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Directory:Jon Awbrey/Papers/Differential Analytic Turing Automata (view source)

Revision as of 13:38, 10 March 2009

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Figure 1.1.  f = ((u)(v))
 
Figure 1.1.  f = ((u)(v))
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Figure 1.2.  Ef = ((u + du)(v + dv))
 
Figure 1.2.  Ef = ((u + du)(v + dv))
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Figure 1.3.  Df = f + Ef
 
Figure 1.3.  Df = f + Ef
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<pre>
 
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Figure 2.1.  g = ((u, v))
 
Figure 2.1.  g = ((u, v))
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|```````````````o%%%%%%%o```````````````|
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|``````````````/`\%%%%%/`\``````````````|
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|```````````o```````o```````o```````````|
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|           o       o       o           |
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|   o       o       o       o       o   |
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|   |\     /%\     /%\     /%\     /|   |
−
|```|`\```/%%%\```/%%%\```/%%%\```/`|```|
+
|   | \   /%%%\   /%%%\   /%%%\   / |   |
−
|```|``\`/%%%%%\`/%%%%%\`/%%%%%\`/``|```|
+
|   | \ /%%%%%\ /%%%%%\ /%%%%%\ / |   |
−
|```|```o%%%%%%%o%%%%%%%o%%%%%%%o```|```|
+
|   |   o%%%%%%%o%%%%%%%o%%%%%%%o   |   |
−
|```|```|\%%%%%/`\%%%%%/`\%%%%%/|```|```|
+
|   |   |\%%%%%/ \%%%%%/ \%%%%%/|   |   |
−
|```|```|`\%%%/```\%%%/```\%%%/`|```|```|
+
|   |   | \%%%/   \%%%/   \%%%/ |   |   |
−
|```|`u`|``\%/`````\%/`````\%/``|`v`|```|
+
|   | u | \%/     \%/     \%/ | v |   |
−
|```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
 
Figure 2.2.  Eg = ((u + du, v + dv))
 
Figure 2.2.  Eg = ((u + du, v + dv))
Line 931: Line 931:  
<pre>
 
<pre>
 
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/`\``````````````````|
+
|                 / \                 |
−
|`````````````````/```\`````````````````|
+
|                 /   \                 |
−
|````````````````/`````\````````````````|
+
|               /     \               |
−
|```````````````o```````o```````````````|
+
|               o       o               |
−
|``````````````/%\`````/%\``````````````|
+
|             /%\     /%\             |
−
|`````````````/%%%\```/%%%\`````````````|
+
|             /%%%\   /%%%\             |
−
|````````````/%%%%%\`/%%%%%\````````````|
+
|           /%%%%%\ /%%%%%\           |
−
|```````````o%%%%%%%o%%%%%%%o```````````|
+
|           o%%%%%%%o%%%%%%%o           |
−
|``````````/%\%%%%%/`\%%%%%/%\``````````|
+
|         /%\%%%%%/ \%%%%%/%\         |
−
|`````````/%%%\%%%/```\%%%/%%%\`````````|
+
|         /%%%\%%%/   \%%%/%%%\         |
−
|````````/%%%%%\%/`````\%/%%%%%\````````|
+
|       /%%%%%\%/     \%/%%%%%\       |
−
|```````o%%%%%%%o```````o%%%%%%%o```````|
+
|       o%%%%%%%o       o%%%%%%%o       |
−
|``````/`\%%%%%/`\`````/`\%%%%%/`\``````|
+
|     / \%%%%%/ \     / \%%%%%/ \     |
−
|`````/```\%%%/```\```/```\%%%/```\`````|
+
|     /   \%%%/   \   /   \%%%/   \     |
−
|````/`````\%/`````\`/`````\%/`````\````|
+
|   /     \%/     \ /     \%/     \   |
−
|```o```````o```````o```````o```````o```|
+
|   o       o       o       o       o   |
−
|```|\`````/%\`````/`\`````/%\`````/|```|
+
|   |\     /%\     / \     /%\     /|   |
−
|```|`\```/%%%\```/```\```/%%%\```/`|```|
+
|   | \   /%%%\   /   \   /%%%\   / |   |
−
|```|``\`/%%%%%\`/`````\`/%%%%%\`/``|```|
+
|   | \ /%%%%%\ /     \ /%%%%%\ / |   |
−
|```|```o%%%%%%%o```````o%%%%%%%o```|```|
+
|   |   o%%%%%%%o       o%%%%%%%o   |   |
−
|```|```|\%%%%%/%\`````/%\%%%%%/|```|```|
+
|   |   |\%%%%%/%\     /%\%%%%%/|   |   |
−
|```|```|`\%%%/%%%\```/%%%\%%%/`|```|```|
+
|   |   | \%%%/%%%\   /%%%\%%%/ |   |   |
−
|```|`u`|``\%/%%%%%\`/%%%%%\%/``|`v`|```|
+
|   | u | \%/%%%%%\ /%%%%%\%/ | v |   |
−
|```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
 
