Contents
1 Differential Logic and Dynamic Systems
1.1 Table 1. Syntax & Semantics of a Calculus for Propositional Logic 1.2 Table 2. Fundamental Notations for Propositional Calculus 1.3 Table 3. Analogy of Real and Boolean Types 1.4 Table 4. An Equivalence Based on the Propositions as Types Analogy 1.5 Table 5. A Bridge Over Troubled Waters 1.6 Table 6. Propositional Forms on One Variable 1.7 Table 7. Propositional Forms on Two Variables 1.8 Table 8. Notation for the Differential Extension of Propositional Calculus 1.9 Table 9. Higher Order Differential Features 1.10 Table 10. A Realm of Intentional Features 1.11 Formula Display 1 1.12 Table 11. A Pair of Commodious Trajectories 1.13 Figure 12. The Anchor 1.14 Figure 13. The Tiller 1.15 Table 14. Differential Propositions 1.16 Table 15. Tacit Extension of [A ] to [A , dA ] 1.17 Figure 16-a. A Couple of Fourth Gear Orbits: 1 1.18 Figure 16-b. A Couple of Fourth Gear Orbits: 2 1.19 Formula Display 2 1.20 Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1 1.21 Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2 1.22 Figure 18-a. Extension from 1 to 2 Dimensions: Areal 1.23 Figure 18-b. Extension from 1 to 2 Dimensions: Bundle 1.24 Figure 18-c. Extension from 1 to 2 Dimensions: Compact 1.25 Figure 18-d. Extension from 1 to 2 Dimensions: Digraph 1.26 Figure 19-a. Extension from 2 to 4 Dimensions: Areal 1.27 Figure 19-b. Extension from 2 to 4 Dimensions: Bundle 1.28 Figure 19-c. Extension from 2 to 4 Dimensions: Compact 1.29 Figure 19-d. Extension from 2 to 4 Dimensions: Digraph 1.30 Figure 20-i. Thematization of Conjunction (Stage 1) 1.31 Figure 20-ii. Thematization of Conjunction (Stage 2) 1.32 Figure 20-iii. Thematization of Conjunction (Stage 3) 1.33 Figure 21. Thematization of Disjunction and Equality 1.34 Table 22. Disjunction f and Equality g 1.35 Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1) 1.36 Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2) 1.37 Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3) 1.38 Tables 26-i and 26-ii. Tacit Extension and Thematization 1.39 Table 27. Thematization of Bivariate Propositions 1.40 Table 28. Propositions on Two Variables 1.41 Table 29. Thematic Extensions of Bivariate Propositions 1.42 Figure 30. Generic Frame of a Logical Transformation 1.43 Formula Display 3 1.44 Figure 31. Operator Diagram (1) 1.45 Figure 32. Operator Diagram (2) 1.46 Figure 33-i. Analytic Diagram (1) 1.47 Figure 33-ii. Analytic Diagram (2) 1.48 Formula Display 4 1.49 Formula Display 5 1.50 Formula Display 6 1.51 Formula Display 7 1.52 Figure 34. Tangent Functor Diagram 1.53 Figure 35. Conjunction as Transformation 1.54 Table 36. Computation of !e!J 1.55 Figure 37-a. Tacit Extension of J (Areal) 1.56 Figure 37-b. Tacit Extension of J (Bundle) 1.57 Figure 37-c. Tacit Extension of J (Compact) 1.58 Figure 37-d. Tacit Extension of J (Digraph) 1.59 Table 38. Computation of EJ (Method 1) 1.60 Table 39. Computation of EJ (Method 2) 1.61 Figure 40-a. Enlargement of J (Areal) 1.62 Figure 40-b. Enlargement of J (Bundle) 1.63 Figure 40-c. Enlargement of J (Compact) 1.64 Figure 40-d. Enlargement of J (Digraph) 1.65 Table 41. Computation of DJ (Method 1) 1.66 Table 42. Computation of DJ (Method 2) 1.67 Table 43. Computation of DJ (Method 3) 1.68 Formula Display 8 1.69 Figure 44-a. Difference Map of J (Areal) 1.70 Figure 44-b. Difference Map of J (Bundle) 1.71 Figure 44-c. Difference Map of J (Compact) 1.72 Figure 44-d. Difference Map of J (Digraph) 1.73 Table 45. Computation of dJ 1.74 Figure 46-a. Differential of J (Areal) 1.75 Figure 46-b. Differential of J (Bundle) 1.76 Figure 46-c. Differential of J (Compact) 1.77 Figure 46-d. Differential of J (Digraph) 1.78 Table 47. Computation of rJ 1.79 Figure 48-a. Remainder of J (Areal) 1.80 Figure 48-b. Remainder of J (Bundle) 1.81 Figure 48-c. Remainder of J (Compact) 1.82 Figure 48-d. Remainder of J (Digraph) 1.83 Table 49. Computation Summary for J 1.84 Table 50. Computation of an Analytic Series in Terms of Coordinates 1.85 Formula Display 9 1.86 Formula Display 10 1.87 Table 51. Computation of an Analytic Series in Symbolic Terms 1.88 Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ) 1.89 Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ) 1.90 Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators 1.91 Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes 1.92 Figure 56-a1. Radius Map of the Conjunction J = uv 1.93 Figure 56-a2. Secant Map of the Conjunction J = uv 1.94 Figure 56-a3. Chord Map of the Conjunction J = uv 1.95 Figure 56-a4. Tangent Map of the Conjunction J = uv 1.96 Figure 56-b1. Radius Map of the Conjunction J = uv 1.97 Figure 56-b2. Secant Map of the Conjunction J = uv 1.98 Figure 56-b3. Chord Map of the Conjunction J = uv 1.99 Figure 56-b4. Tangent Map of the Conjunction J = uv 1.100 Figure 57-1. Radius Operator Diagram for the Conjunction J = uv 1.101 Figure 57-2. Secant Operator Diagram for the Conjunction J = uv 1.102 Figure 57-3. Chord Operator Diagram for the Conjunction J = uv 1.103 Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv 1.104 Formula Display 11 1.105 Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators 1.106 Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes 1.107 Formula Display 12 1.108 Formula Display 13 1.109 Table 60. Propositional Transformation 1.110 Figure 61. Propositional Transformation 1.111 Figure 62. Propositional Transformation (Short Form) 1.112 Figure 63. Transformation of Positions 1.113 Table 64. Transformation of Positions 1.114 Table 65. Induced Transformation on Propositions 1.115 Formula Display 14 1.116 Formula Display 15 1.117 Formula Display 16 1.118 Formula Display 17 1.119 Table 66-i. Computation Summary for f‹u, v› = ((u)(v)) 1.120 Table 66-ii. Computation Summary for g‹u, v› = ((u, v)) 1.121 Table 67. Computation of an Analytic Series in Terms of Coordinates 1.122 Table 68. Computation of an Analytic Series in Symbolic Terms 1.123 Formula Display 18 1.124 Figure 69. Difference Map of F = ‹f, g› = ‹((u)(v)), ((u, v))› 1.125 Formula Display 19 1.126 Figure 70-a. Tangent Functor Diagram for F‹u, v› = ‹((u)(v)), ((u, v))› 1.127 Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›
Differential Logic and Dynamic Systems Table 1. Syntax & Semantics of a Calculus for Propositional Logic 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 |
| | 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 |
| | A, B, C | A'B C' v |
| | 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 |
| | | 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 |
| | Genus Q comprises | Q A'B'C |
| | species A, B, C. | |
o-------------------o-------------------o-------------------o
Table 1. Syntax and Semantics of a Calculus for Propositional Logic
Expression
Interpretation
Other Notations
" "
True.
