | u | \ / \ / \ / | v | |
| |
| − | | o---+---o o o---+---o |
| |
| − | | | \ / \ / | |
| |
| − | | | \ / \ / | |
| |
| − | | | du \ / \ / dv | |
| |
| − | | o-------o o-------o |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | o |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Figure 37-a. Tacit Extension of J (Areal)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 37-b. Tacit Extension of J (Bundle)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------o
| |
| − | | |
| |
| − | | o-----o o-----o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | o o o o |
| |
| − | @ | du | | dv | |
| |
| − | /| o o o o |
| |
| − | / | \ \ / / |
| |
| − | / | \ o / |
| |
| − | / | \ / \ / |
| |
| − | / | o-----o o-----o |
| |
| − | / | |
| |
| − | / o-----------------------------o
| |
| − | /
| |
| − | o----------------------------------------/----o o-----------------------------o
| |
| − | | / | | |
| |
| − | | @ | | o-----o o-----o |
| |
| − | | | | / \ / \ |
| |
| − | | o---------o o---------o | | / o \ |
| |
| − | | / \ / \ | | / / \ \ |
| |
| − | | / o \ | | o o o o |
| |
| − | | / /`\ @------\-----------@ | du | | dv | |
| |
| − | | / /```\ \ | | o o o o |
| |
| − | | / /`````\ \ | | \ \ / / |
| |
| − | | / /```````\ \ | | \ o / |
| |
| − | | o o`````````o o | | \ / \ / |
| |
| − | | | |````@````| | | | o-----o o-----o |
| |
| − | | | |`````\```| | | | |
| |
| − | | | |``````\``| | | o-----------------------------o
| |
| − | | | u |```````\`| v | |
| |
| − | | | |````````\| | | o-----------------------------o
| |
| − | | | |`````````| | | | |
| |
| − | | | |`````````|\ | | | o-----o o-----o |
| |
| − | | o o`````````o \ o | | / \ / \ |
| |
| − | | \ \```````/ \ / | | / o \ |
| |
| − | | \ \`````/ \ / | | / / \ \ |
| |
| − | | \ \```/ \ / | | o o o o |
| |
| − | | \ @------\-/---------\---------------@ | du | | dv | |
| |
| − | | \ o \ / | | o o o o |
| |
| − | | \ / \ / | | \ \ / / |
| |
| − | | o---------o o---------o \ | | \ o / |
| |
| − | | \ | | \ / \ / |
| |
| − | | \ | | o-----o o-----o |
| |
| − | | \ | | |
| |
| − | o----------------------------------------\----o o-----------------------------o
| |
| − | \
| |
| − | \ o-----------------------------o
| |
| − | \ |`````````````````````````````|
| |
| − | \ |````` o-----o```o-----o``````|
| |
| − | \ |`````/```````\`/```````\`````|
| |
| − | \ |````/`````````o`````````\````|
| |
| − | \ |```/`````````/`\`````````\```|
| |
| − | \|``o`````````o```o`````````o``|
| |
| − | @``|```du````|```|````dv```|``|
| |
| − | |``o`````````o```o`````````o``|
| |
| − | |```\`````````\`/`````````/```|
| |
| − | |````\`````````o`````````/````|
| |
| − | |`````\```````/`\```````/`````|
| |
| − | |``````o-----o```o-----o``````|
| |
| − | |`````````````````````````````|
| |
| − | o-----------------------------o
| |
| − | Figure 37-b. Tacit Extension of J (Bundle)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 37-c. Tacit Extension of J (Compact)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | o-------------------o o-------------------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | o o (du).(dv) o o |
| |
| − | | | | -->-- | | |
| |
| − | | | | \ / | | |
| |
| − | | | dv .(du) | \ / | du .(dv) | |
| |
| − | | | u <---------------@---------------> v | |
| |
| − | | | | | | | |
| |
| − | | | | | | | |
| |
| − | | | | | | | |
| |
| − | | o o | o o |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \|/ / |
| |
| − | | \ | / |
| |
| − | | \ /|\ / |
| |
| − | | o-------------------o | o-------------------o |
| |
| − | | | |
| |
| − | | du . dv |
| |
| − | | | |
| |
| − | | V |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 37-c. Tacit Extension of J (Compact)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 37-d. Tacit Extension of J (Digraph)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | (du).(dv) |
| |
| − | | --->--- |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | u @ v |
| |
| − | | /|\ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | (du) dv / | \ du (dv) |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | v | v |
| |
| − | | @ | @ |
| |
| − | | u (v) | (u) v |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | du . dv |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | v |
| |
| − | | @ |
| |
| − | | |
| |
| − | | (u).(v) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 37-d. Tacit Extension of J (Digraph)
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 38. Computation of EJ (Method 1)===
| |
| − |
| |
| − | <pre>
| |
| − | Table 38. Computation of EJ (Method 1)
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | EJ = J<u + du, v + dv> |
| |
| − | | |
| |
| − | | = (u, du)(v, dv) |
| |
| − | | |
| |
| − | | = u v J<1 + du, 1 + dv> + |
| |
| − | | |
| |
| − | | u (v) J<1 + du, 0 + dv> + |
| |
| − | | |
| |
| − | | (u) v J<0 + du, 1 + dv> + |
| |
| − | | |
| |
| − | | (u)(v) J<0 + du, 0 + dv> |
| |
| − | | |
| |
| − | | = u v J<(du), (dv)> + |
| |
| − | | |
| |
| − | | u (v) J<(du), dv > + |
| |
| − | | |
| |
| − | | (u) v J< du , (dv)> + |
| |
| − | | |
| |
| − | | (u)(v) J< du , dv > |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | EJ = u v (du)(dv) |
| |
| − | | + u (v)(du) dv |
| |
| − | | + (u) v du (dv) |
| |
| − | | + (u)(v) du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 39. Computation of EJ (Method 2)===
| |
| − |
| |
| − | <pre>
| |
| − | Table 39. Computation of EJ (Method 2)
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | EJ = <u + du> <v + dv> |
| |
| − | | |
| |
| − | | = u v + u dv + v du + du dv |
| |
| − | | |
| |
| − | | EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 40-a. Enlargement of J (Areal)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | o |
| |
| − | | /%\ |
| |
| − | | /%%%\ |
| |
| − | | /%%%%%\ |
| |
| − | | o%%%%%%%o |
| |
| − | | / \%%%%%/ \ |
| |
| − | | / \%%%/ \ |
| |
| − | | / \%/ \ |
| |
| − | | o o o |
| |
| − | | /%\ / \ /%\ |
| |
| − | | /%%%\ / \ /%%%\ |
| |
| − | | /%%%%%\ / \ /%%%%%\ |
| |
| − | | o%%%%%%%o o%%%%%%%o |
| |
| − | | / \%%%%%/ \ / \%%%%%/ \ |
| |
| − | | / \%%%/ \ / \%%%/ \ |
| |
| − | | / \%/ \ / \%/ \ |
| |
| − | | o o o o o |
| |
| − | | |\ / \ /%\ / \ /| |
| |
| − | | | \ / \ /%%%\ / \ / | |
| |
| − | | | \ / \ /%%%%%\ / \ / | |
| |
| − | | | o o%%%%%%%o o | |
| |
| − | | | |\ / \%%%%%/ \ /| | |
| |
| − | | | | \ / \%%%/ \ / | | |
| |
| − | | | u | \ / \%/ \ / | v | |
| |
| − | | o---+---o o o---+---o |
| |
| − | | | \ / \ / | |
| |
| − | | | \ / \ / | |
| |
| − | | | du \ / \ / dv | |
| |
| − | | o-------o o-------o |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | o |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Figure 40-a. Enlargement of J (Areal)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 40-b. Enlargement of J (Bundle)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------o
| |
| − | | |
| |
| − | | o-----o o-----o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / /%\ \ |
| |
| − | | o o%%%o o |
| |
| − | @ | du |%%%| dv | |
| |
| − | /| o o%%%o o |
| |
| − | / | \ \%/ / |
| |
| − | / | \ o / |
| |
| − | / | \ / \ / |
| |
| − | / | o-----o o-----o |
| |
| − | / | |
| |
| − | / o-----------------------------o
| |
| − | /
| |
| − | o----------------------------------------/----o o-----------------------------o
| |
| − | | / | | |
| |
| − | | @ | | o-----o o-----o |
| |
| − | | | | /%%%%%%%\ / \ |
| |
| − | | o---------o o---------o | | /%%%%%%%%%o \ |
| |
| − | | / \ / \ | | /%%%%%%%%%/ \ \ |
| |
| − | | / o \ | | o%%%%%%%%%o o o |
| |
| − | | / /`\ @------\-----------@ |%% du %%%| | dv | |
| |
| − | | / /```\ \ | | o%%%%%%%%%o o o |
| |
| − | | / /`````\ \ | | \%%%%%%%%%\ / / |
| |
| − | | / /```````\ \ | | \%%%%%%%%%o / |
| |
| − | | o o`````````o o | | \%%%%%%%/ \ / |
| |
| − | | | |````@````| | | | o-----o o-----o |
| |
| − | | | |`````\```| | | | |
| |
| − | | | |``````\``| | | o-----------------------------o
| |
| − | | | u |```````\`| v | |
| |
| − | | | |````````\| | | o-----------------------------o
| |
| − | | | |`````````| | | | |
| |
| − | | | |`````````|\ | | | o-----o o-----o |
| |
| − | | o o`````````o \ o | | / \ /%%%%%%%\ |
| |
| − | | \ \```````/ \ / | | / o%%%%%%%%%\ |
| |
| − | | \ \`````/ \ / | | / / \%%%%%%%%%\ |
| |
| − | | \ \```/ \ / | | o o o%%%%%%%%%o |
| |
| − | | \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| |
| − | | \ o \ / | | o o o%%%%%%%%%o |
| |
| − | | \ / \ / | | \ \ /%%%%%%%%%/ |
| |
| − | | o---------o o---------o \ | | \ o%%%%%%%%%/ |
| |
| − | | \ | | \ / \%%%%%%%/ |
| |
| − | | \ | | o-----o o-----o |
| |
| − | | \ | | |
| |
| − | o----------------------------------------\----o o-----------------------------o
| |
| − | \
| |
| − | \ o-----------------------------o
| |
| − | \ |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
| |
| − | \ |%%%%%%o-----o%%%o-----o%%%%%%|
| |
| − | \ |%%%%%/ \%/ \%%%%%|
| |
| − | \ |%%%%/ o \%%%%|
| |
| − | \ |%%%/ / \ \%%%|
| |
| − | \|%%o o o o%%|
| |
| − | @%%| du | | dv |%%|
| |
| − | |%%o o o o%%|
| |
| − | |%%%\ \ / /%%%|
| |
| − | |%%%%\ o /%%%%|
| |
| − | |%%%%%\ /%\ /%%%%%|
| |
| − | |%%%%%%o-----o%%%o-----o%%%%%%|
| |
| − | |%%%%%%%%%%%%%%%%%%%%%%%%%%%%%|
| |
| − | o-----------------------------o
| |
| − | Figure 40-b. Enlargement of J (Bundle)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 40-c. Enlargement of J (Compact)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | o-------------------o o-------------------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | o o (du).(dv) o o |
| |
| − | | | | -->-- | | |
| |
| − | | | | \ / | | |
| |
| − | | | dv .(du) | \ / | du .(dv) | |
| |
| − | | | u o---------------->@<----------------o v | |
| |
| − | | | | ^ | | |
| |
| − | | | | | | | |
| |
| − | | | | | | | |
| |
| − | | o o | o o |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \|/ / |
| |
| − | | \ | / |
| |
| − | | \ /|\ / |
| |
| − | | o-------------------o | o-------------------o |
| |
| − | | | |
| |
| − | | du . dv |
| |
| − | | | |
| |
| − | | o |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 40-c. Enlargement of J (Compact)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 40-d. Enlargement of J (Digraph)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | (du).(dv) |
| |
| − | | --->--- |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | u @ v |
| |
| − | | ^^^ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | (du) dv / | \ du (dv) |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | @ | @ |
| |
| − | | u (v) | (u) v |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | du . dv |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | @ |
| |
| − | | |
| |
| − | | (u).(v) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 40-d. Enlargement of J (Digraph)
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 41. Computation of DJ (Method 1)===
| |
| − |
| |
| − | <pre>
| |
| − | Table 41. Computation of DJ (Method 1)
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | DJ = EJ + !e!J |
| |
| − | | |
| |
| − | | = J<u + du, v + dv> + J<u, v> |
| |
| − | | |
| |
| − | | = (u, du)(v, dv) + u v |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | DJ = 0 |
| |
| − | | |
| |
| − | | + u v (du) dv + u (v)(du) dv |
| |
| − | | |
| |
| − | | + u v du (dv) + (u) v du (dv) |
| |
| − | | |
| |
| − | | + u v du dv + (u)(v) du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 42. Computation of DJ (Method 2)===
| |
| − |
| |
| − | <pre>
| |
| − | Table 42. Computation of DJ (Method 2)
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | DJ = !e!J + EJ |
| |
| − | | |
| |
| − | | = J<u, v> + J<u + du, v + dv> |
| |
| − | | |
| |
| − | | = u v + (u, du)(v, dv) |
| |
| − | | |
| |
| − | | = 0 + u dv + v du + du dv |
| |
| − | | |
| |
| − | | = 0 + u (du) dv + v du (dv) + ((u, v)) du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 43. Computation of DJ (Method 3)===
| |
| − |
| |
| − | <pre>