Figure 2.3.  Dg = g + Eg
 
Figure 2.3.  Dg = g + Eg
Line 1,063: Line 1,063:     
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/`\``````````````````|
+
|                 / \                 |
−
|`````````````````/```\`````````````````|
+
|                 /   \                 |
−
|````````````````/`````\````````````````|
+
|               /     \               |
−
|```````````````o```````o```````````````|
+
|               o       o               |
−
|``````````````/`\`````/`\``````````````|
+
|             / \     / \             |
−
|`````````````/```\```/```\`````````````|
+
|             /   \   /   \             |
−
|````````````/`````\`/`````\````````````|
+
|           /     \ /     \           |
−
|```````````o```````o```````o```````````|
+
|           o       o       o           |
−
|``````````/`\`````/`\`````/`\``````````|
+
|         / \     / \     / \         |
−
|`````````/```\```/```\```/```\`````````|
+
|         /   \   /   \   /   \         |
−
|````````/`````\`/`````\`/`````\````````|
+
|       /     \ /     \ /     \       |
−
|```````o```````o```````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   |
−
|```````|````\%%%%%/`\%%%%%/````|```````|
+
|       |   \%%%%%/ \%%%%%/   |       |
−
|```````|`````\%%%/```\%%%/`````|```````|
+
|       |     \%%%/   \%%%/     |       |
−
|```````|`du```\%/`````\%/```dv`|```````|
+
|       | du   \%/     \%/   dv |       |
−
|```````o-------o```````o-------o```````|
+
|       o-------o       o-------o       |
−
|````````````````\`````/````````````````|
+
|               \     /               |
−
|`````````````````\```/`````````````````|
+
|                 \   /                 |
−
|``````````````````\`/``````````````````|
+
|                 \ /                 |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|```````````````````````````````````````|
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
 
Figure 1.4.  df = linear approx to Df
 
Figure 1.4.  df = linear approx to Df
Line 1,110: Line 1,110:     
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/`\``````````````````|
+
|                 / \                 |
−
|`````````````````/```\`````````````````|
+
|                 /   \                 |
−
|````````````````/`````\````````````````|
+
|               /     \               |
−
|```````````````o```````o```````````````|
+
|               o       o               |
−
|``````````````/%\`````/%\``````````````|
+
|             /%\     /%\             |
−
|`````````````/%%%\```/%%%\`````````````|
+
|             /%%%\   /%%%\             |
−
|````````````/%%%%%\`/%%%%%\````````````|
+
|           /%%%%%\ /%%%%%\           |
−
|```````````o%%%%%%%o%%%%%%%o```````````|
+
|           o%%%%%%%o%%%%%%%o           |
−
|``````````/%\%%%%%/`\%%%%%/%\``````````|
+
|         /%\%%%%%/ \%%%%%/%\         |
−
|`````````/%%%\%%%/```\%%%/%%%\`````````|
+
|         /%%%\%%%/   \%%%/%%%\         |
−
|````````/%%%%%\%/`````\%/%%%%%\````````|
+
|       /%%%%%\%/     \%/%%%%%\       |
−
|```````o%%%%%%%o```````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   |
−
|```````|````\%%%%%/`\%%%%%/````|```````|
+
|       |   \%%%%%/ \%%%%%/   |       |
−
|```````|`````\%%%/```\%%%/`````|```````|
+
|       |     \%%%/   \%%%/     |       |
−
|```````|`du```\%/`````\%/```dv`|```````|
+
|       | du   \%/     \%/   dv |       |
−
|```````o-------o```````o-------o```````|
+
|       o-------o       o-------o       |
−
|````````````````\`````/````````````````|
+
|               \     /               |
−
|`````````````````\```/`````````````````|
+
|                 \   /                 |
−
|``````````````````\`/``````````````````|
+
|                 \ /                 |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|```````````````````````````````````````|
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
 
