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
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
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
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
φ01
φ02
φ03
φ04
φ05
φ06
φ07
φ08
φ09
φ10
φ11
φ12
φ13
φ14
φ15 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
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
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 ]
W :
U • → EU • ,
X • → EX • ,
(U • → X • )
→
(EU • → EX • ) ,
for each W in the set:
{\(\epsilon\), \(\eta\), E, D, d}
[B 2 ] → [B 2 × D 2 ] ,
[B 1 ] → [B 1 × D 1 ] ,
([B 2 ] → [B 1 ])
→
([B 2 × D 2 ] → [B 1 × D 1 ])
\(\epsilon\)
\(\eta\)
E
D
d
Tacit Extension Operator
\(\epsilon\)
Trope Extension Operator
\(\eta\)
Enlargement Operator
E
Difference Operator
D
Differential Operator
d
W U • → T U • = EU • ,
X • → T X • = EX • ,
(U • → X • )
→
(T U • → T X • ) ,
for each W {e E D T
[B 2 ] → [B 2 × D 2 ] ,
[B 1 ] → [B 1 × D 1 ] ,
([B 2 ] → [B 1 ])
→
([B 2 × D 2 ] → [B 1 × D 1 ])
Radius Operator
e Secant Operator
E Chord Operator
D Tangent Functor
T
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
Table 55. Synopsis of Terminology: Restrictive and Alternative Subtypes
Operator
Proposition
Map
\(\epsilon\) :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → X • )
\(\epsilon\)J :
〈u , v , du , dv 〉 → B
B 2 × D 2 → B
\(\epsilon\)J :
[u , v , du , dv ] → [x ]
[B 2 × D 2 ] → [B 1 ]
\(\eta\) :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
\(\eta\)J :
〈u , v , du , dv 〉 → D
B 2 × D 2 → D
\(\eta\)J :
[u , v , du , dv ] → [dx ]
[B 2 × D 2 ] → [D 1 ]
E :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
EJ :
〈u , v , du , dv 〉 → D
B 2 × D 2 → D
EJ :
[u , v , du , dv ] → [dx ]
[B 2 × D 2 ] → [D 1 ]
D :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
DJ :
〈u , v , du , dv 〉 → D
B 2 × D 2 → D
DJ :
[u , v , du , dv ] → [dx ]
[B 2 × D 2 ] → [D 1 ]
d :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
dJ :
〈u , v , du , dv 〉 → D
B 2 × D 2 → D
dJ :
[u , v , du , dv ] → [dx ]
[B 2 × D 2 ] → [D 1 ]
r :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
rJ :
〈u , v , du , dv 〉 → D
B 2 × D 2 → D
rJ :
[u , v , du , dv ] → [dx ]
[B 2 × D 2 ] → [D 1 ]
e U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → EX • )
e J :
[u , v , du , dv ] → [x , dx ]
[B 2 × D 2 ] → [B × D ]
E U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → EX • )
E J :
[u , v , du , dv ] → [x , dx ]
[B 2 × D 2 ] → [B × D ]
D U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → EX • )
D J :
[u , v , du , dv ] → [x , dx ]
[B 2 × D 2 ] → [B × D ]
T U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → EX • )
dJ :
〈u , v , du , dv 〉 → D
B 2 × D 2 → D
T J :
[u , v , du , dv ] → [x , dx ]
[B 2 × D 2 ] → [B × D ]
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 58. Cast of Characters: Expansive Subtypes of Objects and Operators
Item
Notation
Description
Type
U • = [u , v ]
Source Universe
[B n X •
Target Universe
[B k EU •
= [u , v , du , dv ]
Extended Source Universe
[B n D n EX •
= [x , y , dx , dy ]
= [f , g , df , dg ]
Extended Target Universe
[B k D k F
F = ‹f , g › : U • → X • Transformation, or Mapping
[B n B k
f , g : U → B
f : U → [x ] ⊆ X • g : U → [y ] ⊆ X •
B n B
∈ (B n B n B )
= (B n B ) = [B n
W :
U • → EU • ,
X • → EX • ,
(U • → X • )
→
(EU • → EX • ) ,
for each W in the set:
{\(\epsilon\), \(\eta\), E, D, d}
[B n B n D n [B k B k D k ([B n B k →
([B n D n B k D k
\(\epsilon\)
\(\eta\)
E
D
d