| |
| − | Table 43. Computation of DJ (Method 3)
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | DJ = !e!J + EJ |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | !e!J = u v (du)(dv) + u v (du) dv + u v du (dv) + u v du dv |
| |
| − | | |
| |
| − | | EJ = u v (du)(dv) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | DJ = 0 . (du)(dv) + u . (du) dv + v . du (dv) + ((u, v)) du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 8===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | !e!J = {Dispositions from J to J } + {Dispositions from J to (J)} |
| |
| − | | |
| |
| − | | EJ = {Dispositions from J to J } + {Dispositions from (J) to J } |
| |
| − | | |
| |
| − | | DJ = (!e!J, EJ) |
| |
| − | | |
| |
| − | | DJ = {Dispositions from J to (J)} + {Dispositions from (J) to J } |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 44-a. Difference Map of J (Areal)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | o |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | o o |
| |
| − | | /%\ /%\ |
| |
| − | | /%%%\ /%%%\ |
| |
| − | | /%%%%%\ /%%%%%\ |
| |
| − | | o%%%%%%%o%%%%%%%o |
| |
| − | | /%\%%%%%/%\%%%%%/%\ |
| |
| − | | /%%%\%%%/%%%\%%%/%%%\ |
| |
| − | | /%%%%%\%/%%%%%\%/%%%%%\ |
| |
| − | | o%%%%%%%o%%%%%%%o%%%%%%%o |
| |
| − | | / \%%%%%/ \%%%%%/ \%%%%%/ \ |
| |
| − | | / \%%%/ \%%%/ \%%%/ \ |
| |
| − | | / \%/ \%/ \%/ \ |
| |
| − | | o o o o o |
| |
| − | | |\ / \ /%\ / \ /| |
| |
| − | | | \ / \ /%%%\ / \ / | |
| |
| − | | | \ / \ /%%%%%\ / \ / | |
| |
| − | | | o o%%%%%%%o o | |
| |
| − | | | |\ / \%%%%%/ \ /| | |
| |
| − | | | | \ / \%%%/ \ / | | |
| |
| − | | | u | \ / \%/ \ / | v | |
| |
| − | | o---+---o o o---+---o |
| |
| − | | | \ / \ / | |
| |
| − | | | \ / \ / | |
| |
| − | | | du \ / \ / dv | |
| |
| − | | o-------o o-------o |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | o |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Figure 44-a. Difference Map of J (Areal)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 44-b. Difference Map of J (Bundle)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------o
| |
| − | | |
| |
| − | | o-----o o-----o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / /%\ \ |
| |
| − | | o o%%%o o |
| |
| − | @ | du |%%%| dv | |
| |
| − | /| o o%%%o o |
| |
| − | / | \ \%/ / |
| |
| − | / | \ o / |
| |
| − | / | \ / \ / |
| |
| − | / | o-----o o-----o |
| |
| − | / | |
| |
| − | / o-----------------------------o
| |
| − | /
| |
| − | o----------------------------------------/----o o-----------------------------o
| |
| − | | / | | |
| |
| − | | @ | | o-----o o-----o |
| |
| − | | | | /%%%%%%%\ / \ |
| |
| − | | o---------o o---------o | | /%%%%%%%%%o \ |
| |
| − | | / \ / \ | | /%%%%%%%%%/ \ \ |
| |
| − | | / o \ | | o%%%%%%%%%o o o |
| |
| − | | / /`\ @------\-----------@ |%% du %%%| | dv | |
| |
| − | | / /```\ \ | | o%%%%%%%%%o o o |
| |
| − | | / /`````\ \ | | \%%%%%%%%%\ / / |
| |
| − | | / /```````\ \ | | \%%%%%%%%%o / |
| |
| − | | o o`````````o o | | \%%%%%%%/ \ / |
| |
| − | | | |````@````| | | | o-----o o-----o |
| |
| − | | | |`````\```| | | | |
| |
| − | | | |``````\``| | | o-----------------------------o
| |
| − | | | u |```````\`| v | |
| |
| − | | | |````````\| | | o-----------------------------o
| |
| − | | | |`````````| | | | |
| |
| − | | | |`````````|\ | | | o-----o o-----o |
| |
| − | | o o`````````o \ o | | / \ /%%%%%%%\ |
| |
| − | | \ \```````/ \ / | | / o%%%%%%%%%\ |
| |
| − | | \ \`````/ \ / | | / / \%%%%%%%%%\ |
| |
| − | | \ \```/ \ / | | o o o%%%%%%%%%o |
| |
| − | | \ @------\-/---------\---------------@ | du | |%%% dv %%| |
| |
| − | | \ o \ / | | o o o%%%%%%%%%o |
| |
| − | | \ / \ / | | \ \ /%%%%%%%%%/ |
| |
| − | | o---------o o---------o \ | | \ o%%%%%%%%%/ |
| |
| − | | \ | | \ / \%%%%%%%/ |
| |
| − | | \ | | o-----o o-----o |
| |
| − | | \ | | |
| |
| − | o----------------------------------------\----o o-----------------------------o
| |
| − | \
| |
| − | \ o-----------------------------o
| |
| − | \ | |
| |
| − | \ | o-----o o-----o |
| |
| − | \ | /%%%%%%%\ /%%%%%%%\ |
| |
| − | \ | /%%%%%%%%%o%%%%%%%%%\ |
| |
| − | \ | /%%%%%%%%%/%\%%%%%%%%%\ |
| |
| − | \| o%%%%%%%%%o%%%o%%%%%%%%%o |
| |
| − | @ |%% du %%%|%%%|%%% dv %%| |
| |
| − | | o%%%%%%%%%o%%%o%%%%%%%%%o |
| |
| − | | \%%%%%%%%%\%/%%%%%%%%%/ |
| |
| − | | \%%%%%%%%%o%%%%%%%%%/ |
| |
| − | | \%%%%%%%/ \%%%%%%%/ |
| |
| − | | o-----o o-----o |
| |
| − | | |
| |
| − | o-----------------------------o
| |
| − | Figure 44-b. Difference Map of J (Bundle)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 44-c. Difference Map of J (Compact)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | o-------------------o o-------------------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | o o o o |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | | dv .(du) | | du .(dv) | |
| |
| − | | | u @<--------------->@<--------------->@ v | |
| |
| − | | | | ^ | | |
| |
| − | | | | | | | |
| |
| − | | | | | | | |
| |
| − | | o o | o o |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \|/ / |
| |
| − | | \ | / |
| |
| − | | \ /|\ / |
| |
| − | | o-------------------o | o-------------------o |
| |
| − | | | |
| |
| − | | du . dv |
| |
| − | | | |
| |
| − | | v |
| |
| − | | @ |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 44-c. Difference Map of J (Compact)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 44-d. Difference Map of J (Digraph)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | u v |
| |
| − | | |
| |
| − | | @ |
| |
| − | | ^^^ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | (du) dv / | \ du (dv) |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | / | \ |
| |
| − | | v | v |
| |
| − | | @ | @ |
| |
| − | | u (v) | (u) v |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | du | dv |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | v |
| |
| − | | @ |
| |
| − | | |
| |
| − | | (u) (v) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 44-d. Difference Map of J (Digraph)
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 45. Computation of dJ===
| |
| − |
| |
| − | <pre>
| |
| − | Table 45. Computation of dJ
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| − | | |
| |
| − | | => |
| |
| − | | |
| |
| − | | dj = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 46-a. Differential of J (Areal)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | o |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | o o |
| |
| − | | /%\ /%\ |
| |
| − | | /%%%\ /%%%\ |
| |
| − | | /%%%%%\ /%%%%%\ |
| |
| − | | o%%%%%%%o%%%%%%%o |
| |
| − | | /%\%%%%%/ \%%%%%/%\ |
| |
| − | | /%%%\%%%/ \%%%/%%%\ |
| |
| − | | /%%%%%\%/ \%/%%%%%\ |
| |
| − | | o%%%%%%%o o%%%%%%%o |
| |
| − | | / \%%%%%/%\ /%\%%%%%/ \ |
| |
| − | | / \%%%/%%%\ /%%%\%%%/ \ |
| |
| − | | / \%/%%%%%\ /%%%%%\%/ \ |
| |
| − | | o o%%%%%%%o%%%%%%%o o |
| |
| − | | |\ / \%%%%%/ \%%%%%/ \ /| |
| |
| − | | | \ / \%%%/ \%%%/ \ / | |
| |
| − | | | \ / \%/ \%/ \ / | |
| |
| − | | | o o o o | |
| |
| − | | | |\ / \ / \ /| | |
| |
| − | | | | \ / \ / \ / | | |
| |
| − | | | u | \ / \ / \ / | v | |
| |
| − | | o---+---o o o---+---o |
| |
| − | | | \ / \ / | |
| |
| − | | | \ / \ / | |
| |
| − | | | du \ / \ / dv | |
| |
| − | | o-------o o-------o |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | o |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Figure 46-a. Differential of J (Areal)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 46-b. Differential of J (Bundle)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------o
| |
| − | | |
| |
| − | | o-----o o-----o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | o o o o |
| |
| − | @ | du | | dv | |
| |
| − | /| o o o o |
| |
| − | / | \ \ / / |
| |
| − | / | \ o / |
| |
| − | / | \ / \ / |
| |
| − | / | o-----o o-----o |
| |
| − | / | |
| |
| − | / o-----------------------------o
| |
| − | /
| |
| − | o----------------------------------------/----o o-----------------------------o
| |
| − | | / | | |
| |
| − | | @ | | o-----o o-----o |
| |
| − | | | | /%%%%%%%\ / \ |
| |
| − | | o---------o o---------o | | /%%%%%%%%%o \ |
| |
| − | | / \ / \ | | /%%%%%%%%%/%\ \ |
| |
| − | | / o \ | | o%%%%%%%%%o%%%o o |
| |
| − | | / /`\ @------\-----------@ |%% du %%%|%%%| dv | |
| |
| − | | / /```\ \ | | o%%%%%%%%%o%%%o o |
| |
| − | | / /`````\ \ | | \%%%%%%%%%\%/ / |
| |
| − | | / /```````\ \ | | \%%%%%%%%%o / |
| |
| − | | o o`````````o o | | \%%%%%%%/ \ / |
| |
| − | | | |````@````| | | | o-----o o-----o |
| |
| − | | | |`````\```| | | | |
| |
| − | | | |``````\``| | | o-----------------------------o
| |
| − | | | u |```````\`| v | |
| |
| − | | | |````````\| | | o-----------------------------o
| |
| − | | | |`````````| | | | |
| |
| − | | | |`````````|\ | | | o-----o o-----o |
| |
| − | | o o`````````o \ o | | / \ /%%%%%%%\ |
| |
| − | | \ \```````/ \ / | | / o%%%%%%%%%\ |
| |
| − | | \ \`````/ \ / | | / /%\%%%%%%%%%\ |
| |
| − | | \ \```/ \ / | | o o%%%o%%%%%%%%%o |
| |
| − | | \ @------\-/---------\---------------@ | du |%%%|%%% dv %%| |
| |
| − | | \ o \ / | | o o%%%o%%%%%%%%%o |
| |
| − | | \ / \ / | | \ \%/%%%%%%%%%/ |
| |
| − | | o---------o o---------o \ | | \ o%%%%%%%%%/ |
| |
| − | | \ | | \ / \%%%%%%%/ |
| |
| − | | \ | | o-----o o-----o |
| |
| − | | \ | | |
| |
| − | o----------------------------------------\----o o-----------------------------o
| |
| − | \
| |
| − | \ o-----------------------------o
| |
| − | \ | |
| |
| − | \ | o-----o o-----o |
| |
| − | \ | /%%%%%%%\ /%%%%%%%\ |
| |
| − | \ | /%%%%%%%%%o%%%%%%%%%\ |
| |
| − | \ | /%%%%%%%%%/ \%%%%%%%%%\ |
| |
| − | \| o%%%%%%%%%o o%%%%%%%%%o |
| |
| − | @ |%% du %%%| |%%% dv %%| |
| |
| − | | o%%%%%%%%%o o%%%%%%%%%o |
| |
| − | | \%%%%%%%%%\ /%%%%%%%%%/ |
| |
| − | | \%%%%%%%%%o%%%%%%%%%/ |
| |
| − | | \%%%%%%%/ \%%%%%%%/ |
| |
| − | | o-----o o-----o |
| |
| − | | |
| |
| − | o-----------------------------o
| |
| − | Figure 46-b. Differential of J (Bundle)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 46-c. Differential of J (Compact)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | o-------------------o o-------------------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / @ \ \ |
| |
| − | | / / ^ ^ \ \ |
| |
| − | | o o / \ o o |
| |
| − | | | | / \ | | |
| |
| − | | | | / \ | | |
| |
| − | | | |/ \| | |
| |
| − | | | u (du)/ dv du \(dv) v | |
| |
| − | | | /| |\ | |
| |
| − | | | / | | \ | |
| |
| − | | | / | | \ | |
| |
| − | | o / o o \ o |
| |
| − | | \ / \ / \ / |
| |
| − | | \ v \ du dv / v / |
| |
| − | | \ @<----------------------->@ / |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o-------------------o o-------------------o |
| |
| − | | |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 46-c. Differential of J (Compact)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 46-d. Differential of J (Digraph)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | u v |
| |
| − | | @ |
| |
| − | | ^ ^ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | (du) dv / \ du (dv) |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | v v |
| |
| − | | u (v) @<--------------------->@ (u) v |
| |
| − | | du dv |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | @ |
| |
| − | | (u) (v) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 46-d. Differential of J (Digraph)
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 47. Computation of rJ===
| |
| − |
| |
| − | <pre>
| |
| − | Table 47. Computation of rJ
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | rJ = DJ + dJ |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | DJ = u v ((du)(dv)) + u (v)(du) dv + (u) v du (dv) + (u)(v) du dv |
| |
| − | | |
| |
| − | | dJ = u v (du, dv) + u (v) dv + (u) v du + (u)(v) . 0 |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | rJ = u v du dv + u (v) du dv + (u) v du dv + (u)(v) du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 48-a. Remainder of J (Areal)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------o