1
( )
False.
0
A
A.
A
(A)
Not A.
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.
A ⇒ B
(A, B)
A not equal to B.
A ≠ B
((A, B))
A is equal to B.
A = B
(A, B, C)
Just one of
A’B C ∨
((A),(B),(C))
Just one of Partition all
A B’C’ ∨
((A, B), C)
Oddly many of
A + B + C
(Q, (A),(B),(C))
Partition Q Genus Q comprises
Q’A’B’C’ ∨
Table 2. Fundamental Notations for Propositional Calculus 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
Table 2. Fundamental Notations for Propositional Calculus
Symbol
Notation
Description
Type
A
{a 1 , …, a n
Alphabet
[n ] = n
A i {(a i a i
Dimension i
B
A
〈A 〉a 1 , …, a n a 1 , …, a n A 1 × … × A n i A i
Set of cells,
B n
A *
(hom : A → B )
Linear functions
(B n B n
A ^
(A → B )
Boolean functions
B n B
A •
[A ]A , A ^)A +→ B )A , (A → B ))a 1 , …, a n
Universe of discoursea 1 , …, a n
(B n B n B ))B n B )B n
Table 3. Analogy of Real and Boolean Types 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
Table 3. Analogy of Real and Boolean Types
Real Domain R
←→
Boolean Domain B
R n Basic Space
B n
R n R
Function Space
B n B
(R n R ) → R
Tangent Vector
(B n B ) → B
R n R n R )→R )
Vector Field
B n B n B )→B )
(R n R n R )) → R
ditto
(B n B n B )) → B
((R n R ) × R n R
ditto
((B n B ) × B n B
(R n R ) → (R n R )
Derivation
(B n B ) → (B n B )
R n R m Basic Transformation
B n B m
(R n R ) → (R m R )
Function Transformation
(B n B ) → (B m B )
...
...
...
Table 4. An Equivalence Based on the Propositions as Types Analogy 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
Table 4. An Equivalence Based on the Propositions as Types Analogy
Pattern
Construction
Instance
X → (Y → Z )
Vector Field
K n K n K ) → K )
(X × Y ) → Z
(K n K n K )) → K
(Y × X ) → Z
((K n K ) × K n K
Y → (X → Z )
Derivation
(K n K ) → (K n K )
Table 5. A Bridge Over Troubled Waters 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
Table 5. A Bridge Over Troubled Waters
Linear Space
Liminal Space
Logical Space
X x 1 , …, x n n
X x 1 , …, x n n
A a 1 , …, a n n
X i x i K
X i x i x i B
A i a i a i B
X X 〉x 1 , …, x n x 1 , …, x n X 1 × … × X n i X i K n
X X x 1 , …, x n x 1 , …, x n X 1 × … × X n i X i B n
A A 〉a 1 , …, a n a 1 , …, a n A 1 × … × A n i A i B n
X *X → K )K n
X X B )B n
A *A → B )B n
X ^X → K )K n K )
X X B )B n B )
A ^A → B )B n B )
X • X ]x 1 , …, x n X , X ^)X +→ K )X , (X → K ))K n K n K ))K n K )K n
X • X x 1 , …, x n X X X B )X X B ))B n B n B ))B n B )B n
A • A ]a 1 , …, a n A , A ^)A +→ B )A , (A → B ))B n B n B ))B n B )B n
Table 6. Propositional Forms on One Variable 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
Table 6. Propositional Forms on One Variable
L1
L2
L3
L4
L5
L6
x :
1 0
f0
f00
0 0
( )
false
0
f1
f01
0 1
(x)
not x
~x
f2
f10
1 0
x
x
x
f3
f11
1 1
(( ))
true
1
Table 7. Propositional Forms on Two Variables 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
Table 7. Propositional Forms on Two Variables
L1
L2
L3
L4
L5
L6
x :
1 1 0 0
y :
1 0 1 0
f0
f0000
0 0 0 0
( )
false
0
f1
f0001
0 0 0 1
(x)(y)
neither x nor y
¬x ∧ ¬y
f2
f0010
0 0 1 0
(x) y
y and not x
¬x ∧ y
f3
f0011
0 0 1 1
(x)
not x
¬x
f4
f0100
0 1 0 0
x (y)
x and not y
x ∧ ¬y
f5
f0101
0 1 0 1
(y)
not y
¬y
f6
f0110
0 1 1 0
(x, y)
x not equal to y
x ≠ y
f7
f0111
0 1 1 1
(x y)
not both x and y
¬x ∨ ¬y
f8
f1000
1 0 0 0
x y
x and y
x ∧ y
f9
f1001
1 0 0 1
((x, y))
x equal to y
x = y
f10
f1010
1 0 1 0
y
y
y
f11
f1011
1 0 1 1
(x (y))
not x without y
x → y
f12
f1100
1 1 0 0
x
x
x
f13
f1101
1 1 0 1
((x) y)
not y without x
x ← y
f14
f1110
1 1 1 0
((x)(y))
x or y
x ∨ y
f15
f1111
1 1 1 1
(( ))
true
1
Table 8. Notation for the Differential Extension of Propositional Calculus 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
Table 8. Notation for the Differential Extension of Propositional Calculus
Symbol
Notation
Description
Type
dA
{da 1 , …, da n
Alphabet of
[n ] = n
dA i
{(da i a i
Differentiali
D
dA
〈dA 〉a 1 , …, da n a 1 , …, da n A 1 × … × dA n i A i
Tangent space
D n
dA *
(hom : dA → B )
Linear functionsA
(D n D n
dA ^
(dA → B )
Boolean functionsA
D n B
dA •
[dA ]A , dA ^)A +→ B )A , (dA → B ))a 1 , …, da n
Tangent universeA • ,a 1 , …, da n
(D n D n B ))D n B )D n
Table 9. Higher Order Differential Features 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
Table 9. Higher Order Differential Features
A = d0 A = {a 1 , …, a n A = d1 A = {da 1 , …, da n k A = {dk a 1 k a n * A = {d0 A , …, dk A , …}
E0 A = d0 A 1 A = d0 A ∪ d1 A k A = d0 A ∪ … ∪ dk A ∞ A = ∪ d* A
Table 9. Higher Order Differential Features
A
=
d0 A
=
{a 1 ,
…,
a n
dA
=
d1 A
=
{da 1 ,
…,
da n
dk A
=
{dk a 1
…,
dk a n
d* A
=
{d0 A ,
…,
dk A ,
…}
E0 A
=
d0 A
E1 A
=
d0 A ∪ d1 A
Ek A
=
d0 A ∪ … ∪ dk A
E∞ A
=
∪ d* A
Table 10. A Realm of Intentional Features 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
Table 10. A Realm of Intentional Features
p0 A
=
A
=
{a 1 ,
…,
a n
p1 A
=
A ′
=
{a 1 ′,
…,
a n
p2 A
=
A ″
=
{a 1 ″,
…,
a n
...
