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ONT Re: Differential Logic


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DLOG.  Note D70

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Operator Diagrams 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

            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

            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

            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

Jon Awbrey

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