| |
| − | | |
| |
| − | | o |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | / \ |
| |
| − | | o o |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | / \ / \ |
| |
| − | | o o o |
| |
| − | | / \ /%\ / \ |
| |
| − | | / \ /%%%\ / \ |
| |
| − | | / \ /%%%%%\ / \ |
| |
| − | | o o%%%%%%%o o |
| |
| − | | / \ /%\%%%%%/%\ / \ |
| |
| − | | / \ /%%%\%%%/%%%\ / \ |
| |
| − | | / \ /%%%%%\%/%%%%%\ / \ |
| |
| − | | o o%%%%%%%o%%%%%%%o o |
| |
| − | | |\ / \%%%%%/%\%%%%%/ \ /| |
| |
| − | | | \ / \%%%/%%%\%%%/ \ / | |
| |
| − | | | \ / \%/%%%%%\%/ \ / | |
| |
| − | | | o o%%%%%%%o o | |
| |
| − | | | |\ / \%%%%%/ \ /| | |
| |
| − | | | | \ / \%%%/ \ / | | |
| |
| − | | | u | \ / \%/ \ / | v | |
| |
| − | | o---+---o o o---+---o |
| |
| − | | | \ / \ / | |
| |
| − | | | \ / \ / | |
| |
| − | | | du \ / \ / dv | |
| |
| − | | o-------o o-------o |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | \ / |
| |
| − | | o |
| |
| − | | |
| |
| − | o---------------------------------------o
| |
| − | Figure 48-a. Remainder of J (Areal)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 48-b. Remainder of J (Bundle)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------o
| |
| − | | |
| |
| − | | o-----o o-----o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / /%\ \ |
| |
| − | | o o%%%o o |
| |
| − | @ | du |%%%| dv | |
| |
| − | /| o o%%%o o |
| |
| − | / | \ \%/ / |
| |
| − | / | \ o / |
| |
| − | / | \ / \ / |
| |
| − | / | o-----o o-----o |
| |
| − | / | |
| |
| − | / o-----------------------------o
| |
| − | /
| |
| − | o----------------------------------------/----o o-----------------------------o
| |
| − | | / | | |
| |
| − | | @ | | o-----o o-----o |
| |
| − | | | | / \ / \ |
| |
| − | | o---------o o---------o | | / o \ |
| |
| − | | / \ / \ | | / /%\ \ |
| |
| − | | / o \ | | o o%%%o o |
| |
| − | | / /`\ @------\-----------@ | du |%%%| dv | |
| |
| − | | / /```\ \ | | o o%%%o o |
| |
| − | | / /`````\ \ | | \ \%/ / |
| |
| − | | / /```````\ \ | | \ o / |
| |
| − | | o o`````````o o | | \ / \ / |
| |
| − | | | |````@````| | | | o-----o o-----o |
| |
| − | | | |`````\```| | | | |
| |
| − | | | |``````\``| | | o-----------------------------o
| |
| − | | | u |```````\`| v | |
| |
| − | | | |````````\| | | o-----------------------------o
| |
| − | | | |`````````| | | | |
| |
| − | | | |`````````|\ | | | o-----o o-----o |
| |
| − | | o o`````````o \ o | | / \ / \ |
| |
| − | | \ \```````/ \ / | | / o \ |
| |
| − | | \ \`````/ \ / | | / /%\ \ |
| |
| − | | \ \```/ \ / | | o o%%%o o |
| |
| − | | \ @------\-/---------\---------------@ | du |%%%| dv | |
| |
| − | | \ o \ / | | o o%%%o o |
| |
| − | | \ / \ / | | \ \%/ / |
| |
| − | | o---------o o---------o \ | | \ o / |
| |
| − | | \ | | \ / \ / |
| |
| − | | \ | | o-----o o-----o |
| |
| − | | \ | | |
| |
| − | o----------------------------------------\----o o-----------------------------o
| |
| − | \
| |
| − | \ o-----------------------------o
| |
| − | \ | |
| |
| − | \ | o-----o o-----o |
| |
| − | \ | / \ / \ |
| |
| − | \ | / o \ |
| |
| − | \ | / /%\ \ |
| |
| − | \| o o%%%o o |
| |
| − | @ | du |%%%| dv | |
| |
| − | | o o%%%o o |
| |
| − | | \ \%/ / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o-----o o-----o |
| |
| − | | |
| |
| − | o-----------------------------o
| |
| − | Figure 48-b. Remainder of J (Bundle)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 48-c. Remainder of J (Compact)===
| |
| − |
| |
| − | <pre>
| |
| − | o---------------------------------------------------------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | o-------------------o o-------------------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | o o o o |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | | | du dv | | |
| |
| − | | | u @<------------------------->@ v | |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | o o @ o o |
| |
| − | | \ \ ^ / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \ | / / |
| |
| − | | \ \|/ / |
| |
| − | | \ du | dv / |
| |
| − | | \ /|\ / |
| |
| − | | o-------------------o | o-------------------o |
| |
| − | | | |
| |
| − | | | |
| |
| − | | v |
| |
| − | | @ |
| |
| − | | |
| |
| − | o---------------------------------------------------------------------o
| |
| − | Figure 48-c. Remainder of J (Compact)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 48-d. Remainder of J (Digraph)===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | u v |
| |
| − | | @ |
| |
| − | | ^ |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | du | dv |
| |
| − | | u (v) @<----------|---------->@ (u) v |
| |
| − | | du | dv |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | | |
| |
| − | | v |
| |
| − | | @ |
| |
| − | | (u) (v) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | Figure 48-d. Remainder of J (Digraph)
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 49. Computation Summary for J===
| |
| − |
| |
| − | <pre>
| |
| − | Table 49. Computation Summary for J
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | !e!J = uv . 1 + u(v) . 0 + (u)v . 0 + (u)(v) . 0 |
| |
| − | | |
| |
| − | | EJ = uv . (du)(dv) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv |
| |
| − | | |
| |
| − | | DJ = uv . ((du)(dv)) + u(v) . (du)dv + (u)v . du(dv) + (u)(v) . du dv |
| |
| − | | |
| |
| − | | dJ = uv . (du, dv) + u(v) . dv + (u)v . du + (u)(v) . 0 |
| |
| − | | |
| |
| − | | rJ = uv . du dv + u(v) . du dv + (u)v . du dv + (u)(v) . du dv |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 50. Computation of an Analytic Series in Terms of Coordinates===
| |
| − |
| |
| − | <pre>
| |
| − | Table 50. Computation of an Analytic Series in Terms of Coordinates
| |
| − | o-----------o-------------o-------------oo-------------o---------o-------------o
| |
| − | | u v | du dv | u' v' || !e!J EJ | DJ | dJ d^2.J |
| |
| − | o-----------o-------------o-------------oo-------------o---------o-------------o
| |
| − | | | | || | | |
| |
| − | | 0 0 | 0 0 | 0 0 || 0 0 | 0 | 0 0 |
| |
| − | | | | || | | |
| |
| − | | | 0 1 | 0 1 || 0 | 0 | 0 0 |
| |
| − | | | | || | | |
| |
| − | | | 1 0 | 1 0 || 0 | 0 | 0 0 |
| |
| − | | | | || | | |
| |
| − | | | 1 1 | 1 1 || 1 | 1 | 0 1 |
| |
| − | | | | || | | |
| |
| − | o-----------o-------------o-------------oo-------------o---------o-------------o
| |
| − | | | | || | | |
| |
| − | | 0 1 | 0 0 | 0 1 || 0 0 | 0 | 0 0 |
| |
| − | | | | || | | |
| |
| − | | | 0 1 | 0 0 || 0 | 0 | 0 0 |
| |
| − | | | | || | | |
| |
| − | | | 1 0 | 1 1 || 1 | 1 | 1 0 |
| |
| − | | | | || | | |
| |
| − | | | 1 1 | 1 0 || 0 | 0 | 1 1 |
| |
| − | | | | || | | |
| |
| − | o-----------o-------------o-------------oo-------------o---------o-------------o
| |
| − | | | | || | | |
| |
| − | | 1 0 | 0 0 | 1 0 || 0 0 | 0 | 0 0 |
| |
| − | | | | || | | |
| |
| − | | | 0 1 | 1 1 || 1 | 1 | 1 0 |
| |
| − | | | | || | | |
| |
| − | | | 1 0 | 0 0 || 0 | 0 | 0 0 |
| |
| − | | | | || | | |
| |
| − | | | 1 1 | 0 1 || 0 | 0 | 1 1 |
| |
| − | | | | || | | |
| |
| − | o-----------o-------------o-------------oo-------------o---------o-------------o
| |
| − | | | | || | | |
| |
| − | | 1 1 | 0 0 | 1 1 || 1 1 | 0 | 0 0 |
| |
| − | | | | || | | |
| |
| − | | | 0 1 | 1 0 || 0 | 1 | 1 0 |
| |
| − | | | | || | | |
| |
| − | | | 1 0 | 0 1 || 0 | 1 | 1 0 |
| |
| − | | | | || | | |
| |
| − | | | 1 1 | 0 0 || 0 | 1 | 0 1 |
| |
| − | | | | || | | |
| |
| − | o-----------o-------------o-------------oo-------------o---------o-------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 9===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | u' = u + du = (u, du) |
| |
| − | | |
| |
| − | | v' = v + du = (v, dv) |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 10===
| |
| − |
| |
| − | <pre>
| |
| − | o--------------------------------------------------------------o
| |
| − | | |
| |
| − | | EJ<u, v, du, dv> = J<u + du, v + dv> = J<u', v'> |
| |
| − | | |
| |
| − | o--------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 51. Computation of an Analytic Series in Symbolic Terms===
| |
| − |
| |
| − | <pre>
| |
| − | Table 51. Computation of an Analytic Series in Symbolic Terms
| |
| − | o-----------o---------o------------o------------o------------o-----------o
| |
| − | | u v | J | EJ | DJ | dJ | d^2.J |
| |
| − | o-----------o---------o------------o------------o------------o-----------o
| |
| − | | | | | | | |
| |
| − | | 0 0 | 0 | du dv | du dv | () | du dv |
| |
| − | | | | | | | |
| |
| − | | 0 1 | 0 | du (dv) | du (dv) | du | du dv |
| |
| − | | | | | | | |
| |
| − | | 1 0 | 0 | (du) dv | (du) dv | dv | du dv |
| |
| − | | | | | | | |
| |
| − | | 1 1 | 1 | (du)(dv) | ((du)(dv)) | (du, dv) | du dv |
| |
| − | | | | | | | |
| |
| − | o-----------o---------o------------o------------o------------o-----------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)===
| |
| − |
| |
| − | <pre>
| |
| − | o o o
| |
| − | /%\ /%\ / \
| |
| − | /%%%\ /%%%\ / \
| |
| − | o%%%%%o o%%%%%o o o
| |
| − | / \%%%/ \ /%\%%%/%\ /%\ /%\
| |
| − | / \%/ \ /%%%\%/%%%\ /%%%\ /%%%\
| |
| − | o o o o%%%%%o%%%%%o o%%%%%o%%%%%o
| |
| − | /%\ / \ /%\ / \%%%/%\%%%/ \ /%\%%%/%\%%%/%\
| |
| − | /%%%\ / \ /%%%\ / \%/%%%\%/ \ /%%%\%/%%%\%/%%%\
| |
| − | o%%%%%o o%%%%%o o o%%%%%o o o%%%%%o%%%%%o%%%%%o
| |
| − | / \%%%/ \ / \%%%/ \ / \ / \%%%/ \ / \ / \%%%/ \%%%/ \%%%/ \
| |
| − | / \%/ \ / \%/ \ / \ / \%/ \ / \ / \%/ \%/ \%/ \
| |
| − | o o o o o o o o o o o o o o o
| |
| − | |\ / \ /%\ / \ /| |\ / \ / \ / \ /| |\ / \ /%\ / \ /|
| |
| − | | \ / \ /%%%\ / \ / | | \ / \ / \ / \ / | | \ / \ /%%%\ / \ / |
| |
| − | | o o%%%%%o o | | o o o o | | o o%%%%%o o |
| |
| − | | |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| |
| |
| − | |u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v|
| |
| − | o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o
| |
| − | | \ / \ / | | \ / \ / | | \ / \ / |
| |
| − | | du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv |
| |
| − | o-----o o-----o o-----o o-----o o-----o o-----o
| |
| − | \ / \ / \ /
| |
| − | \ / \ / \ /
| |
| − | o o o
| |
| − |
| |
| − | EJ = J + DJ
| |
| − |
| |
| − | o-----------------------o o-----------------------o o-----------------------o
| |
| − | | | | | | |
| |
| − | | o--o o--o | | o--o o--o | | o--o o--o |
| |
| − | | / \ / \ | | / \ / \ | | / \ / \ |
| |
| − | | / o \ | | / o \ | | / o \ |
| |
| − | | / u / \ v \ | | / u / \ v \ | | / u / \ v \ |
| |
| − | | o /->-\ o | | o /->-\ o | | o / \ o |
| |
| − | | | o \ / o | | | | o \ / o | | | | o o | |
| |
| − | | | @--|->@<-|--@ | | | | @<-|--@--|->@ | | | | @<-|->@<-|->@ | |
| |
| − | | | o ^ o | | | | o | o | | | | o ^ o | |
| |
| − | | o \ | / o | | o \ | / o | | o \ | / o |
| |
| − | | \ \|/ / | | \ \|/ / | | \ \|/ / |
| |
| − | | \ | / | | \ | / | | \ | / |
| |
| − | | \ /|\ / | | \ /|\ / | | \ /|\ / |
| |
| − | | o--o | o--o | | o--o v o--o | | o--o v o--o |
| |
| − | | @ | | @ | | @ |
| |
| − | o-----------------------o o-----------------------o o-----------------------o
| |
| − | Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)===
| |
| − |
| |
| − | <pre>
| |
| − | o o o
| |
| − | / \ / \ / \
| |
| − | / \ / \ / \
| |
| − | o o o o o o
| |
| − | /%\ /%\ /%\ /%\ / \ / \
| |
| − | /%%%\ /%%%\ /%%%\%/%%%\ / \ / \
| |
| − | o%%%%%o%%%%%o o%%%%%o%%%%%o o o o
| |
| − | /%\%%%/%\%%%/%\ /%\%%%/ \%%%/%\ / \ /%\ / \
| |
| − | /%%%\%/%%%\%/%%%\ /%%%\%/ \%/%%%\ / \ /%%%\ / \
| |
| − | o%%%%%o%%%%%o%%%%%o o%%%%%o o%%%%%o o o%%%%%o o
| |
| − | / \%%%/ \%%%/ \%%%/ \ / \%%%/%\ /%\%%%/ \ / \ /%\%%%/%\ / \
| |
| − | / \%/ \%/ \%/ \ / \%/%%%\ /%%%\%/ \ / \ /%%%\%/%%%\ / \
| |
| − | o o o o o o o%%%%%o%%%%%o o o o%%%%%o%%%%%o o
| |
| − | |\ / \ /%\ / \ /| |\ / \%%%/ \%%%/ \ /| |\ / \%%%/%\%%%/ \ /|
| |
| − | | \ / \ /%%%\ / \ / | | \ / \%/ \%/ \ / | | \ / \%/%%%\%/ \ / |
| |
| − | | o o%%%%%o o | | o o o o | | o o%%%%%o o |