...
pk A
=
{pk a 1 ,
…,
pk a n
Q0 A
=
A
Q1 A
=
A ∪ A ′
Q2 A
=
A ∪ A ′ ∪ A ″
...
...
Qk A
=
A ∪ A ′ ∪ … ∪ pk A
Formula Display 1 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
From
(A )
and
(dA )
infer
(A )
next.
From
(A )
and
dA
infer
A next.
From
A and
(dA )
infer
A next.
From
A and
dA
infer
(A )
next.
Table 11. A Pair of Commodious Trajectories 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
Table 11. A Pair of Commodious Trajectories
Time
Trajectory 1
Trajectory 2
A dA
(d2 A )
(A )
dA
d2 A
A (dA )
(d2 A )
A (dA )
(d2 A )
"
"
"
(A )
(dA )
d2 A
(A )
dA
d2 A
A (dA )
(d2 A )
A (dA )
(d2 A )
"
"
"
Figure 12. The Anchor 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
Figure 13. The Tiller 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
Table 14. Differential Propositions 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
Table 14. Differential Propositions
A :
1 1 0 0
dA :
1 0 1 0
f0
g0
0 0 0 0
( )
False
0
g1
0 0 0 1
(A)(dA)
Neither A nor dA
¬A ∧ ¬dA
g2
0 0 1 0
(A) dA
Not A but dA
¬A ∧ dA
g4
0 1 0 0
A (dA)
A but not dA
A ∧ ¬dA
g8
1 0 0 0
A dA
A and dA
A ∧ dA
f1
g3
0 0 1 1
(A)
Not A
¬A
f2
g12
1 1 0 0
A
A
A
g6
0 1 1 0
(A, dA)
A not equal to dA
A ≠ dA
g9
1 0 0 1
((A, dA))
A equal to dA
A = dA
g5
0 1 0 1
(dA)
Not dA
¬dA
g10
1 0 1 0
dA
dA
dA
g7
0 1 1 1
(A dA)
Not both A and dA
¬A ∨ ¬dA
g11
1 0 1 1
(A (dA))
Not A without dA
A → dA
g13
1 1 0 1
((A) dA)
Not dA without A
A ← dA
g14
1 1 1 0
((A)(dA))
A or dA
A ∨ dA
f3
g15
1 1 1 1
(( ))
True
1
Table 14. Differential Propositions
A :
1 1 0 0
dA :
1 0 1 0
f0
g0
0 0 0 0
( )
False
0
0 0 0 1
(A)(dA)
Neither A nor dA
¬A ∧ ¬dA
A not equal to dA
0 1 1 1
(A dA)
Not both A and dA
¬A ∨ ¬dA
f3
g15
1 1 1 1
(( ))
True
1
Table 15. Tacit Extension of [A ] to [A , dA ] 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
Table 15. Tacit Extension of [A ] to [A , dA ]
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
Figure 16-a. A Couple of Fourth Gear Orbits: 1 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
Figure 16-b. A Couple of Fourth Gear Orbits: 2 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
Formula Display 2 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
r (q )
=
∑k d k -k
=
∑k k A (q ) · 2-k
=
s (q )/t
=
(∑k d k (m -k ) ) / 2m
=
(∑k k A (q ) · 2(m -k ) ) / 2m
\(r(q)\!\)
\(=\)
\(\sum_k d_k \cdot 2^{-k}\)
\(=\)
\(\sum_k \mbox{d}^k A(q) \cdot 2^{-k}\)
\(=\)
\(\frac{s(q)}{t}\)
\(=\)
\(\frac{\sum_k d_k \cdot 2^{(m-k)}}{2^m}\)
\(=\)
\(\frac{\sum_k \mbox{d}^k A(q) \cdot 2^{(m-k)}}{2^m}\)
Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1 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
Table 17-a. A Couple of Orbits in Fourth Gear: Orbit 1
Time
State
A
dA
p i q j d0 A
d1 A
d2 A
d3 A
d4 A
q 01
q 03
q 05
q 15
q 17
q 19
q 21
q 31
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
Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2 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
Table 17-b. A Couple of Orbits in Fourth Gear: Orbit 2
Time
State
A
dA
p i q j d0 A
d1 A
d2 A
d3 A
d4 A
q 25
q 11
q 29
q 07
q 09
q 27
q 13
q 23
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
Figure 18-a. Extension from 1 to 2 Dimensions: Areal 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
Figure 18-b. Extension from 1 to 2 Dimensions: Bundle 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
Figure 18-c. Extension from 1 to 2 Dimensions: Compact 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
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph o-----------------------------------------------------------o
| |
| |
| dx |
| .--. .---------->----------. .--. |
| | \ / \ / | |
| (dx) ^ @ x (x) @ v (dx) |
| | / \ / \ | |
| *--* *----------<----------* *--* |
| dx |
| |
| |
o-----------------------------------------------------------o
Figure 18-d. Extension from 1 to 2 Dimensions: Digraph
Figure 19-a. Extension from 2 to 4 Dimensions: Areal 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
Figure 19-b. Extension from 2 to 4 Dimensions: Bundle 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
Figure 19-c. Extension from 2 to 4 Dimensions: Compact 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
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph o-----------------------------------------------------------o
| |
| .->-. |
| | | |
| * * |
| \ / |
| .-->--@--<--. |
| / / \ \ |
| / / \ \ |
| / . . \ |
| / | | \ |
| / | | \ |
| / | | \ |
| . | | . |
| | | | | |
| v | | v |
| .--. | .---------->----------. | .--. |
| | \|/ | | \|/ | |
| ^ @ ^ v @ v |
| | /|\ | | /|\ | |
| *--* | *----------<----------* | *--* |
| ^ | | ^ |
| | | | | |
| * | | * |
| \ | | / |
| \ | | / |
| \ | | / |
| \ . . / |
| \ \ / / |
| \ \ / / |
| *-->--@--<--* |
| / \ |