Tacit Extension Operator
\(\epsilon\)
Trope Extension Operator
\(\eta\)
Enlargement Operator
E
Difference Operator
D
Differential Operator
d
W U • → T U • = EU • ,
X • → T X • = EX • ,
(U • → X • )
→
(T U • → T X • ) ,
for each W {e E D T
[B n B n D n [B k B k D k ([B n B k →
([B n D n B k D k
Radius Operator
e Secant Operator
E Chord Operator
D Tangent Functor
T
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
Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
Operator
Proposition
Transformation
Operand
F = ‹F 1 , F 2 ›
F = ‹f , g › : U → X
F i u , v 〉 → B
F i B n B
F : [u , v ] → [x , y ]
F : B n B k
\(\epsilon\) :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → X • )
\(\epsilon\)F i 〈u , v , du , dv 〉 → B
B n D n B
\(\epsilon\)F :
[u , v , du , dv ] → [x , y ]
[B n D n B k
\(\eta\) :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
\(\eta\)F i 〈u , v , du , dv 〉 → D
B n D n D
\(\eta\)F :
[u , v , du , dv ] → [dx , dy ]
[B n D n D k
E :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
EF i 〈u , v , du , dv 〉 → D
B n D n D
EF :
[u , v , du , dv ] → [dx , dy ]
[B n D n D k
D :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
DF i 〈u , v , du , dv 〉 → D
B n D n D
DF :
[u , v , du , dv ] → [dx , dy ]
[B n D n D k
d :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
dF i 〈u , v , du , dv 〉 → D
B n D n D
dF :
[u , v , du , dv ] → [dx , dy ]
[B n D n D k
r :
U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → dX • )
rF i 〈u , v , du , dv 〉 → D
B n D n D
rF :
[u , v , du , dv ] → [dx , dy ]
[B n D n D k
e U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → EX • )
e F :
[u , v , du , dv ] → [x , y , dx , dy ]
[B n D n B k D k
E U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → EX • )
E F :
[u , v , du , dv ] → [x , y , dx , dy ]
[B n D n B k D k
D U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → EX • )
D F :
[u , v , du , dv ] → [x , y , dx , dy ]
[B n D n B k D k
T U • → EU • , X • → EX • ,
(U • → X • ) → (EU • → EX • )
dF i 〈u , v , du , dv 〉 → D
B n D n D
T F :
[u , v , du , dv ] → [x , y , dx , dy ]
[B n D n B k D k
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
Table 60. Propositional Transformation
u
v
f
g
((u )(v ))
((u , v ))
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 64. Transformation of Positions
u v
x
y
x y
x (y )
(x ) y
(x )(y )
X • = [x , y ]
((u )(v ))
((u , v ))
u v
(u , v )
(u )(v )
( )
U • = [u , v ]
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
Table 65. Induced Transformation on Propositions
X • ← F = ‹f , g › ←
U • f i x , y ›
f j u , v ›
()
(x)(y)
(x) y
(x)
x (y)
(y)
(x, y)
(x y)
0 0 0 0
0 0 0 1
0 0 1 0
0 0 1 1
0 1 0 0
0 1 0 1
0 1 1 0
0 1 1 1
()
()
(u)(v)
(u)(v)
(u, v)
(u, v)
(u v)
(u v)
x y
((x, y))
y
(x (y))
x
((x) y)
((x)(y))
(())
1 0 0 0
1 0 0 1
1 0 1 0
1 0 1 1
1 1 0 0
1 1 0 1
1 1 1 0
1 1 1 1
u v
u v
((u, v))
((u, v))
((u)(v))
((u)(v))
(())
(())
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 67. Computation of an Analytic Series in Terms of Coordinates
\(\epsilon\)f
\(\epsilon\)g
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 | rg |
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
Table 68. Computation of an Analytic Series in Symbolic Terms
u v
f g
Df
Dg
df
dg
d2 f
d2 g
((du )(dv ))
(du ) dv
du (dv )
du dv
(du , dv )
(du , dv )
(du , dv )
(du , dv )
(du , dv )
(du , dv )
(du , dv )
(du , dv )
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))>
Figure 70-b. Tangent Functor Ferris Wheel for F‹u, v› = ‹((u)(v)), ((u, v))›