| |
| − | | |\ / \%%%/ \ /| | | |\ / \ / \ /| | | |\ / \%%%/ \ /| |
| |
| − | |u | \ / \%/ \ / | v| |u | \ / \ / \ / | v| |u | \ / \%/ \ / | v|
| |
| − | o--+--o o o--+--o o--+--o o o--+--o o--+--o o o--+--o
| |
| − | | \ / \ / | | \ / \ / | | \ / \ / |
| |
| − | | du \ / \ / dv | | du \ / \ / dv | | du \ / \ / dv |
| |
| − | o-----o o-----o o-----o o-----o o-----o o-----o
| |
| − | \ / \ / \ /
| |
| − | \ / \ / \ /
| |
| − | o o o
| |
| − |
| |
| − | DJ = dJ + ddJ
| |
| − |
| |
| − | o-----------------------o o-----------------------o o-----------------------o
| |
| − | | | | | | |
| |
| − | | o--o o--o | | o--o o--o | | o--o o--o |
| |
| − | | / \ / \ | | / \ / \ | | / \ / \ |
| |
| − | | / o \ | | / o \ | | / o \ |
| |
| − | | / u / \ v \ | | / u / \ v \ | | / u / \ v \ |
| |
| − | | o / \ o | | o / \ o | | o / \ o |
| |
| − | | | o o | | | | o o | | | | o o | |
| |
| − | | | @<-|->@<-|->@ | | | | @<-|->@<-|->@ | | | | @<-|-----|->@ | |
| |
| − | | | o ^ o | | | | ^ o o ^ | | | | o @ o | |
| |
| − | | o \ | / o | | o \ \ / / o | | o \ ^ / o |
| |
| − | | \ \|/ / | | \ --\-/-- / | | \ \|/ / |
| |
| − | | \ | / | | \ o / | | \ | / |
| |
| − | | \ /|\ / | | \ / \ / | | \ /|\ / |
| |
| − | | o--o v o--o | | o--o o--o | | o--o v o--o |
| |
| − | | @ | | @ | | @ |
| |
| − | o-----------------------o o-----------------------o o-----------------------o
| |
| − | Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ)
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators===
| |
| − |
| |
| − | <pre>
| |
| − | Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | Item | Notation | Description | Type |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | U% | = [u, v] | Source Universe | [B^2] |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | X% | = [x] | Target Universe | [B^1] |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | EU% | = [u, v, du, dv] | Extended | [B^2 x D^2] |
| |
| − | | | | Source Universe | |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | EX% | = [x, dx] | Extended | [B^1 x D^1] |
| |
| − | | | | Target Universe | |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | J | J : U -> B | Proposition | (B^2 -> B) c [B^2] |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | J | J : U% -> X% | Transformation, | [B^2] -> [B^1] |
| |
| − | | | | or Mapping | |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | W | W : | Operator | |
| |
| − | | | U% -> EU%, | | [B^2] -> [B^2 x D^2], |
| |
| − | | | X% -> EX%, | | [B^1] -> [B^1 x D^1], |
| |
| − | | | (U%->X%)->(EU%->EX%), | | ([B^2] -> [B^1]) |
| |
| − | | | for each W among: | | -> |
| |
| − | | | e!, !h!, E, D, d | | ([B^2 x D^2]->[B^1 x D^1]) |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | |
| |
| − | | !e! | | Tacit Extension Operator !e! |
| |
| − | | !h! | | Trope Extension Operator !h! |
| |
| − | | E | | Enlargement Operator E |
| |
| − | | D | | Difference Operator D |
| |
| − | | d | | Differential Operator d |
| |
| − | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | $W$ | $W$ : | Operator | |
| |
| − | | | U% -> $T$U% = EU%, | | [B^2] -> [B^2 x D^2], |
| |
| − | | | X% -> $T$X% = EX%, | | [B^1] -> [B^1 x D^1], |
| |
| − | | | (U%->X%)->($T$U%->$T$X%)| | ([B^2] -> [B^1]) |
| |
| − | | | for each $W$ among: | | -> |
| |
| − | | | $e$, $E$, $D$, $T$ | | ([B^2 x D^2]->[B^1 x D^1]) |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | |
| |
| − | | $e$ | | Radius Operator $e$ = <!e!, !h!> |
| |
| − | | $E$ | | Secant Operator $E$ = <!e!, E > |
| |
| − | | $D$ | | Chord Operator $D$ = <!e!, D > |
| |
| − | | $T$ | | Tangent Functor $T$ = <!e!, d > |
| |
| − | | | | |
| |
| − | o------o-------------------------o-----------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes===
| |
| − |
| |
| − | <pre>
| |
| − | Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | Operator | Proposition | Map |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Tacit | !e! : | !e!J : | !e!J : |
| |
| − | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x] |
| |
| − | | | (U%->X%)->(EU%->X%) | B^2 x D^2 -> B | [B^2 x D^2]->[B^1] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Trope | !h! : | !h!J : | !h!J : |
| |
| − | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Enlargement | E : | EJ : | EJ : |
| |
| − | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Difference | D : | DJ : | DJ : |
| |
| − | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Differential | d : | dJ : | dJ : |
| |
| − | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Remainder | r : | rJ : | rJ : |
| |
| − | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D | [B^2 x D^2]->[D^1] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Radius | $e$ = <!e!, !h!> : | | $e$J : |
| |
| − | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| |
| − | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Secant | $E$ = <!e!, E> : | | $E$J : |
| |
| − | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| |
| − | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Chord | $D$ = <!e!, D> : | | $D$J : |
| |
| − | | Operator | U%->EU%, X%->EX%, | | [u,v,du,dv]->[x, dx] |
| |
| − | | | (U%->X%)->(EU%->EX%) | | [B^2 x D^2]->[B x D] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Tangent | $T$ = <!e!, d> : | dJ : | $T$J : |
| |
| − | | Functor | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] |
| |
| − | | | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D | [B^2 x D^2]->[B x D] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 56-a1. Radius Map of the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o
| |
| − | /X\
| |
| − | /XXX\
| |
| − | oXXXXXo
| |
| − | /X\XXX/X\
| |
| − | /XXX\X/XXX\
| |
| − | oXXXXXoXXXXXo
| |
| − | / \XXX/X\XXX/ \
| |
| − | / \X/XXX\X/ \
| |
| − | o oXXXXXo o
| |
| − | / \ / \XXX/ \ / \
| |
| − | / \ / \X/ \ / \
| |
| − | o o o o o
| |
| − | =|\ / \ / \ / \ /|=
| |
| − | = | \ / \ / \ / \ / | =
| |
| − | = | o o o o | =
| |
| − | = | |\ / \ / \ /| | =
| |
| − | = |u | \ / \ / \ / | v| =
| |
| − | o o--+--o o o--+--o o
| |
| − | //\ | \ / \ / | /\\
| |
| − | ////\ | du \ / \ / dv | /\\\\
| |
| − | o/////o o-----o o-----o o\\\\\o
| |
| − | //\/////\ \ / /\\\\\/\\
| |
| − | ////\/////\ \ / /\\\\\/\\\\
| |
| − | o/////o/////o o o\\\\\o\\\\\o
| |
| − | / \/////\//// \ = = / \\\\/\\\\\/ \
| |
| − | / \/////\// \ = = / \\/\\\\\/ \
| |
| − | o o/////o o = = o o\\\\\o o
| |
| − | / \ / \//// \ / \ = = / \ / \\\\/ \ / \
| |
| − | / \ / \// \ / \ = = / \ / \\/ \ / \
| |
| − | o o o o o o o o o o
| |
| − | |\ / \ / \ / \ /| |\ / \ / \ / \ /|
| |
| − | | \ / \ / \ / \ / | | \ / \ / \ / \ / |
| |
| − | | o o o o | | o o o o |
| |
| − | | |\ / \ / \ /| | | |\ / \ / \ /| |
| |
| − | |u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
| |
| − | o--+--o o o--+--o o o--+--o o o--+--o
| |
| − | . | \ / \ / | /X\ | \ / \ / | .
| |
| − | .| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
| |
| − | o-----o o-----o /XXXXX\ o-----o o-----o
| |
| − | . \ / /XXXXXXX\ \ / .
| |
| − | . \ / /XXXXXXXXX\ \ / .
| |
| − | . o oXXXXXXXXXXXo o .
| |
| − | . //\XXXXXXXXX/\\ .
| |
| − | . ////\XXXXXXX/\\\\ .
| |
| − | !e!J //////\XXXXX/\\\\\\ !h!J
| |
| − | . ////////\XXX/\\\\\\\\ .
| |
| − | . //////////\X/\\\\\\\\\\ .
| |
| − | . o///////////o\\\\\\\\\\\o .
| |
| − | . |\////////// \\\\\\\\\\/| .
| |
| − | . | \//////// \\\\\\\\/ | .
| |
| − | . | \////// \\\\\\/ | .
| |
| − | . | \//// \\\\/ | .
| |
| − | .| x \// \\/ dx |.
| |
| − | o-----o o-----o
| |
| − | \ /
| |
| − | \ /
| |
| − | x = uv \ / dx = uv
| |
| − | \ /
| |
| − | \ /
| |
| − | o
| |
| − |
| |
| − | Figure 56-a1. Radius Map of the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 56-a2. Secant Map of the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o
| |
| − | /X\
| |
| − | /XXX\
| |
| − | oXXXXXo
| |
| − | //\XXX//\
| |
| − | ////\X////\
| |
| − | o/////o/////o
| |
| − | /\\/////\////\\
| |
| − | /\\\\/////\//\\\\
| |
| − | o\\\\\o/////o\\\\\o
| |
| − | / \\\\/ \//// \\\\/ \
| |
| − | / \\/ \// \\/ \
| |
| − | o o o o o
| |
| − | =|\ / \ /\\ / \ /|=
| |
| − | = | \ / \ /\\\\ / \ / | =
| |
| − | = | o o\\\\\o o | =
| |
| − | = | |\ / \\\\/ \ /| | =
| |
| − | = |u | \ / \\/ \ / | v| =
| |
| − | o o--+--o o o--+--o o
| |
| − | //\ | \ / \ / | /\\
| |
| − | ////\ | du \ / \ / dv | /\\\\
| |
| − | o/////o o-----o o-----o o\\\\\o
| |
| − | //\/////\ \ / / \\\\/ \
| |
| − | ////\/////\ \ / / \\/ \
| |
| − | o/////o/////o o o o o
| |
| − | / \/////\//// \ = = /\\ / \ /\\
| |
| − | / \/////\// \ = = /\\\\ / \ /\\\\
| |
| − | o o/////o o = = o\\\\\o o\\\\\o
| |
| − | / \ / \//// \ / \ = = / \\\\/ \ / \\\\/ \
| |
| − | / \ / \// \ / \ = = / \\/ \ / \\/ \
| |
| − | o o o o o o o o o o
| |
| − | |\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| |
| − | | \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| |
| − | | o o o o | | o o\\\\\o o |
| |
| − | | |\ / \ / \ /| | | |\ / \\\\/ \ /| |
| |
| − | |u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
| |
| − | o--+--o o o--+--o o o--+--o o o--+--o
| |
| − | . | \ / \ / | /X\ | \ / \ / | .
| |
| − | .| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
| |
| − | o-----o o-----o /XXXXX\ o-----o o-----o
| |
| − | . \ / /XXXXXXX\ \ / .
| |
| − | . \ / /XXXXXXXXX\ \ / .
| |
| − | . o oXXXXXXXXXXXo o .
| |
| − | . //\XXXXXXXXX/\\ .
| |
| − | . ////\XXXXXXX/\\\\ .
| |
| − | !e!J //////\XXXXX/\\\\\\ EJ
| |
| − | . ////////\XXX/\\\\\\\\ .
| |
| − | . //////////\X/\\\\\\\\\\ .
| |
| − | . o///////////o\\\\\\\\\\\o .
| |
| − | . |\////////// \\\\\\\\\\/| .
| |
| − | . | \//////// \\\\\\\\/ | .
| |
| − | . | \////// \\\\\\/ | .
| |
| − | . | \//// \\\\/ | .
| |
| − | .| x \// \\/ dx |.
| |
| − | o-----o o-----o
| |
| − | \ /
| |
| − | \ / dx = (u, du)(v, dv)
| |
| − | x = uv \ /
| |
| − | \ / dx = uv + u dv + v du + du dv
| |
| − | \ /
| |
| − | o
| |
| − |
| |
| − | Figure 56-a2. Secant Map of the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 56-a3. Chord Map of the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o
| |
| − | //\
| |
| − | ////\
| |
| − | o/////o
| |
| − | /X\////X\
| |
| − | /XXX\//XXX\
| |
| − | oXXXXXoXXXXXo
| |
| − | /\\XXX/X\XXX/\\
| |
| − | /\\\\X/XXX\X/\\\\
| |
| − | o\\\\\oXXXXXo\\\\\o
| |
| − | / \\\\/ \XXX/ \\\\/ \
| |
| − | / \\/ \X/ \\/ \
| |
| − | o o o o o
| |
| − | =|\ / \ /\\ / \ /|=
| |
| − | = | \ / \ /\\\\ / \ / | =
| |
| − | = | o o\\\\\o o | =
| |
| − | = | |\ / \\\\/ \ /| | =
| |
| − | = |u | \ / \\/ \ / | v| =
| |
| − | o o--+--o o o--+--o o
| |
| − | //\ | \ / \ / | / \
| |
| − | ////\ | du \ / \ / dv | / \
| |
| − | o/////o o-----o o-----o o o
| |
| − | //\/////\ \ / /\\ /\\
| |
| − | ////\/////\ \ / /\\\\ /\\\\
| |
| − | o/////o/////o o o\\\\\o\\\\\o
| |
| − | / \/////\//// \ = = /\\\\\/\\\\\/\\
| |
| − | / \/////\// \ = = /\\\\\/\\\\\/\\\\
| |
| − | o o/////o o = = o\\\\\o\\\\\o\\\\\o
| |
| − | / \ / \//// \ / \ = = / \\\\/ \\\\/ \\\\/ \
| |
| − | / \ / \// \ / \ = = / \\/ \\/ \\/ \
| |
| − | o o o o o o o o o o
| |
| − | |\ / \ / \ / \ /| |\ / \ /\\ / \ /|
| |
| − | | \ / \ / \ / \ / | | \ / \ /\\\\ / \ / |
| |
| − | | o o o o | | o o\\\\\o o |
| |
| − | | |\ / \ / \ /| | | |\ / \\\\/ \ /| |
| |
| − | |u | \ / \ / \ / | v| |u | \ / \\/ \ / | v|
| |
| − | o--+--o o o--+--o o o--+--o o o--+--o
| |
| − | . | \ / \ / | /X\ | \ / \ / | .
| |
| − | .| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
| |
| − | o-----o o-----o /XXXXX\ o-----o o-----o
| |
| − | . \ / /XXXXXXX\ \ / .
| |
| − | . \ / /XXXXXXXXX\ \ / .
| |
| − | . o oXXXXXXXXXXXo o .
| |
| − | . //\XXXXXXXXX/\\ .
| |
| − | . ////\XXXXXXX/\\\\ .
| |
| − | !e!J //////\XXXXX/\\\\\\ DJ
| |
| − | . ////////\XXX/\\\\\\\\ .
| |
| − | . //////////\X/\\\\\\\\\\ .
| |
| − | . o///////////o\\\\\\\\\\\o .
| |
| − | . |\////////// \\\\\\\\\\/| .
| |
| − | . | \//////// \\\\\\\\/ | .
| |
| − | . | \////// \\\\\\/ | .
| |
| − | . | \//// \\\\/ | .
| |
| − | .| x \// \\/ dx |.