| . . |
| | | |
| *-<-* |
| |
o-----------------------------------------------------------o
Figure 19-d. Extension from 2 to 4 Dimensions: Digraph
Figure 20-i. Thematization of Conjunction (Stage 1) 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)
Figure 20-ii. Thematization of Conjunction (Stage 2) 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)
Figure 20-iii. Thematization of Conjunction (Stage 3) 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)
Figure 21. Thematization of Disjunction and Equality 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
Table 22. Disjunction f and Equality g 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
Table 22. Disjunction f and Equality g
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1) 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
Tables 23-i and 23-ii. Thematics of Disjunction and Equality (1)
Table 23-i. Disjunction f
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2) 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
Tables 24-i and 24-ii. Thematics of Disjunction and Equality (2)
Table 24-i. Disjunction f
0
0
→
0
0
0
1
0
1
0
0
1
→
1
1
0
0
1
0
→
1
1
1
0
1
1
→
1
Table 24-ii. Equality g
0
0
0
0
0
→
1
0
1
→
0
0
1
1
1
0
→
0
1
0
1
1
1
0
1
1
→
1
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3) 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
Tables 25-i and 25-ii. Thematics of Disjunction and Equality (3)
Table 25-i. Disjunction f
0
0
→
0
0
1
0
1
0
0
1
1
0
0
0
1
0
1
→
1
1
0
→
1
1
1
→
1
Table 25-ii. Equality g
0
0
0
0
1
→
0
1
0
→
0
1
1
0
0
0
→
1
0
1
1
1
0
1
1
1
→
1
Tables 26-i and 26-ii. Tacit Extension and Thematization 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
Tables 26-i and 26-ii. Tacit Extension and Thematization
Table 26-i. Disjunction f
Table 27. Thematization of Bivariate Propositions 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
Table 27. Thematization of Bivariate Propositions
0000
0001
0010
0011
0100
0101
0110
0111
()
(u)(v)
(u) v
(u)
u (v)
(v)
(u, v)
(u v)
(( f , () ))
(( f , (u)(v) ))
(( f , (u) v ))
(( f , (u) ))
(( f , u (v) ))
(( f , (v) ))
(( f , (u, v) ))
(( f , (u v) ))
f + 1
f + u + v + uv
f + v + uv + 1
f + u
f + u + uv + 1
f + v
f + u + v + 1
f + uv
f8
f9
f10
f11
f12
f13
f14
f15
1000
1001
1010
1011
1100
1101
1110
1111
u v
((u, v))
v
(u (v))
u
((u) v)
((u)(v))
(())
(( f , u v ))
(( f , ((u, v)) ))
(( f , v ))
(( f , (u (v)) ))
(( f , u ))
(( f , ((u) v) ))
(( f , ((u)(v)) ))
(( f , (()) ))
f + uv + 1
f + u + v
f + v + 1
f + u + uv
f + u + 1
f + v + uv
f + u + v + uv + 1
f
Table 28. Propositions on Two Variables 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
Table 28. Propositions on Two Variables
u v
f0
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
f11
f12
f13
f14
f15
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
Table 28. Propositions on Two Variables
u v
f0
f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
f11
f12
f13
f14
f15
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
Table 28. Propositions on Two Variables
u
v
f00
f01
f02
f03
f04
f05
f06
f07
f08
f09
f10
f11
f12
f13
f14
f15
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
Table 29. Thematic Extensions of Bivariate Propositions 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
Table 29. Thematic Extensions of Bivariate Propositions
u
v
f¢
φ00
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Figure 30. Generic Frame of a Logical Transformation 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
Formula Display 3 o-------------------------------------------------o
| |
| x = f<u, v> |
| |
| y = g<u, v> |
| |
| z = h<u, v> |
| |
o-------------------------------------------------o
x
=
f ‹u , v ›
y =
g ‹u , v ›
z =
h ‹u , v ›
Figure 31. Operator Diagram (1) 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)
Figure 32. Operator Diagram (2) 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)
Figure 33-i. Analytic Diagram (1) 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)
Figure 33-ii. Analytic Diagram (2) 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)
Formula Display 4 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
x 1
=
\(\epsilon\)F 1 ‹u 1 , …, u n u 1 , …, du n
=
F 1 ‹u 1 , …, u n
...
x k =
\(\epsilon\)F k u 1 , …, u n u 1 , …, du n
=
F k u 1 , …, u n
dx 1
=
EF 1 ‹u 1 , …, u n u 1 , …, du n
=
F 1 ‹u 1 + du 1 , …, u n u n
...
dx k
=
EF k u 1 , …, u n u 1 , …, du n
=
F k u 1 + du 1 , …, u n u n
Formula Display 5 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
x 1
=
\(\epsilon\)F 1 ‹u 1 , …, u n u 1 , …, du n
=
F 1 ‹u 1 , …, u n
...
x k =
\(\epsilon\)F k u 1 , …, u n u 1 , …, du n
=
F k u 1 , …, u n
dx 1
=
\(\epsilon\)F 1 ‹u 1 , …, u n u 1 , …, du n
=
F 1 ‹u 1 , …, u n
...