| |
| − | o-----o o-----o
| |
| − | \ /
| |
| − | \ / dx = (u, du)(v, dv) - uv
| |
| − | x = uv \ /
| |
| − | \ / dx = u dv + v du + du dv
| |
| − | \ /
| |
| − | o
| |
| − |
| |
| − | Figure 56-a3. Chord Map of the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 56-a4. Tangent Map of the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o
| |
| − | //\
| |
| − | ////\
| |
| − | o/////o
| |
| − | /X\////X\
| |
| − | /XXX\//XXX\
| |
| − | oXXXXXoXXXXXo
| |
| − | /\\XXX//\XXX/\\
| |
| − | /\\\\X////\X/\\\\
| |
| − | o\\\\\o/////o\\\\\o
| |
| − | / \\\\/\\////\\\\\/ \
| |
| − | / \\/\\\\//\\\\\/ \
| |
| − | o o\\\\\o\\\\\o o
| |
| − | =|\ / \\\\/ \\\\/ \ /|=
| |
| − | = | \ / \\/ \\/ \ / | =
| |
| − | = | o o o o | =
| |
| − | = | |\ / \ / \ /| | =
| |
| − | = |u | \ / \ / \ / | v| =
| |
| − | o o--+--o o o--+--o o
| |
| − | //\ | \ / \ / | / \
| |
| − | ////\ | du \ / \ / dv | / \
| |
| − | o/////o o-----o o-----o o o
| |
| − | //\/////\ \ / /\\ /\\
| |
| − | ////\/////\ \ / /\\\\ /\\\\
| |
| − | o/////o/////o o o\\\\\o\\\\\o
| |
| − | / \/////\//// \ = = /\\\\\/ \\\\/\\
| |
| − | / \/////\// \ = = /\\\\\/ \\/\\\\
| |
| − | o o/////o o = = o\\\\\o o\\\\\o
| |
| − | / \ / \//// \ / \ = = / \\\\/\\ /\\\\\/ \
| |
| − | / \ / \// \ / \ = = / \\/\\\\ /\\\\\/ \
| |
| − | o o o o o o o\\\\\o\\\\\o o
| |
| − | |\ / \ / \ / \ /| |\ / \\\\/ \\\\/ \ /|
| |
| − | | \ / \ / \ / \ / | | \ / \\/ \\/ \ / |
| |
| − | | o o o o | | o o o o |
| |
| − | | |\ / \ / \ /| | | |\ / \ / \ /| |
| |
| − | |u | \ / \ / \ / | v| |u | \ / \ / \ / | v|
| |
| − | o--+--o o o--+--o o o--+--o o o--+--o
| |
| − | . | \ / \ / | /X\ | \ / \ / | .
| |
| − | .| du \ / \ / dv | /XXX\ | du \ / \ / dv |.
| |
| − | o-----o o-----o /XXXXX\ o-----o o-----o
| |
| − | . \ / /XXXXXXX\ \ / .
| |
| − | . \ / /XXXXXXXXX\ \ / .
| |
| − | . o oXXXXXXXXXXXo o .
| |
| − | . //\XXXXXXXXX/\\ .
| |
| − | . ////\XXXXXXX/\\\\ .
| |
| − | !e!J //////\XXXXX/\\\\\\ dJ
| |
| − | . ////////\XXX/\\\\\\\\ .
| |
| − | . //////////\X/\\\\\\\\\\ .
| |
| − | . o///////////o\\\\\\\\\\\o .
| |
| − | . |\////////// \\\\\\\\\\/| .
| |
| − | . | \//////// \\\\\\\\/ | .
| |
| − | . | \////// \\\\\\/ | .
| |
| − | . | \//// \\\\/ | .
| |
| − | .| x \// \\/ dx |.
| |
| − | o-----o o-----o
| |
| − | \ /
| |
| − | \ /
| |
| − | x = uv \ / dx = u dv + v du
| |
| − | \ /
| |
| − | \ /
| |
| − | o
| |
| − |
| |
| − | Figure 56-a4. Tangent Map of the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 56-b1. Radius Map of the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | o--o o--o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / du / \ dv \ |
| |
| − | | o / \ o |
| |
| − | | | o o | |
| |
| − | | | | | | |
| |
| − | | | o o | |
| |
| − | | o \ / o |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o--o o--o |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------@
| |
| − | \
| |
| − | o-----------------------o \
| |
| − | | | \
| |
| − | | | \
| |
| − | | | \
| |
| − | | o--o o--o | \
| |
| − | | / \ / \ | \
| |
| − | | / o \ | \
| |
| − | | / du / \ dv \ | \
| |
| − | | o / \ o | \
| |
| − | | | o o | @ \
| |
| − | | | | | | |\ \
| |
| − | | | o o | | \ \
| |
| − | | o \ / o | \ \
| |
| − | | \ \ / / | \ \
| |
| − | | \ o / | \ \
| |
| − | | \ / \ / | \ \
| |
| − | | o--o o--o | \ \
| |
| − | | | \ \
| |
| − | | | \ \
| |
| − | | | \ \
| |
| − | o-----------------------o \ \
| |
| − | \ \
| |
| − | o-----------------------@ o--------\----------\---o o-----------------------o
| |
| − | | |\ | \ \ | |```````````````````````|
| |
| − | | | \ | \ @ | |```````````````````````|
| |
| − | | | \| \ | |```````````````````````|
| |
| − | | o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| |
| − | | / \ / \ | |\ / \ /\ \ | |`````/````\`/````\`````|
| |
| − | | / o \ | | \ / o @ \ | |````/``````o``````\````|
| |
| − | | / du / \ dv \ | | \/ du /`\ dv \ | |```/``du``/`\``dv``\```|
| |
| − | | o / \ o | | o\ /```\ o | |``o``````/```\``````o``|
| |
| − | | | o o | | | | \ o`````o | | |``|`````o`````o`````|``|
| |
| − | | | | | | | | | @ |``@--|-----|------@``|`````|`````|`````|``|
| |
| − | | | o o | | | | o`````o | | |``|`````o`````o`````|``|
| |
| − | | o \ / o | | o \```/ o | |``o``````\```/``````o``|
| |
| − | | \ \ / / | | \ \`/ / | |```\``````\`/``````/```|
| |
| − | | \ o / | | \ o / | |````\``````o``````/````|
| |
| − | | \ / \ / | | \ / \ / | |`````\````/`\````/`````|
| |
| − | | o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| |
| − | | | | | |```````````````````````|
| |
| − | | | | | |```````````````````````|
| |
| − | | | | | |```````````````````````|
| |
| − | o-----------------------o o-----------------------o o-----------------------o
| |
| − | \ / \ / \ /
| |
| − | \ !h!J / \ J / \ !h!J /
| |
| − | \ / \ / \ /
| |
| − | \ / o----------\---------/----------o \ /
| |
| − | \ / | \ / | \ /
| |
| − | \ / | \ / | \ /
| |
| − | \ / | o-----o-----o | \ /
| |
| − | \ / | /`````````````\ | \ /
| |
| − | \ / | /```````````````\ | \ /
| |
| − | o------\---/------o | /`````````````````\ | o------\---/------o
| |
| − | | \ / | | /```````````````````\ | | \ / |
| |
| − | | o--o--o | | /`````````````````````\ | | o--o--o |
| |
| − | | /```````\ | | o```````````````````````o | | /```````\ |
| |
| − | | /`````````\ | | |```````````````````````| | | /`````````\ |
| |
| − | | o```````````o | | |```````````````````````| | | o```````````o |
| |
| − | | |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| |
| − | | o```````````o | | |```````````````````````| | | o```````````o |
| |
| − | | \`````````/ | | |```````````````````````| | | \`````````/ |
| |
| − | | \```````/ | | o```````````````````````o | | \```````/ |
| |
| − | | o-----o | | \`````````````````````/ | | o-----o |
| |
| − | | | | \```````````````````/ | | |
| |
| − | o-----------------o | \`````````````````/ | o-----------------o
| |
| − | | \```````````````/ |
| |
| − | | \`````````````/ |
| |
| − | | o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------o
| |
| − |
| |
| − | Figure 56-b1. Radius Map of the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 56-b2. Secant Map of the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | o--o o--o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / du /`\ dv \ |
| |
| − | | o /```\ o |
| |
| − | | | o`````o | |
| |
| − | | | |`````| | |
| |
| − | | | o`````o | |
| |
| − | | o \```/ o |
| |
| − | | \ \`/ / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o--o o--o |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------@
| |
| − | \
| |
| − | o-----------------------o \
| |
| − | | | \
| |
| − | | | \
| |
| − | | | \
| |
| − | | o--o o--o | \
| |
| − | | /````\ / \ | \
| |
| − | | /``````o \ | \
| |
| − | | /``du``/ \ dv \ | \
| |
| − | | o``````/ \ o | \
| |
| − | | |`````o o | @ \
| |
| − | | |`````| | | |\ \
| |
| − | | |`````o o | | \ \
| |
| − | | o``````\ / o | \ \
| |
| − | | \``````\ / / | \ \
| |
| − | | \``````o / | \ \
| |
| − | | \````/ \ / | \ \
| |
| − | | o--o o--o | \ \
| |
| − | | | \ \
| |
| − | | | \ \
| |
| − | | | \ \
| |
| − | o-----------------------o \ \
| |
| − | \ \
| |
| − | o-----------------------@ o--------\----------\---o o-----------------------o
| |
| − | | |\ | \ \ | |```````````````````````|
| |
| − | | | \ | \ @ | |```````````````````````|
| |
| − | | | \| \ | |```````````````````````|
| |
| − | | o--o o--o | \ o--o \o--o | |``````o--o```o--o``````|
| |
| − | | / \ /````\ | |\ / \ /\ \ | |`````/ \`/ \`````|
| |
| − | | / o``````\ | | \ / o @ \ | |````/ o \````|
| |
| − | | / du / \``dv``\ | | \/ du /`\ dv \ | |```/ du / \ dv \```|
| |
| − | | o / \``````o | | o\ /```\ o | |``o / \ o``|
| |
| − | | | o o`````| | | | \ o`````o | | |``| o o |``|
| |
| − | | | | |`````| | | | @ |``@--|-----|------@``| | | |``|
| |
| − | | | o o`````| | | | o`````o | | |``| o o |``|
| |
| − | | o \ /``````o | | o \```/ o | |``o \ / o``|
| |
| − | | \ \ /``````/ | | \ \`/ / | |```\ \ / /```|
| |
| − | | \ o``````/ | | \ o / | |````\ o /````|
| |
| − | | \ / \````/ | | \ / \ / | |`````\ /`\ /`````|
| |
| − | | o--o o--o | | o--o o--o | |``````o--o```o--o``````|
| |
| − | | | | | |```````````````````````|
| |
| − | | | | | |```````````````````````|
| |
| − | | | | | |```````````````````````|
| |
| − | o-----------------------o o-----------------------o o-----------------------o
| |
| − | \ / \ / \ /
| |
| − | \ EJ / \ J / \ EJ /
| |
| − | \ / \ / \ /
| |
| − | \ / o----------\---------/----------o \ /
| |
| − | \ / | \ / | \ /
| |
| − | \ / | \ / | \ /
| |
| − | \ / | o-----o-----o | \ /
| |
| − | \ / | /`````````````\ | \ /
| |
| − | \ / | /```````````````\ | \ /
| |
| − | o------\---/------o | /`````````````````\ | o------\---/------o
| |
| − | | \ / | | /```````````````````\ | | \ / |
| |
| − | | o--o--o | | /`````````````````````\ | | o--o--o |
| |
| − | | /```````\ | | o```````````````````````o | | /```````\ |
| |
| − | | /`````````\ | | |```````````````````````| | | /`````````\ |
| |
| − | | o```````````o | | |```````````````````````| | | o```````````o |
| |
| − | | |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| |
| − | | o```````````o | | |```````````````````````| | | o```````````o |
| |
| − | | \`````````/ | | |```````````````````````| | | \`````````/ |
| |
| − | | \```````/ | | o```````````````````````o | | \```````/ |
| |
| − | | o-----o | | \`````````````````````/ | | o-----o |
| |
| − | | | | \```````````````````/ | | |
| |
| − | o-----------------o | \`````````````````/ | o-----------------o
| |
| − | | \```````````````/ |
| |
| − | | \`````````````/ |
| |
| − | | o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------o
| |
| − |
| |
| − | Figure 56-b2. Secant Map of the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 56-b3. Chord Map of the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | o--o o--o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / du /`\ dv \ |
| |
| − | | o /```\ o |
| |
| − | | | o`````o | |
| |
| − | | | |`````| | |
| |
| − | | | o`````o | |
| |
| − | | o \```/ o |
| |
| − | | \ \`/ / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o--o o--o |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------@
| |
| − | \
| |
| − | o-----------------------o \
| |
| − | | | \
| |
| − | | | \
| |
| − | | | \
| |
| − | | o--o o--o | \
| |
| − | | /````\ / \ | \
| |
| − | | /``````o \ | \
| |
| − | | /``du``/ \ dv \ | \
| |
| − | | o``````/ \ o | \
| |
| − | | |`````o o | @ \
| |
| − | | |`````| | | |\ \
| |
| − | | |`````o o | | \ \
| |
| − | | o``````\ / o | \ \
| |
| − | | \``````\ / / | \ \
| |
| − | | \``````o / | \ \
| |
| − | | \````/ \ / | \ \
| |
| − | | o--o o--o | \ \
| |
| − | | | \ \
| |
| − | | | \ \
| |
| − | | | \ \
| |
| − | o-----------------------o \ \
| |
| − | \ \
| |
| − | o-----------------------@ o--------\----------\---o o-----------------------o
| |
| − | | |\ | \ \ | | |
| |
| − | | | \ | \ @ | | |
| |
| − | | | \| \ | | |
| |
| − | | o--o o--o | \ o--o \o--o | | o--o o--o |
| |
| − | | / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| |
| − | | / o``````\ | | \ / o @ \ | | /``````o``````\ |
| |
| − | | / du / \``dv``\ | | \/ du /`\ dv \ | | /``du``/`\``dv``\ |
| |
| − | | o / \``````o | | o\ /```\ o | | o``````/```\``````o |
| |
| − | | | o o`````| | | | \ o`````o | | | |`````o`````o`````| |
| |
| − | | | | |`````| | | | @ |``@--|-----|------@ |`````|`````|`````| |
| |
| − | | | o o`````| | | | o`````o | | | |`````o`````o`````| |
| |
| − | | o \ /``````o | | o \```/ o | | o``````\```/``````o |
| |
| − | | \ \ /``````/ | | \ \`/ / | | \``````\`/``````/ |
| |
| − | | \ o``````/ | | \ o / | | \``````o``````/ |
| |
| − | | \ / \````/ | | \ / \ / | | \````/ \````/ |
| |
| − | | o--o o--o | | o--o o--o | | o--o o--o |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | o-----------------------o o-----------------------o o-----------------------o
| |
| − | \ / \ / \ /
| |
| − | \ DJ / \ J / \ DJ /
| |
| − | \ / \ / \ /
| |
| − | \ / o----------\---------/----------o \ /
| |
| − | \ / | \ / | \ /
| |
| − | \ / | \ / | \ /
| |
| − | \ / | o-----o-----o | \ /
| |
| − | \ / | /`````````````\ | \ /
| |
| − | \ / | /```````````````\ | \ /
| |
| − | o------\---/------o | /`````````````````\ | o------\---/------o
| |
| − | | \ / | | /```````````````````\ | | \ / |
| |
| − | | o--o--o | | /`````````````````````\ | | o--o--o |
| |
| − | | /```````\ | | o```````````````````````o | | /```````\ |
| |
| − | | /`````````\ | | |```````````````````````| | | /`````````\ |
| |
| − | | o```````````o | | |```````````````````````| | | o```````````o |
| |
| − | | |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| |
| − | | o```````````o | | |```````````````````````| | | o```````````o |
| |
| − | | \`````````/ | | |```````````````````````| | | \`````````/ |
| |
| − | | \```````/ | | o```````````````````````o | | \```````/ |
| |
| − | | o-----o | | \`````````````````````/ | | o-----o |
| |
| − | | | | \```````````````````/ | | |
| |
| − | o-----------------o | \`````````````````/ | o-----------------o
| |
| − | | \```````````````/ |
| |
| − | | \`````````````/ |
| |
| − | | o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------o
| |
| − |
| |
| − | Figure 56-b3. Chord Map of the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 56-b4. Tangent Map of the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------o
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | o--o o--o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / du / \ dv \ |
| |
| − | | o / \ o |
| |
| − | | | o o | |
| |
| − | | | | | | |
| |
| − | | | o o | |
| |
| − | | o \ / o |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o--o o--o |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------@
| |
| − | \
| |
| − | o-----------------------o \
| |
| − | | | \
| |
| − | | | \
| |
| − | | | \
| |
| − | | o--o o--o | \
| |
| − | | /````\ / \ | \
| |
| − | | /``````o \ | \
| |
| − | | /``du``/`\ dv \ | \
| |
| − | | o``````/```\ o | \
| |
| − | | |`````o`````o | @ \
| |
| − | | |`````|`````| | |\ \
| |
| − | | |`````o`````o | | \ \
| |
| − | | o``````\```/ o | \ \
| |
| − | | \``````\`/ / | \ \
| |
| − | | \``````o / | \ \
| |
| − | | \````/ \ / | \ \
| |
| − | | o--o o--o | \ \
| |
| − | | | \ \
| |
| − | | | \ \
| |
| − | | | \ \
| |
| − | o-----------------------o \ \
| |
| − | \ \
| |
| − | o-----------------------@ o--------\----------\---o o-----------------------o
| |
| − | | |\ | \ \ | | |
| |
| − | | | \ | \ @ | | |
| |
| − | | | \| \ | | |
| |
| − | | o--o o--o | \ o--o \o--o | | o--o o--o |
| |
| − | | / \ /````\ | |\ / \ /\ \ | | /````\ /````\ |
| |
| − | | / o``````\ | | \ / o @ \ | | /``````o``````\ |
| |
| − | | / du /`\``dv``\ | | \/ du /`\ dv \ | | /``du``/ \``dv``\ |
| |
| − | | o /```\``````o | | o\ /```\ o | | o``````/ \``````o |
| |
| − | | | o`````o`````| | | | \ o`````o | | | |`````o o`````| |
| |
| − | | | |`````|`````| | | | @ |``@--|-----|------@ |`````| |`````| |
| |
| − | | | o`````o`````| | | | o`````o | | | |`````o o`````| |
| |
| − | | o \```/``````o | | o \```/ o | | o``````\ /``````o |
| |
| − | | \ \`/``````/ | | \ \`/ / | | \``````\ /``````/ |
| |
| − | | \ o``````/ | | \ o / | | \``````o``````/ |
| |
| − | | \ / \````/ | | \ / \ / | | \````/ \````/ |
| |
| − | | o--o o--o | | o--o o--o | | o--o o--o |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | o-----------------------o o-----------------------o o-----------------------o
| |
| − | \ / \ / \ /
| |
| − | \ dJ / \ J / \ dJ /
| |
| − | \ / \ / \ /
| |
| − | \ / o----------\---------/----------o \ /
| |
| − | \ / | \ / | \ /
| |
| − | \ / | \ / | \ /
| |
| − | \ / | o-----o-----o | \ /
| |
| − | \ / | /`````````````\ | \ /
| |
| − | \ / | /```````````````\ | \ /
| |
| − | o------\---/------o | /`````````````````\ | o------\---/------o
| |
| − | | \ / | | /```````````````````\ | | \ / |
| |
| − | | o--o--o | | /`````````````````````\ | | o--o--o |
| |
| − | | /```````\ | | o```````````````````````o | | /```````\ |
| |
| − | | /`````````\ | | |```````````````````````| | | /`````````\ |
| |
| − | | o```````````o | | |```````````````````````| | | o```````````o |
| |
| − | | |````dx`````| @----@ |```````````x`````@-----|------@ |``` dx ````| |
| |
| − | | o```````````o | | |```````````````````````| | | o```````````o |
| |
| − | | \`````````/ | | |```````````````````````| | | \`````````/ |
| |
| − | | \```````/ | | o```````````````````````o | | \```````/ |
| |
| − | | o-----o | | \`````````````````````/ | | o-----o |
| |
| − | | | | \```````````````````/ | | |
| |
| − | o-----------------o | \`````````````````/ | o-----------------o
| |
| − | | \```````````````/ |
| |
| − | | \`````````````/ |
| |
| − | | o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-------------------------------o
| |
| − |
| |
| − | Figure 56-b4. Tangent Map of the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 57-1. Radius Operator Diagram for the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o o
| |
| − | //\ /X\
| |
| − | ////\ /XXX\
| |
| − | //////\ oXXXXXo
| |
| − | ////////\ /X\XXX/X\
| |
| − | //////////\ /XXX\X/XXX\
| |
| − | o///////////o oXXXXXoXXXXXo
| |
| − | / \////////// \ / \XXX/X\XXX/ \
| |
| − | / \//////// \ / \X/XXX\X/ \
| |
| − | / \////// \ o oXXXXXo o
| |
| − | / \//// \ / \ / \XXX/ \ / \
| |
| − | / \// \ / \ / \X/ \ / \
| |
| − | o o o o o o o o
| |
| − | |\ / \ /| |\ / \ / \ / \ /|
| |
| − | | \ / \ / | | \ / \ / \ / \ / |
| |
| − | | \ / \ / | | o o o o |
| |
| − | | \ / \ / | | |\ / \ / \ /| |
| |
| − | | u \ / \ / v | |u | \ / \ / \ / | v|
| |
| − | o-----o o-----o o--+--o o o--+--o
| |
| − | \ / | \ / \ / |
| |
| − | \ / | du \ / \ / dv |
| |
| − | \ / o-----o o-----o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o o
| |
| − | U% $e$ $E$U%
| |
| − | o------------------>o
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | J | | $e$J
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | v v
| |
| − | o------------------>o
| |
| − | X% $e$ $E$X%
| |
| − | o o
| |
| − | //\ /X\
| |
| − | ////\ /XXX\
| |
| − | //////\ /XXXXX\
| |
| − | ////////\ /XXXXXXX\
| |
| − | //////////\ /XXXXXXXXX\
| |
| − | ////////////o oXXXXXXXXXXXo
| |
| − | ///////////// \ //\XXXXXXXXX/\\
| |
| − | ///////////// \ ////\XXXXXXX/\\\\
| |
| − | ///////////// \ //////\XXXXX/\\\\\\
| |
| − | ///////////// \ ////////\XXX/\\\\\\\\
| |
| − | ///////////// \ //////////\X/\\\\\\\\\\
| |
| − | o//////////// o o///////////o\\\\\\\\\\\o
| |
| − | |\////////// / |\////////// \\\\\\\\\\/|
| |
| − | | \//////// / | \//////// \\\\\\\\/ |
| |
| − | | \////// / | \////// \\\\\\/ |
| |
| − | | \//// / | \//// \\\\/ |
| |
| − | | x \// / | x \// \\/ dx |
| |
| − | o-----o / o-----o o-----o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o o
| |
| − |
| |
| − | Figure 57-1. Radius Operator Diagram for the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 57-2. Secant Operator Diagram for the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o o
| |
| − | //\ /X\
| |
| − | ////\ /XXX\
| |
| − | //////\ oXXXXXo
| |
| − | ////////\ //\XXX//\
| |
| − | //////////\ ////\X////\
| |
| − | o///////////o o/////o/////o
| |
| − | / \////////// \ /\\/////\////\\
| |
| − | / \//////// \ /\\\\/////\//\\\\
| |
| − | / \////// \ o\\\\\o/////o\\\\\o
| |
| − | / \//// \ / \\\\/ \//// \\\\/ \
| |
| − | / \// \ / \\/ \// \\/ \
| |
| − | o o o o o o o o
| |
| − | |\ / \ /| |\ / \ /\\ / \ /|
| |
| − | | \ / \ / | | \ / \ /\\\\ / \ / |
| |
| − | | \ / \ / | | o o\\\\\o o |
| |
| − | | \ / \ / | | |\ / \\\\/ \ /| |
| |
| − | | u \ / \ / v | |u | \ / \\/ \ / | v|
| |
| − | o-----o o-----o o--+--o o o--+--o
| |
| − | \ / | \ / \ / |
| |
| − | \ / | du \ / \ / dv |
| |
| − | \ / o-----o o-----o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o o
| |
| − | U% $E$ $E$U%
| |
| − | o------------------>o
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | J | | $E$J
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | v v
| |
| − | o------------------>o
| |
| − | X% $E$ $E$X%
| |
| − | o o
| |
| − | //\ /X\
| |
| − | ////\ /XXX\
| |
| − | //////\ /XXXXX\
| |
| − | ////////\ /XXXXXXX\
| |
| − | //////////\ /XXXXXXXXX\
| |
| − | ////////////o oXXXXXXXXXXXo
| |
| − | ///////////// \ //\XXXXXXXXX/\\
| |
| − | ///////////// \ ////\XXXXXXX/\\\\
| |
| − | ///////////// \ //////\XXXXX/\\\\\\
| |
| − | ///////////// \ ////////\XXX/\\\\\\\\
| |
| − | ///////////// \ //////////\X/\\\\\\\\\\
| |
| − | o//////////// o o///////////o\\\\\\\\\\\o
| |
| − | |\////////// / |\////////// \\\\\\\\\\/|
| |
| − | | \//////// / | \//////// \\\\\\\\/ |
| |
| − | | \////// / | \////// \\\\\\/ |
| |
| − | | \//// / | \//// \\\\/ |
| |
| − | | x \// / | x \// \\/ dx |
| |
| − | o-----o / o-----o o-----o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o o
| |
| − |
| |
| − | Figure 57-2. Secant Operator Diagram for the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 57-3. Chord Operator Diagram for the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o o
| |
| − | //\ //\
| |
| − | ////\ ////\
| |
| − | //////\ o/////o
| |
| − | ////////\ /X\////X\
| |
| − | //////////\ /XXX\//XXX\
| |
| − | o///////////o oXXXXXoXXXXXo
| |
| − | / \////////// \ /\\XXX/X\XXX/\\
| |
| − | / \//////// \ /\\\\X/XXX\X/\\\\
| |
| − | / \////// \ o\\\\\oXXXXXo\\\\\o
| |
| − | / \//// \ / \\\\/ \XXX/ \\\\/ \
| |
| − | / \// \ / \\/ \X/ \\/ \
| |
| − | o o o o o o o o
| |
| − | |\ / \ /| |\ / \ /\\ / \ /|
| |
| − | | \ / \ / | | \ / \ /\\\\ / \ / |
| |
| − | | \ / \ / | | o o\\\\\o o |
| |
| − | | \ / \ / | | |\ / \\\\/ \ /| |
| |
| − | | u \ / \ / v | |u | \ / \\/ \ / | v|
| |
| − | o-----o o-----o o--+--o o o--+--o
| |
| − | \ / | \ / \ / |
| |
| − | \ / | du \ / \ / dv |
| |
| − | \ / o-----o o-----o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o o
| |
| − | U% $D$ $E$U%
| |
| − | o------------------>o
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | J | | $D$J
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | v v
| |
| − | o------------------>o
| |
| − | X% $D$ $E$X%
| |
| − | o o
| |
| − | //\ /X\
| |
| − | ////\ /XXX\
| |
| − | //////\ /XXXXX\
| |
| − | ////////\ /XXXXXXX\
| |
| − | //////////\ /XXXXXXXXX\
| |
| − | ////////////o oXXXXXXXXXXXo
| |
| − | ///////////// \ //\XXXXXXXXX/\\
| |
| − | ///////////// \ ////\XXXXXXX/\\\\
| |
| − | ///////////// \ //////\XXXXX/\\\\\\
| |
| − | ///////////// \ ////////\XXX/\\\\\\\\
| |
| − | ///////////// \ //////////\X/\\\\\\\\\\
| |
| − | o//////////// o o///////////o\\\\\\\\\\\o
| |
| − | |\////////// / |\////////// \\\\\\\\\\/|
| |
| − | | \//////// / | \//////// \\\\\\\\/ |
| |
| − | | \////// / | \////// \\\\\\/ |
| |
| − | | \//// / | \//// \\\\/ |
| |
| − | | x \// / | x \// \\/ dx |
| |
| − | o-----o / o-----o o-----o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o o
| |
| − |
| |
| − | Figure 57-3. Chord Operator Diagram for the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv===
| |
| − |
| |
| − | <pre>
| |
| − | o o
| |
| − | //\ //\
| |
| − | ////\ ////\
| |
| − | //////\ o/////o
| |
| − | ////////\ /X\////X\
| |
| − | //////////\ /XXX\//XXX\
| |
| − | o///////////o oXXXXXoXXXXXo
| |
| − | / \////////// \ /\\XXX//\XXX/\\
| |
| − | / \//////// \ /\\\\X////\X/\\\\
| |
| − | / \////// \ o\\\\\o/////o\\\\\o
| |
| − | / \//// \ / \\\\/\\////\\\\\/ \
| |
| − | / \// \ / \\/\\\\//\\\\\/ \
| |
| − | o o o o o\\\\\o\\\\\o o
| |
| − | |\ / \ /| |\ / \\\\/ \\\\/ \ /|
| |
| − | | \ / \ / | | \ / \\/ \\/ \ / |
| |
| − | | \ / \ / | | o o o o |
| |
| − | | \ / \ / | | |\ / \ / \ /| |
| |
| − | | u \ / \ / v | |u | \ / \ / \ / | v|
| |
| − | o-----o o-----o o--+--o o o--+--o
| |
| − | \ / | \ / \ / |
| |
| − | \ / | du \ / \ / dv |
| |
| − | \ / o-----o o-----o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o o
| |
| − | U% $T$ $E$U%
| |
| − | o------------------>o
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | J | | $T$J
| |
| − | | |
| |
| − | | |
| |
| − | | |
| |
| − | v v
| |
| − | o------------------>o
| |
| − | X% $T$ $E$X%
| |
| − | o o
| |
| − | //\ /X\
| |
| − | ////\ /XXX\
| |
| − | //////\ /XXXXX\
| |
| − | ////////\ /XXXXXXX\
| |
| − | //////////\ /XXXXXXXXX\
| |
| − | ////////////o oXXXXXXXXXXXo
| |
| − | ///////////// \ //\XXXXXXXXX/\\
| |
| − | ///////////// \ ////\XXXXXXX/\\\\
| |
| − | ///////////// \ //////\XXXXX/\\\\\\
| |
| − | ///////////// \ ////////\XXX/\\\\\\\\
| |
| − | ///////////// \ //////////\X/\\\\\\\\\\
| |
| − | o//////////// o o///////////o\\\\\\\\\\\o
| |
| − | |\////////// / |\////////// \\\\\\\\\\/|
| |
| − | | \//////// / | \//////// \\\\\\\\/ |
| |
| − | | \////// / | \////// \\\\\\/ |
| |
| − | | \//// / | \//// \\\\/ |
| |
| − | | x \// / | x \// \\/ dx |
| |
| − | o-----o / o-----o o-----o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o o
| |
| − |
| |
| − | Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 11===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | F = <f, g> = <F_1, F_2> : [u, v] -> [x, y] |
| |
| − | | |
| |
| − | | where f = F_1 : [u, v] -> [x] |
| |
| − | | |
| |
| − | | and g = F_2 : [u, v] -> [y] |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators===
| |
| − |
| |
| − | <pre>
| |
| − | Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | Item | Notation | Description | Type |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | U% | = [u, v] | Source Universe | [B^n] |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | X% | = [x, y] | Target Universe | [B^k] |