dx k
=
\(\epsilon\)F k u 1 , …, u n u 1 , …, du n
=
F k u 1 , …, u n
Formula Display 6 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
dx 1
=
\(\epsilon\)F 1 ‹u 1 , …, u n u 1 , …, du n
=
F 1 ‹u 1 , …, u n
...
dx k
=
\(\epsilon\)F k u 1 , …, u n u 1 , …, du n
=
F k u 1 , …, u n
Formula Display 7 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
D =
E e
=
r 0
=
d 1 + r 1
=
d 1 + … + d m r m
=
∑ (i = 1 … m ) d i r m
Figure 34. Tangent Functor Diagram 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
Figure 35. Conjunction as Transformation 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
Table 36. Computation of !e!J 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
Table 36. Computation of \(\epsilon\)J
\(\epsilon\)J
=
J ‹u , v ›
=
u v
=
u v (du )(dv )+
u v (du ) dv +
u v du (dv )+
u v du dv
\(\epsilon\)J
=
u v (du )(dv )
+
u v (du ) dv
+
u v du (dv )
+
u v du dv
Figure 37-a. Tacit Extension of J (Areal) 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)
Figure 37-b. Tacit Extension of J (Bundle) 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)
Figure 37-c. Tacit Extension of J (Compact) 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)
Figure 37-d. Tacit Extension of J (Digraph) 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)
Table 38. Computation of EJ (Method 1) 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
Table 38. Computation of EJ (Method 1)
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 ›
EJ
= u v (du )(dv )
+ u (v ) (du ) dv
+ (u ) v du (dv )
+ (u )(v ) du dv
Table 39. Computation of EJ (Method 2) 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
Table 39. Computation of EJ (Method 2)
EJ
= ‹u + du › \(\cdot\) ‹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
Figure 40-a. Enlargement of J (Areal) 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)
Figure 40-b. Enlargement of J (Bundle) 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)
Figure 40-c. Enlargement of J (Compact) 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)
Figure 40-d. Enlargement of J (Digraph) 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)
Table 41. Computation of DJ (Method 1) 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
Table 41. Computation of DJ (Method 1)
DJ
=
EJ
+
\(\epsilon\)J
=
J ‹u + du , v + dv ›
+
J ‹u , v ›
=
(u , du )(v , dv )
+
u v
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
DJ
=
u v ((du )(dv ))
+ u (v )(du ) dv
+ (u ) v du (dv )
+ (u )(v ) du dv
Table 42. Computation of DJ (Method 2) 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
Table 42. Computation of DJ (Method 2)
DJ
=
\(\epsilon\)J
+
EJ
=
J ‹u , v ›
+
J ‹u + du , v + dv ›
=
u v
+
(u , du )(v , dv )
=
0
+
u dv
+
v du
+
du dv
DJ
=
0
+
u (du ) dv
+
v du (dv )
+
((u , v )) du dv
Table 43. Computation of DJ (Method 3) 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
Table 43. Computation of DJ (Method 3)
\(\epsilon\)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
DJ
= 0 \(\cdot\) (du )(dv )
+ u \(\cdot\) (du ) dv
+ v \(\cdot\) du (dv )
+ ((u , v )) du dv
Formula Display 8 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
\(\epsilon\)J
= {Dispositions from J to J }
+ {Dispositions from J to (J ) }
EJ
= {Dispositions from J to J }
+ {Dispositions from (J ) to J }
DJ
= (\(\epsilon\)J , EJ )
DJ
= {Dispositions from J to (J ) }
+ {Dispositions from (J ) to J }
Figure 44-a. Difference Map of J (Areal) 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)
Figure 44-b. Difference Map of J (Bundle) 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)
Figure 44-c. Difference Map of J (Compact) 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)
Figure 44-d. Difference Map of J (Digraph) 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)
Table 45. Computation of dJ 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
Table 45. Computation of dJ
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 ) \(\cdot\) 0
Figure 46-a. Differential of J (Areal) 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)
Figure 46-b. Differential of J (Bundle) 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)
Figure 46-c. Differential of J (Compact) 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)
Figure 46-d. Differential of J (Digraph) 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)
Table 47. Computation of rJ 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
Table 47. Computation of rJ
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 ) \(\cdot\) 0
rJ
= u v du dv
+ u (v ) du dv
+ (u ) v du dv
+ (u )(v ) du dv
Figure 48-a. Remainder of J (Areal) 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)
Figure 48-b. Remainder of J (Bundle) 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)
Figure 48-c. Remainder of J (Compact) 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)
Figure 48-d. Remainder of J (Digraph) o-----------------------------------------------------------o
| |
| u v |
| @ |
| ^ |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| du | dv |
| u (v) @<----------|---------->@ (u) v |
| du | dv |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| | |
| v |
| @ |
| (u) (v) |
| |
o-----------------------------------------------------------o
Figure 48-d. Remainder of J (Digraph)
Table 49. Computation Summary for J 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
Table 49. Computation Summary for J