| |
| − | | | = [f, g] | | |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | EU% | = [u, v, du, dv] | Extended | [B^n x D^n] |
| |
| − | | | | Source Universe | |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | EX% | = [x, y, dx, dy] | Extended | [B^k x D^k] |
| |
| − | | | = [f, g, df, dg] | Target Universe | |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | F | F = <f, g> : U% -> X% | Transformation, | [B^n] -> [B^k] |
| |
| − | | | | or Mapping | |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | | f, g : U -> B | Proposition, | B^n -> B |
| |
| − | | | | special case | |
| |
| − | | f | f : U -> [x] c X% | of a mapping, | c (B^n, B^n -> B) |
| |
| − | | | | or component | |
| |
| − | | g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | W | W : | Operator | |
| |
| − | | | U% -> EU%, | | [B^n] -> [B^n x D^n], |
| |
| − | | | X% -> EX%, | | [B^k] -> [B^k x D^k], |
| |
| − | | | (U%->X%)->(EU%->EX%), | | ([B^n] -> [B^k]) |
| |
| − | | | for each W among: | | -> |
| |
| − | | | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | |
| |
| − | | !e! | | Tacit Extension Operator !e! |
| |
| − | | !h! | | Trope Extension Operator !h! |
| |
| − | | E | | Enlargement Operator E |
| |
| − | | D | | Difference Operator D |
| |
| − | | d | | Differential Operator d |
| |
| − | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | | |
| |
| − | | $W$ | $W$ : | Operator | |
| |
| − | | | U% -> $T$U% = EU%, | | [B^n] -> [B^n x D^n], |
| |
| − | | | X% -> $T$X% = EX%, | | [B^k] -> [B^k x D^k], |
| |
| − | | | (U%->X%)->($T$U%->$T$X%)| | ([B^n] -> [B^k]) |
| |
| − | | | for each $W$ among: | | -> |
| |
| − | | | $e$, $E$, $D$, $T$ | | ([B^n x D^n]->[B^k x D^k]) |
| |
| − | | | | | |
| |
| − | o------o-------------------------o------------------o----------------------------o
| |
| − | | | | |
| |
| − | | $e$ | | Radius Operator $e$ = <!e!, !h!> |
| |
| − | | $E$ | | Secant Operator $E$ = <!e!, E > |
| |
| − | | $D$ | | Chord Operator $D$ = <!e!, D > |
| |
| − | | $T$ | | Tangent Functor $T$ = <!e!, d > |
| |
| − | | | | |
| |
| − | o------o-------------------------o-----------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
| |
| − |
| |
| − | <pre>
| |
| − | Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | Operator | Proposition | Transformation |
| |
| − | | | or | or | or |
| |
| − | | | Operand | Component | Mapping |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
| |
| − | | | | | |
| |
| − | | | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Tacit | !e! : | !e!F_i : | !e!F : |
| |
| − | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
| |
| − | | | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Trope | !h! : | !h!F_i : | !h!F : |
| |
| − | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Enlargement | E : | EF_i : | EF : |
| |
| − | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Difference | D : | DF_i : | DF : |
| |
| − | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Differential | d : | dF_i : | dF : |
| |
| − | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Remainder | r : | rF_i : | rF : |
| |
| − | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| |
| − | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Radius | $e$ = <!e!, !h!> : | | $e$F : |
| |
| − | | Operator | | | |
| |
| − | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| |
| − | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| |
| − | | | | | |
| |
| − | | | | | [B^n x D^n] -> |
| |
| − | | | | | [B^k x D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Secant | $E$ = <!e!, E> : | | $E$F : |
| |
| − | | Operator | | | |
| |
| − | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| |
| − | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| |
| − | | | | | |
| |
| − | | | | | [B^n x D^n] -> |
| |
| − | | | | | [B^k x D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Chord | $D$ = <!e!, D> : | | $D$F : |
| |
| − | | Operator | | | |
| |
| − | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| |
| − | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| |
| − | | | | | |
| |
| − | | | | | [B^n x D^n] -> |
| |
| − | | | | | [B^k x D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | | | | | |
| |
| − | | Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |
| |
| − | | Functor | | | |
| |
| − | | | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> |
| |
| − | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| |
| − | | | | | |
| |
| − | | | | B^n x D^n -> D | [B^n x D^n] -> |
| |
| − | | | | | [B^k x D^k] |
| |
| − | | | | | |
| |
| − | o--------------o----------------------o--------------------o----------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 12===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | x = f(u, v) = ((u)(v)) |
| |
| − | | |
| |
| − | | y = g(u, v) = ((u, v)) |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 13===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------------o
| |
| − | | |
| |
| − | | <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
| |
| − | | |
| |
| − | o-----------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 60. Propositional Transformation===
| |
| − |
| |
| − | <pre>
| |
| − | Table 60. Propositional Transformation
| |
| − | o-------------o-------------o-------------o-------------o
| |
| − | | u | v | f | g |
| |
| − | o-------------o-------------o-------------o-------------o
| |
| − | | | | | |
| |
| − | | 0 | 0 | 0 | 1 |
| |
| − | | | | | |
| |
| − | | 0 | 1 | 1 | 0 |
| |
| − | | | | | |
| |
| − | | 1 | 0 | 1 | 0 |
| |
| − | | | | | |
| |
| − | | 1 | 1 | 1 | 1 |
| |
| − | | | | | |
| |
| − | o-------------o-------------o-------------o-------------o
| |
| − | | | | ((u)(v)) | ((u, v)) |
| |
| − | o-------------o-------------o-------------o-------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 61. Propositional Transformation===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------o
| |
| − | | U |
| |
| − | | |
| |
| − | | o-----------o o-----------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | o o o o |
| |
| − | | | | | | |
| |
| − | | | u | | v | |
| |
| − | | | | | | |
| |
| − | | o o o o |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o-----------o o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------------------------------------o
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | / \ / \
| |
| − | o-------------------------o o-------------------------o
| |
| − | | U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| |
| − | | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| |
| − | | //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| |
| − | | ////////o///////\ | |\\\\/ o \\\\\|
| |
| − | | //////////\///////\ | |\\\/ /\\ \\\\|
| |
| − | | o///////o///o///////o | |\\o o\\\o o\\|
| |
| − | | |// u //|///|// v //| | |\\| u |\\\| v |\\|
| |
| − | | o///////o///o///////o | |\\o o\\\o o\\|
| |
| − | | \///////\////////// | |\\\\ \\/ /\\\|
| |
| − | | \///////o//////// | |\\\\\ o /\\\\|
| |
| − | | \////// \////// | |\\\\\\ /\\ /\\\\\|
| |
| − | | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| |
| − | | | |\\\\\\\\\\\\\\\\\\\\\\\\\|
| |
| − | o-------------------------o o-------------------------o
| |
| − | \ | | /
| |
| − | \ | | /
| |
| − | \ | | /
| |
| − | \ f | | g /
| |
| − | \ | | /
| |
| − | \ | | /
| |
| − | \ | | /
| |
| − | \ | | /
| |
| − | \ | | /
| |
| − | \ | | /
| |
| − | o-------\----|---------------------------|----/-------o
| |
| − | | X \ | | / |
| |
| − | | \| |/ |
| |
| − | | o-----------o o-----------o |
| |
| − | | //////////////\ /\\\\\\\\\\\\\\ |
| |
| − | | ////////////////o\\\\\\\\\\\\\\\\ |
| |
| − | | /////////////////X\\\\\\\\\\\\\\\\\ |
| |
| − | | /////////////////XXX\\\\\\\\\\\\\\\\\ |
| |
| − | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |
| − | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |
| − | | |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |
| − | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |
| − | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |
| − | | \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| |
| − | | \///////////////\X/\\\\\\\\\\\\\\\/ |
| |
| − | | \///////////////o\\\\\\\\\\\\\\\/ |
| |
| − | | \////////////// \\\\\\\\\\\\\\/ |
| |
| − | | o-----------o o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------------------------------------o
| |
| − | Figure 61. Propositional Transformation
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 62. Propositional Transformation (Short Form)===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------o o-------------------------o
| |
| − | | U | |\U \\\\\\\\\\\\\\\\\\\\\\|
| |
| − | | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| |
| − | | //////\ //////\ | |\\\\\/ \\/ \\\\\\|
| |
| − | | ////////o///////\ | |\\\\/ o \\\\\|
| |
| − | | //////////\///////\ | |\\\/ /\\ \\\\|
| |
| − | | o///////o///o///////o | |\\o o\\\o o\\|
| |
| − | | |// u //|///|// v //| | |\\| u |\\\| v |\\|
| |
| − | | o///////o///o///////o | |\\o o\\\o o\\|
| |
| − | | \///////\////////// | |\\\\ \\/ /\\\|
| |
| − | | \///////o//////// | |\\\\\ o /\\\\|
| |
| − | | \////// \////// | |\\\\\\ /\\ /\\\\\|
| |
| − | | o---o o---o | |\\\\\\o---o\\\o---o\\\\\\|
| |
| − | | | |\\\\\\\\\\\\\\\\\\\\\\\\\|
| |
| − | o-------------------------o o-------------------------o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ f / \ g /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o---------\-----/---------------------\-----/---------o
| |
| − | | X \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | o-----------o o-----------o |
| |
| − | | //////////////\ /\\\\\\\\\\\\\\ |
| |
| − | | ////////////////o\\\\\\\\\\\\\\\\ |
| |
| − | | /////////////////X\\\\\\\\\\\\\\\\\ |
| |
| − | | /////////////////XXX\\\\\\\\\\\\\\\\\ |
| |
| − | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |
| − | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |
| − | | |////// x //////|XXXXX|\\\\\\ y \\\\\\| |
| |
| − | | |///////////////|XXXXX|\\\\\\\\\\\\\\\| |
| |
| − | | o///////////////oXXXXXo\\\\\\\\\\\\\\\o |
| |
| − | | \///////////////\XXX/\\\\\\\\\\\\\\\/ |
| |
| − | | \///////////////\X/\\\\\\\\\\\\\\\/ |
| |
| − | | \///////////////o\\\\\\\\\\\\\\\/ |
| |
| − | | \////////////// \\\\\\\\\\\\\\/ |
| |
| − | | o-----------o o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------------------------------------o
| |
| − | Figure 62. Propositional Transformation (Short Form)
| |
| − | </pre>
| |
| − |
| |
| − | ===Figure 63. Transformation of Positions===
| |
| − |
| |
| − | <pre>
| |
| − | o-----------------------------------------------------o
| |
| − | |`U` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| |
| − | |` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` `|
| |
| − | |` ` ` ` ` ` o-----------o ` o-----------o ` ` ` ` ` `|
| |
| − | |` ` ` ` ` `/' ' ' ' ' ' '\`/' ' ' ' ' ' '\` ` ` ` ` `|
| |
| − | |` ` ` ` ` / ' ' ' ' ' ' ' o ' ' ' ' ' ' ' \ ` ` ` ` `|
| |
| − | |` ` ` ` `/' ' ' ' ' ' ' '/^\' ' ' ' ' ' ' '\` ` ` ` `|
| |
| − | |` ` ` ` / ' ' ' ' ' ' ' /^^^\ ' ' ' ' ' ' ' \ ` ` ` `|
| |
| − | |` ` ` `o' ' ' ' ' ' ' 'o^^^^^o' ' ' ' ' ' ' 'o` ` ` `|
| |
| − | |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
| |
| − | |` ` ` `|' ' ' ' u ' ' '|^^^^^|' ' ' v ' ' ' '|` ` ` `|
| |
| − | |` ` ` `|' ' ' ' ' ' ' '|^^^^^|' ' ' ' ' ' ' '|` ` ` `|
| |
| − | |` `@` `o' ' ' ' @ ' ' 'o^^@^^o' ' ' @ ' ' ' 'o` ` ` `|
| |
| − | |` ` \ ` \ ' ' ' | ' ' ' \^|^/ ' ' ' | ' ' ' / ` ` ` `|
| |
| − | |` ` `\` `\' ' ' | ' ' ' '\|/' ' ' ' | ' ' '/` ` ` ` `|
| |
| − | |` ` ` \ ` \ ' ' | ' ' ' ' | ' ' ' ' | ' ' / ` ` ` ` `|
| |
| − | |` ` ` `\` `\' ' | ' ' ' '/|\' ' ' ' | ' '/` ` ` ` ` `|
| |
| − | |` ` ` ` \ ` o---|-------o | o-------|---o ` ` ` ` ` `|
| |
| − | |` ` ` ` `\` ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
| |
| − | |` ` ` ` ` \ ` ` | ` ` ` ` | ` ` ` ` | ` ` ` ` ` ` ` `|
| |
| − | o-----------\----|---------|---------|----------------o
| |
| − | " " \ | | | " "
| |
| − | " " \ | | | " "
| |
| − | " " \ | | | " "
| |
| − | " " \| | | " "
| |
| − | o-------------------------o \ | | o-------------------------o
| |
| − | | U | |\ | | |`U```````````````````````|
| |
| − | | o---o o---o | | \ | | |``````o---o```o---o``````|
| |
| − | | /'''''\ /'''''\ | | \ | | |`````/ \`/ \`````|
| |
| − | | /'''''''o'''''''\ | | \ | | |````/ o \````|
| |
| − | | /'''''''/'\'''''''\ | | \ | | |```/ /`\ \```|
| |
| − | | o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| |
| − | | |'''u'''|'''|'''v'''| | | \ | | |``| u |```| v |``|
| |