\(\epsilon\)J
=
uv \(\cdot\)
1
+
u (v )\(\cdot\)
0
+
(u )v
\(\cdot\)
0
+
(u )(v )
\(\cdot\)
0
EJ
=
uv \(\cdot\)
(du )(dv )
+
u (v )\(\cdot\)
(du )dv
+
(u )v
\(\cdot\)
du (dv )
+
(u )(v )
\(\cdot\)
du dv
DJ
=
uv \(\cdot\)
((du )(dv ))
+
u (v )\(\cdot\)
(du )dv
+
(u )v
\(\cdot\)
du (dv )
+
(u )(v )
\(\cdot\)
du dv
dJ
=
uv \(\cdot\)
(du , dv )
+
u (v )\(\cdot\)
dv
+
(u )v
\(\cdot\)
du
+
(u )(v )
\(\cdot\)
0
rJ
=
uv \(\cdot\)
du dv
+
u (v )\(\cdot\)
du dv
+
(u )v
\(\cdot\)
du dv
+
(u )(v )
\(\cdot\)
du dv
Table 50. Computation of an Analytic Series in Terms of Coordinates 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
Table 50. Computation of an Analytic Series in Terms of Coordinates
Formula Display 9 o-------------------------------------------------o
| |
| u' = u + du = (u, du) |
| |
| v' = v + du = (v, dv) |
| |
o-------------------------------------------------o
u ’=
u + du =
(u , du )
v ’=
v + du =
(v , dv )
Formula Display 10 o--------------------------------------------------------------o
| |
| EJ<u, v, du, dv> = J<u + du, v + dv> = J<u', v'> |
| |
o--------------------------------------------------------------o
EJ ‹u , v , du , dv ›
=
J ‹u + du , v + dv ›
=
J ‹u ’, v ’›
Table 51. Computation of an Analytic Series in Symbolic Terms 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
Table 51. Computation of an Analytic Series in Symbolic Terms
du dv
du (dv )
(du ) dv
(du )(dv )
du dv
du (dv )
(du ) dv
((du )(dv ))
Figure 52. Decomposition of the Enlarged Conjunction EJ = (J, DJ) o o o
/%\ /%\ / \
/%%%\ /%%%\ / \
o%%%%%o o%%%%%o o o
/ \%%%/ \ /%\%%%/%\ /%\ /%\
/ \%/ \ /%%%\%/%%%\ /%%%\ /%%%\
o o o o%%%%%o%%%%%o o%%%%%o%%%%%o
/%\ / \ /%\ / \%%%/%\%%%/ \ /%\%%%/%\%%%/%\
/%%%\ / \ /%%%\ / \%/%%%\%/ \ /%%%\%/%%%\%/%%%\
o%%%%%o o%%%%%o o o%%%%%o o o%%%%%o%%%%%o%%%%%o
/ \%%%/ \ / \%%%/ \ / \ / \%%%/ \ / \ / \%%%/ \%%%/ \%%%/ \
/ \%/ \ / \%/ \ / \ / \%/ \ / \ / \%/ \%/ \%/ \
o o o 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)
Figure 53. Decomposition of the Differed Conjunction DJ = (dJ, ddJ) o o o
/ \ / \ / \
/ \ / \ / \
o o o o o o
/%\ /%\ /%\ /%\ / \ / \
/%%%\ /%%%\ /%%%\%/%%%\ / \ / \
o%%%%%o%%%%%o o%%%%%o%%%%%o o o o
/%\%%%/%\%%%/%\ /%\%%%/ \%%%/%\ / \ /%\ / \
/%%%\%/%%%\%/%%%\ /%%%\%/ \%/%%%\ / \ /%%%\ / \
o%%%%%o%%%%%o%%%%%o o%%%%%o o%%%%%o o o%%%%%o o
/ \%%%/ \%%%/ \%%%/ \ / \%%%/%\ /%\%%%/ \ / \ /%\%%%/%\ / \
/ \%/ \%/ \%/ \ / \%/%%%\ /%%%\%/ \ / \ /%%%\%/%%%\ / \
o o o 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)
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators 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
W
Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators
Item
Notation
Description
Type
U •
= [u , v ]
Source Universe
[B 2 ]
X •
= [x ]
Target Universe
[B 1 ]
EU •
= [u , v , du , dv ]
Extended Source Universe
[B 2 × D 2 ]
EX •
= [x , dx ]
Extended Target Universe
[B 1 × D 1 ]
J
J : U → B
Proposition
(B 2 → B ) ∈ [B 2 ]
J
J : U • → X •
Transformation, or Mapping
[B 2 ] → [B 1 ]
A
{a 1 , …, a n
Alphabet
[n ] = n
A i {(a i a i
Dimension i
B
A
〈A 〉a 1 , …, a n a 1 , …, a n A 1 × … × A n i A i
Set of cells,
B n
A *
(hom : A → B )
Linear functions
(B n B n
A ^
(A → B )
Boolean functions
B n B
A •
[A ]A , A ^)A +→ B )A , (A → B ))a 1 , …, a n
Universe of discoursea 1 , …, a n
(B n B n B ))B n B )B n
| 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
| !e! | | Tacit Extension Operator !e!
| !h! | | Trope Extension Operator !h!
| E | | Enlargement Operator E
| D | | Difference Operator D
| d | | Differential Operator d
-------------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
| $e$ || Radius Operator $e$ = <!e!, !h!>
| $E$ || Secant Operator $E$ = <!e!, E >
| $D$ || Chord Operator $D$ = <!e!, D >
| $T$ || Tangent Functor $T$ = <!e!, d >
-------------o
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes 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
Figure 56-a1. Radius Map of the Conjunction J = uv 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
Figure 56-a2. Secant Map of the Conjunction J = uv 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
Figure 56-a3. Chord Map of the Conjunction J = uv 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
Figure 56-a4. Tangent Map of the Conjunction J = uv 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
Figure 56-b1. Radius Map of the Conjunction J = uv 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
Figure 56-b2. Secant Map of the Conjunction J = uv 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
Figure 56-b3. Chord Map of the Conjunction J = uv 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
Figure 56-b4. Tangent Map of the Conjunction J = uv 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
Figure 57-1. Radius Operator Diagram for the Conjunction J = uv 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
Figure 57-2. Secant Operator Diagram for the Conjunction J = uv 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
Figure 57-3. Chord Operator Diagram for the Conjunction J = uv 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
Figure 57-4. Tangent Functor Diagram for the Conjunction J = uv 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
Formula Display 11 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
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 ]
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 ]
Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators 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
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes 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
Formula Display 12 o-----------------------------------------------------------o
| |
| x = f(u, v) = ((u)(v)) |
| |
| y = g(u, v) = ((u, v)) |
| |
o-----------------------------------------------------------o
x
=
f ‹u , v ›
=
((u )(v ))
y
=
g ‹u , v ›
=
((u , v ))
Formula Display 13 o-----------------------------------------------------------o
| |
| <x, y> = F<u, v> = <((u)(v)), ((u, v))> |
| |
o-----------------------------------------------------------o
‹x , y ›
=
F ‹u , v ›
=
‹((u )(v )), ((u , v ))›
‹x , y ›
=
F ‹u , v ›
=
‹((u )(v )), ((u , v ))›
Table 60. Propositional Transformation 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
Figure 61. Propositional Transformation 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
Figure 62. Propositional Transformation (Short Form) 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)
Figure 63. Transformation of Positions 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
Table 64. Transformation of Positions 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
Table 65. Induced Transformation on Propositions 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
Formula Display 14 o-------------------------------------------------o
| |
| EG_i = G_i <u + du, v + dv> |
| |
o-------------------------------------------------o
Formula Display 15 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
DG i
=
G i u , v ›
+
EG i u , v , du , dv ›
=
G i u , v ›
+
G i u + du , v + dv ›
Formula Display 16 o-------------------------------------------------o
| |
| Ef = ((u + du)(v + dv)) |
| |
| Eg = ((u + du, v + dv)) |
| |
o-------------------------------------------------o
Ef
=
((u + du )(v + dv ))
Eg
=
((u + du , v + dv ))
Formula Display 17 o-------------------------------------------------o
| |
| Df = ((u)(v)) + ((u + du)(v + dv)) |
| |
| Dg = ((u, v)) + ((u + du, v + dv)) |
| |
o-------------------------------------------------o
Df
=
((u )(v ))
+
((u + du )(v + dv ))
Dg
=
((u , v ))
+
((u + du , v + dv ))
Table 66-i. Computation Summary for f‹u, v› = ((u)(v)) 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
Table 66-i. Computation Summary for f ‹u , v › = ((u )(v ))
\(\epsilon\)f
=
uv \(\cdot\)
1
+
u (v )\(\cdot\)
1
+
(u )v
\(\cdot\)
1
+
(u )(v )
\(\cdot\)
0
Ef
=
uv \(\cdot\)
(du dv )
+
u (v )\(\cdot\)
(du (d v))
+
(u )v
\(\cdot\)
((du ) dv )
+
(u )(v )
\(\cdot\)
((du )(dv ))
Df
=
uv \(\cdot\)
du dv
+
u (v )\(\cdot\)
du (dv )
+
(u )v
\(\cdot\)
(du ) dv
+
(u )(v )
\(\cdot\)
((du )(dv ))
df
=
uv \(\cdot\)
0
+
u (v )\(\cdot\)
du
+
(u )v
\(\cdot\)
dv
+
(u )(v )
\(\cdot\)
(du , dv )
rf
=
uv \(\cdot\)
du dv
+
u (v )\(\cdot\)
du dv
+
(u )v
\(\cdot\)
du dv
+
(u )(v )
\(\cdot\)
du dv
Table 66-ii. Computation Summary for g‹u, v› = ((u, v)) 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
Table 66-ii. Computation Summary for g‹u , v › = ((u , v ))
\(\epsilon\)g
=
uv \(\cdot\)
1
+
u (v )\(\cdot\)
0
+
(u )v
\(\cdot\)
0
+
(u )(v )
\(\cdot\)
1
Eg
=
uv \(\cdot\)
((du , dv ))
+
u (v )\(\cdot\)
(du , dv )
+
(u )v
\(\cdot\)
(du , dv )
+
(u )(v )
\(\cdot\)
((du , dv ))
Dg
=
uv \(\cdot\)
(du , dv )
+
u (v )\(\cdot\)
(du , dv )
+
(u )v
\(\cdot\)
(du , dv )
+
(u )(v )
\(\cdot\)
(du , dv )
dg
=
uv \(\cdot\)
(du , dv )
+
u (v )\(\cdot\)
(du , dv )
+
(u )v
\(\cdot\)
(du , dv )
+
(u )(v )
\(\cdot\)
(du , dv )
rg
=
uv \(\cdot\)
0
+
u (v )\(\cdot\)
0
+
(u )v
\(\cdot\)
0
+
(u )(v )
\(\cdot\)
0
Table 67. Computation of an Analytic Series in Terms of Coordinates 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
Table 68. Computation of an Analytic Series in Symbolic Terms 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
Formula Display 18 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
Df
=
uv \(\cdot\)
du dv
+
u (v )\(\cdot\)
du (dv )
+
(u )v
\(\cdot\)
(du ) dv
+
(u )(v )
\(\cdot\)
((du )(dv ))
Dg
=
uv \(\cdot\)
(du , dv )
+
u (v )\(\cdot\)
(du , dv )
+
(u )v
\(\cdot\)
(du , dv )
+
(u )(v )
\(\cdot\)
(du , dv )
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 |
| |
| |
| |
o-------------------------------------------------------------------------o
Figure 69. Difference Map of F = <f, g> = <((u)(v)), ((u, v))>
Formula Display 19 o-------------------------------------------------------------------------------o
| |
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v).(du, dv) |
| |
| dg = uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv) |
| |
o-------------------------------------------------------------------------------o
df
=
uv \(\cdot\)
0
+
u (v )\(\cdot\)
du
+
(u )v
\(\cdot\)
dv
+
(u )(v )
\(\cdot\)
(du , dv )
dg
=
uv \(\cdot\)
(du , dv )
+
u (v )\(\cdot\)
(du , dv )
+
(u )v
\(\cdot\)
(du , dv )
+
(u )(v )
\(\cdot\)
(du , dv )
Figure 70-a. Tangent Functor Diagram for F‹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 /@ o
|\ / \ /| |\ / \ / / \ / / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @\ o /@ o |
| \ / \ / | | |\ / \ / \ / \ / \ /| |
| \ / \ / | | | \ / \ / \ / | |
| u \ / O \ / v | | u | \ / O \ / O \ / | v |
o-------o @\ o-------o o---+---o @\ o @\ o---+---o
\ / | \ / \ / \ / \ / |
\ / | \ / \ / |
\ / | du \ / O \ / dv |
\ / o-------o @\ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
U% $T$ $E$U%
o------------------>o
| |
| |
| |
| |
F | | $T$F
| |
| |
| |
v v
o------------------>o
X% $T$ $E$X%
o o
/ \ / \
/ \ / \
/ \ / O \
/ \ o /@\ o
/ \ / \ / \
/ \ / \ / \
/ O \ / O \ / O \
o /@\ o o /@\ o /@\ o
/ \ / \ / \ \ / \ / / \
/ \ / \ / \ / \ / \
/ \ / \ / O \ / O \ / O \
/ \ / \ o /@ o /@\ o @\ o
/ \ / \ / \ \ / \ / \ / \ / / \
/ \ / \ / \ / \ / \ / \
/ O \ / O \ / O \ / O \ / O \ / O \
o /@ o @\ o o /@ o /@ o @\ o @\ o
|\ / \ /| |\ / \ / \ / \ / \ / \ /|
| \ / \ / | | \ / \ / \ / \ / |
| \ / \ / | | \ / O \ / O \ / O \ / |
| \ / \ / | | o /@ o @ o @\ o |
| \ / \ / | | |\ / / \ / \ / \ \ /| |
| \ / \ / | | | \ / \ / \ / | |
| x \ / O \ / y | | x | \ / O \ / O \ / | y |
o-------o @ o-------o o---+---o @ o @ o---+---o
\ / | \ / / \ \ / |
\ / | \ / \ / |
\ / | dx \ / O \ / dy |
\ / o-------o @ o-------o
\ / \ /
\ / \ /
\ / \ /
o o
Figure 70-a. Tangent Functor Diagram for F‹u, v› = <((u)(v)), ((u, v))>
Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))› o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ /////\ | | /XXXX\ /XXXX\ | | /\\\\\ /\\\\\ |
| ///////o//////\ | | /XXXXXXoXXXXXX\ | | /\\\\\\o\\\\\\\ |
| //////// \//////\ | | /XXXXXX/ \XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o/////// \//////o | | oXXXXXX/ \XXXXXXo | | o\\\\\\/ \\\\\\\o |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| |/du//| |//dv/| | | |XXXXX| |XXXXX| | | |\du\\| |\\dv\| |
| |/////o o/////| | | |XXXXXo oXXXXX| | | |\\\\\o o\\\\\| |
| o//////\ ///////o | | oXXXXXX\ /XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \//////\ //////// | | \XXXXXX\ /XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \//////o/////// | | \XXXXXXoXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \///// \///// | | \XXXX/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u)(v) o-----------------------o dv' @ (u)(v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| / \ /////\ | | /\\\\\ /XXXX\ | | /\\\\\ /\\\\\ |
| / o//////\ | | /\\\\\\oXXXXXX\ | | /\\\\\\o\\\\\\\ |
| / //\//////\ | | /\\\\\\//\XXXXXX\ | | /\\\\\\/ \\\\\\\\ |
| o ////\//////o | | o\\\\\\////\XXXXXXo | | o\\\\\\/ \\\\\\\o |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| | du |/////|//dv/| | | |\\\\\|/////|XXXXX| | | |\du\\| |\\dv\| |
| | o/////o/////| | | |\\\\\o/////oXXXXX| | | |\\\\\o o\\\\\| |
| o \//////////o | | o\\\\\\\////XXXXXXo | | o\\\\\\\ /\\\\\\o |
| \ \///////// | | \\\\\\\\//XXXXXX/ | | \\\\\\\\ /\\\\\\/ |
| \ o/////// | | \\\\\\\oXXXXXX/ | | \\\\\\\o\\\\\\/ |
| \ / \///// | | \\\\\/ \XXXX/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ (u) v o-----------------------o dv' @ (u) v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--o | | o--o o--o | | o--o o--o |
| /////\ / \ | | /XXXX\ /\\\\\ | | /\\\\\ /\\\\\ |
| ///////o \ | | /XXXXXXo\\\\\\\ | | /\\\\\\o\\\\\\\ |
| /////////\ \ | | /XXXXXX//\\\\\\\\ | | /\\\\\\/ \\\\\\\\ |
| o//////////\ o | | oXXXXXX////\\\\\\\o | | o\\\\\\/ \\\\\\\o |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| |/du//|/////| dv | | | |XXXXX|/////|\\\\\| | | |\du\\| |\\dv\| |
| |/////o/////o | | | |XXXXXo/////o\\\\\| | | |\\\\\o o\\\\\| |
| o//////\//// o | | oXXXXXX\////\\\\\\o | | o\\\\\\\ /\\\\\\o |
| \//////\// / | | \XXXXXX\//\\\\\\/ | | \\\\\\\\ /\\\\\\/ |
| \//////o / | | \XXXXXXo\\\\\\/ | | \\\\\\\o\\\\\\/ |
| \///// \ / | | \XXXX/ \\\\\/ | | \\\\\/ \\\\\/ |
| o--o o--o | | o--o o--o | | o--o o--o |
| | | | | |
o-----------------------o o-----------------------o o-----------------------o
= du' @ u (v) o-----------------------o dv' @ u (v) =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
o-----------------------o o-----------------------o o-----------------------o
| dU | | dU | | dU |
| o--o o--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\| |
| | o o | | | |\\\\\o o\\\\\| | | |\\\\\o o\\\\\| |
| o \ / o | | o\\\\\\\ /\\\\\\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' @ u v o-----------------------o dv' @ u v =
= | dU' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/du'/|XXXXX|\dv'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////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 \\\\\|
| /////////\//////\ | |\\\////////X\//////\\\\| |\\\/ /\\ \\\\|
| o//////////\//////o | |\\o///////XXX\//////o\\| |\\o /\\\\ o\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| |//u//|/////|//v//| | |\\|//u//|XXXXX|//v//|\\| |\\| u |\\\\\| v |\\|
| |/////o/////o/////| | |\\|/////oXXXXXo/////|\\| |\\| o\\\\\o |\\|
| o//////\//////////o | |\\o//////\XXX///////o\\| |\\o \\\\/ o\\|
| \//////\///////// | |\\\\//////\X////////\\\| |\\\\ \\/ /\\\|
| \//////o/////// | |\\\\\//////o///////\\\\| |\\\\\ o /\\\\|
| \///// \///// | |\\\\\\/////\\/////\\\\\| |\\\\\\ /\\ /\\\\\|
| o--o o--o | |\\\\\\o--o\\\o--o\\\\\\| |\\\\\\o--o\\\o--o\\\\\\|
| | |\\\\\\\\\\\\\\\\\\\\\\\| |\\\\\\\\\\\\\\\\\\\\\\\|
o-----------------------o o-----------------------o o-----------------------o
= u' o-----------------------o v' =
= | U' | =
= | o--o o--o | =
= | /////\ /\\\\\ | =
= | ///////o\\\\\\\ | =
= | ////////X\\\\\\\\ | =
= | o///////XXX\\\\\\\o | =
= | |/////oXXXXXo\\\\\| | =
= = = = = = = = = = =|/u'//|XXXXX|\\v'\|= = = = = = = = = = =
| |/////oXXXXXo\\\\\| |
| o//////\XXX/\\\\\\o |
| \//////\X/\\\\\\/ |
| \//////o\\\\\\/ |
| \///// \\\\\/ |
| o--o o--o |
| |
o-----------------------o
Figure 70-b. Tangent Functor Ferris Wheel for F<u, v> = <((u)(v)), ((u, v))>