| − | | o'''''''o'''o'''''''o | | \ | | |``o o```o o``|
| |
| − | | \'''''''\'/'''''''/ | | \| | |```\ \`/ /```|
| |
| − | | \'''''''o'''''''/ | | \ | |````\ o /````|
| |
| − | | \'''''/ \'''''/ | | |\ | |`````\ /`\ /`````|
| |
| − | | o---o o---o | | | \ | |``````o---o```o---o``````|
| |
| − | | | | | \ * |`````````````````````````|
| |
| − | o-------------------------o | | \ / o-------------------------o
| |
| − | \ | | | \ / | /
| |
| − | \ ((u)(v)) | | | \/ | ((u, v)) /
| |
| − | \ | | | /\ | /
| |
| − | \ | | | / \ | /
| |
| − | \ | | | / \ | /
| |
| − | \ | | | / * | /
| |
| − | \ | | | / | | /
| |
| − | \ | | |/ | | /
| |
| − | \ | | / | | /
| |
| − | \ | | /| | | /
| |
| − | o-------\----|---|-------/-|---------|---|----/-------o
| |
| − | | X \ | | / | | | / |
| |
| − | | \| | / | | |/ |
| |
| − | | o---|----/--o | o-------|---o |
| |
| − | | /' ' | ' / ' '\|/` ` ` ` | ` `\ |
| |
| − | | / ' ' | '/' ' ' | ` ` ` ` | ` ` \ |
| |
| − | | /' ' ' | / ' ' '/|\` ` ` ` | ` ` `\ |
| |
| − | | / ' ' ' |/' ' ' /^|^\ ` ` ` | ` ` ` \ |
| |
| − | | @ o' ' ' ' @ ' ' 'o^^@^^o` ` ` @ ` ` ` `o |
| |
| − | | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| |
| − | | |' ' ' ' f ' ' '|^^^^^|` ` ` g ` ` ` `| |
| |
| − | | |' ' ' ' ' ' ' '|^^^^^|` ` ` ` ` ` ` `| |
| |
| − | | o' ' ' ' ' ' ' 'o^^^^^o` ` ` ` ` ` ` `o |
| |
| − | | \ ' ' ' ' ' ' ' \^^^/ ` ` ` ` ` ` ` / |
| |
| − | | \' ' ' ' ' ' ' '\^/` ` ` ` ` ` ` `/ |
| |
| − | | \ ' ' ' ' ' ' ' o ` ` ` ` ` ` ` / |
| |
| − | | \' ' ' ' ' ' '/ \` ` ` ` ` ` `/ |
| |
| − | | o-----------o o-----------o |
| |
| − | | |
| |
| − | | |
| |
| − | o-----------------------------------------------------o
| |
| − | Figure 63. Transformation of Positions
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 64. Transformation of Positions===
| |
| − |
| |
| − | <pre>
| |
| − | Table 64. Transformation of Positions
| |
| − | o-----o----------o----------o-------o-------o--------o--------o-------------o
| |
| − | | u v | x | y | x y | x(y) | (x)y | (x)(y) | X% = [x, y] |
| |
| − | o-----o----------o----------o-------o-------o--------o--------o-------------o
| |
| − | | | | | | | | | ^ |
| |
| − | | 0 0 | 0 | 1 | 0 | 0 | 1 | 0 | | |
| |
| − | | | | | | | | | |
| |
| − | | 0 1 | 1 | 0 | 0 | 1 | 0 | 0 | F |
| |
| − | | | | | | | | | = |
| |
| − | | 1 0 | 1 | 0 | 0 | 1 | 0 | 0 | <f , g> |
| |
| − | | | | | | | | | |
| |
| − | | 1 1 | 1 | 1 | 1 | 0 | 0 | 0 | ^ |
| |
| − | | | | | | | | | | |
| |
| − | o-----o----------o----------o-------o-------o--------o--------o-------------o
| |
| − | | | ((u)(v)) | ((u, v)) | u v | (u,v) | (u)(v) | 0 | U% = [u, v] |
| |
| − | o-----o----------o----------o-------o-------o--------o--------o-------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 65. Induced Transformation on Propositions===
| |
| − |
| |
| − | <pre>
| |
| − | Table 65. Induced Transformation on Propositions
| |
| − | o------------o---------------------------------o------------o
| |
| − | | X% | <--- F = <f , g> <--- | U% |
| |
| − | o------------o----------o-----------o----------o------------o
| |
| − | | | u = | 1 1 0 0 | = u | |
| |
| − | | | v = | 1 0 1 0 | = v | |
| |
| − | | f_i <x, y> o----------o-----------o----------o f_j <u, v> |
| |
| − | | | x = | 1 1 1 0 | = f<u,v> | |
| |
| − | | | y = | 1 0 0 1 | = g<u,v> | |
| |
| − | o------------o----------o-----------o----------o------------o
| |
| − | | | | | | |
| |
| − | | f_0 | () | 0 0 0 0 | () | f_0 |
| |
| − | | | | | | |
| |
| − | | f_1 | (x)(y) | 0 0 0 1 | () | f_0 |
| |
| − | | | | | | |
| |
| − | | f_2 | (x) y | 0 0 1 0 | (u)(v) | f_1 |
| |
| − | | | | | | |
| |
| − | | f_3 | (x) | 0 0 1 1 | (u)(v) | f_1 |
| |
| − | | | | | | |
| |
| − | | f_4 | x (y) | 0 1 0 0 | (u, v) | f_6 |
| |
| − | | | | | | |
| |
| − | | f_5 | (y) | 0 1 0 1 | (u, v) | f_6 |
| |
| − | | | | | | |
| |
| − | | f_6 | (x, y) | 0 1 1 0 | (u v) | f_7 |
| |
| − | | | | | | |
| |
| − | | f_7 | (x y) | 0 1 1 1 | (u v) | f_7 |
| |
| − | | | | | | |
| |
| − | o------------o----------o-----------o----------o------------o
| |
| − | | | | | | |
| |
| − | | f_8 | x y | 1 0 0 0 | u v | f_8 |
| |
| − | | | | | | |
| |
| − | | f_9 | ((x, y)) | 1 0 0 1 | u v | f_8 |
| |
| − | | | | | | |
| |
| − | | f_10 | y | 1 0 1 0 | ((u, v)) | f_9 |
| |
| − | | | | | | |
| |
| − | | f_11 | (x (y)) | 1 0 1 1 | ((u, v)) | f_9 |
| |
| − | | | | | | |
| |
| − | | f_12 | x | 1 1 0 0 | ((u)(v)) | f_14 |
| |
| − | | | | | | |
| |
| − | | f_13 | ((x) y) | 1 1 0 1 | ((u)(v)) | f_14 |
| |
| − | | | | | | |
| |
| − | | f_14 | ((x)(y)) | 1 1 1 0 | (()) | f_15 |
| |
| − | | | | | | |
| |
| − | | f_15 | (()) | 1 1 1 1 | (()) | f_15 |
| |
| − | | | | | | |
| |
| − | o------------o----------o-----------o----------o------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 14===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | EG_i = G_i <u + du, v + dv> |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 15===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | DG_i = G_i <u, v> + EG_i <u, v, du, dv> |
| |
| − | | |
| |
| − | | = G_i <u, v> + G_i <u + du, v + dv> |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 16===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | Ef = ((u + du)(v + dv)) |
| |
| − | | |
| |
| − | | Eg = ((u + du, v + dv)) |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 17===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------o
| |
| − | | |
| |
| − | | Df = ((u)(v)) + ((u + du)(v + dv)) |
| |
| − | | |
| |
| − | | Dg = ((u, v)) + ((u + du, v + dv)) |
| |
| − | | |
| |
| − | o-------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 66-i. Computation Summary for f‹u, v› = ((u)(v))===
| |
| − |
| |
| − | <pre>
| |
| − | Table 66-i. Computation Summary for f<u, v> = ((u)(v))
| |
| − | o--------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 |
| |
| − | | |
| |
| − | | Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) |
| |
| − | | |
| |
| − | | Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) |
| |
| − | | |
| |
| − | | df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) |
| |
| − | | |
| |
| − | | rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv |
| |
| − | | |
| |
| − | o--------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 66-ii. Computation Summary for g‹u, v› = ((u, v))===
| |
| − |
| |
| − | <pre>
| |
| − | Table 66-ii. Computation Summary for g<u, v> = ((u, v))
| |
| − | o--------------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | !e!g = uv. 1 + u(v). 0 + (u)v. 0 + (u)(v). 1 |
| |
| − | | |
| |
| − | | Eg = uv.((du, dv)) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v).((du, dv)) |
| |
| − | | |
| |
| − | | Dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| |
| − | | |
| |
| − | | dg = uv. (du, dv) + u(v). (du, dv) + (u)v. (du, dv) + (u)(v). (du, dv) |
| |
| − | | |
| |
| − | | rg = uv. 0 + u(v). 0 + (u)v. 0 + (u)(v). 0 |
| |
| − | | |
| |
| − | o--------------------------------------------------------------------------------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 67. Computation of an Analytic Series in Terms of Coordinates===
| |
| − |
| |
| − | <pre>
| |
| − | Table 67. Computation of an Analytic Series in Terms of Coordinates
| |
| − | o--------o-------o-------o--------o-------o-------o-------o-------o
| |
| − | | u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg |
| |
| − | o--------o-------o-------o--------o-------o-------o-------o-------o
| |
| − | | | | | | | | | |
| |
| − | | 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 |
| |
| − | | | | | | | | | |
| |
| − | o--------o-------o-------o--------o-------o-------o-------o-------o
| |
| − | | | | | | | | | |
| |
| − | | 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 |
| |
| − | | | | | | | | | |
| |
| − | o--------o-------o-------o--------o-------o-------o-------o-------o
| |
| − | | | | | | | | | |
| |
| − | | 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 |
| |
| − | | | | | | | | | |
| |
| − | o--------o-------o-------o--------o-------o-------o-------o-------o
| |
| − | | | | | | | | | |
| |
| − | | 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 |
| |
| − | | | | | | | | | |
| |
| − | | | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 |
| |
| − | | | | | | | | | |
| |
| − | o--------o-------o-------o--------o-------o-------o-------o-------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Table 68. Computation of an Analytic Series in Symbolic Terms===
| |
| − |
| |
| − | <pre>
| |
| − | Table 68. Computation of an Analytic Series in Symbolic Terms
| |
| − | o-----o-----o------------o----------o----------o----------o----------o----------o
| |
| − | | u v | f g | Df | Dg | df | dg | rf | rf |
| |
| − | o-----o-----o------------o----------o----------o----------o----------o----------o
| |
| − | | | | | | | | | |
| |
| − | | 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () |
| |
| − | | | | | | | | | |
| |
| − | | 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () |
| |
| − | | | | | | | | | |
| |
| − | | 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () |
| |
| − | | | | | | | | | |
| |
| − | | 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |
| |
| − | | | | | | | | | |
| |
| − | o-----o-----o------------o----------o----------o----------o----------o----------o
| |
| − | </pre>
| |
| − |
| |
| − | ===Formula Display 18===
| |
| − |
| |
| − | <pre>
| |
| − | o-------------------------------------------------------------------------o
| |
| − | | |
| |
| − | | Df = uv. du dv + u(v). du (dv) + (u)v.(du) dv + (u)(v).((du)(dv)) |
| |
| − | | |
| |
| − | | Dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v). (du, dv) |
| |
| − | | |
| |
| − | o-------------------------------------------------------------------------o
| |
| − |
| |
| − | ===Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>===
| |
| − |
| |
| − | o-----------------------------------o o-----------------------------------o
| |
| − | | U | |`U`````````````````````````````````|
| |
| − | | | |```````````````````````````````````|
| |
| − | | ^ | |```````````````````````````````````|
| |
| − | | | | |```````````````````````````````````|
| |
| − | | o-------o | o-------o | |```````o-------o```o-------o```````|
| |
| − | | ^ /`````````\|/`````````\ ^ | | ^ ```/ ^ \`/ ^ \``` ^ |
| |
| − | | \ /```````````|```````````\ / | |``\``/ \ o / \``/``|
| |
| − | | \/`````u`````/|\`````v`````\/ | |```\/ u \/`\/ v \/```|
| |
| − | | /\``````````/`|`\``````````/\ | |```/\ /\`/\ /\```|
| |
| − | | o``\````````o``@``o````````/``o | |``o \ o``@``o / o``|
| |
| − | | |```\```````|`````|```````/```| | |``| \ |`````| / |``|
| |
| − | | |````@``````|`````|``````@````| | |``| @-------->`<--------@ |``|
| |
| − | | |```````````|`````|```````````| | |``| |`````| |``|
| |
| − | | o```````````o` ^ `o```````````o | |``o o`````o o``|
| |
| − | | \```````````\`|`/```````````/ | |```\ \```/ /```|
| |
| − | | \```` ^ ````\|/```` ^ ````/ | |````\ ^ \`/ ^ /````|
| |
| − | | \`````\`````|`````/`````/ | |`````\ \ o / /`````|
| |
| − | | \`````\```/|\```/`````/ | |``````\ \ /`\ / /``````|
| |
| − | | o-----\-o | o-/-----o | |```````o-----\-o```o-/-----o```````|
| |
| − | | \ | / | |``````````````\`````/``````````````|
| |
| − | | \ | / | |```````````````\```/```````````````|
| |
| − | | \|/ | |````````````````\`/````````````````|
| |
| − | | @ | |`````````````````@`````````````````|
| |
| − | o-----------------------------------o o-----------------------------------o
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | \ ((u)(v)) / \ ((u, v)) /
| |
| − | \ / \ /
| |
| − | \ / \ /
| |
| − | o----------\-------------/-----------------------\-------------/----------o
| |
| − | | X \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | \ / \ / |
| |
| − | | o----------------o o----------------o |
| |
| − | | / \ / \ |
| |
| − | | / o \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | / / \ \ |
| |
| − | | o o o o |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | | f | | g | |
| |
| − | | | | | | |
| |
| − | | | | | | |
| |
| − | | o o o o |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ \ / / |
| |
| − | | \ o / |
| |
| − | | \ / \ / |
| |
| − | | o----------------o o----------------o |
| |
| − | | |
| |
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| − | o-------------------------------------------------------------------------o
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| − | Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
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| − | </pre>
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| | ==Inquiry Driven Systems== | | ==Inquiry Driven Systems== |