Figure 2.4.  dg = linear approx to Dg
 
Figure 2.4.  dg = linear approx to Dg
Line 1,246: Line 1,246:     
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/%\``````````````````|
+
|                 /%\                 |
−
|`````````````````/%%%\`````````````````|
+
|                 /%%%\                 |
−
|````````````````/%%%%%\````````````````|
+
|               /%%%%%\               |
−
|```````````````o%%%%%%%o```````````````|
+
|               o%%%%%%%o               |
−
|``````````````/%\%%%%%/%\``````````````|
+
|             /%\%%%%%/%\             |
−
|`````````````/%%%\%%%/%%%\`````````````|
+
|             /%%%\%%%/%%%\             |
−
|````````````/%%%%%\%/%%%%%\````````````|
+
|           /%%%%%\%/%%%%%\           |
−
|```````````o%%%%%%%o%%%%%%%o```````````|
+
|           o%%%%%%%o%%%%%%%o           |
−
|``````````/%\%%%%%/%\%%%%%/%\``````````|
+
|         /%\%%%%%/%\%%%%%/%\         |
−
|`````````/%%%\%%%/%%%\%%%/%%%\`````````|
+
|         /%%%\%%%/%%%\%%%/%%%\         |
−
|````````/%%%%%\%/%%%%%\%/%%%%%\````````|
+
|       /%%%%%\%/%%%%%\%/%%%%%\       |
−
|```````o%%%%%%%o%%%%%%%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   |
−
|```````|````\`````/`\`````/````|```````|
+
|       |   \     / \     /   |       |
−
|```````|`````\```/```\```/`````|```````|
+
|       |     \   /   \   /     |       |
−
|```````|`du```\`/`````\`/```dv`|```````|
+
|       | du   \ /     \ /   dv |       |
−
|```````o-------o```````o-------o```````|
+
|       o-------o       o-------o       |
−
|````````````````\`````/````````````````|
+
|               \     /               |
−
|`````````````````\```/`````````````````|
+
|                 \   /                 |
−
|``````````````````\`/``````````````````|
+
|                 \ /                 |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|```````````````````````````````````````|
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
 
Figure 1.1.  f = ((u)(v))
 
Figure 1.1.  f = ((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       o   |   |
−
|```|```|\`````/%\%%%%%/%\`````/|```|```|
+
|   |   |\     /%\%%%%%/%\     /|   |   |
−
|```|```|`\```/%%%\%%%/%%%\```/`|```|```|
+
|   |   | \   /%%%\%%%/%%%\   / |   |   |
−
|```|`u`|``\`/%%%%%\%/%%%%%\`/``|`v`|```|
+
|   | u | \ /%%%%%\%/%%%%%\ / | v |   |
−
|```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
 
Figure 1.2.  Ef = ((u + du)(v + dv))
 
Figure 1.2.  Ef = ((u + du)(v + 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   |   |
−
|```|```|\%%%%%/%\%%%%%/%\%%%%%/|```|```|
+
|   |   |\%%%%%/%\%%%%%/%\%%%%%/|   |   |
−
|```|```|`\%%%/%%%\%%%/%%%\%%%/`|```|```|
+
|   |   | \%%%/%%%\%%%/%%%\%%%/ |   |   |
−
|```|`u`|``\%/%%%%%\%/%%%%%\%/``|`v`|```|
+
|   | u | \%/%%%%%\%/%%%%%\%/ | v |   |
−
|```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
 
Figure 1.3.  Difference Map Df = f + Ef
 
Figure 1.3.  Difference Map Df = f + Ef
    
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/`\``````````````````|
+
|                 / \                 |
−
|`````````````````/```\`````````````````|
+
|                 /   \                 |
−
|````````````````/`````\````````````````|
+
|               /     \               |
−
|```````````````o```````o```````````````|
+
|               o       o               |
−
|``````````````/`\`````/`\``````````````|
+
|             / \     / \             |
−
|`````````````/```\```/```\`````````````|
+
|             /   \   /   \             |
−
|````````````/`````\`/`````\````````````|
+
|           /     \ /     \           |
−
|```````````o```````o```````o```````````|
+
|           o       o       o           |
−
|``````````/`\`````/`\`````/`\``````````|
+
|         / \     / \     / \         |
−
|`````````/```\```/```\```/```\`````````|
+
|         /   \   /   \   /   \         |
−
|````````/`````\`/`````\`/`````\````````|
+
|       /     \ /     \ /     \       |
−
|```````o```````o```````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   |
−
|```````|````\%%%%%/`\%%%%%/````|```````|
+
|       |   \%%%%%/ \%%%%%/   |       |
−
|```````|`````\%%%/```\%%%/`````|```````|
+
|       |     \%%%/   \%%%/     |       |
−
|```````|`du```\%/`````\%/```dv`|```````|
+
|       | du   \%/     \%/   dv |       |
−
|```````o-------o```````o-------o```````|
+
|       o-------o       o-------o       |
−
|````````````````\`````/````````````````|
+
|               \     /               |
−
|`````````````````\```/`````````````````|
+
|                 \   /                 |
−
|``````````````````\`/``````````````````|
+
|                 \ /                 |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|```````````````````````````````````````|
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
 
Figure 1.4.  Linear Proxy df for Df
 
Figure 1.4.  Linear Proxy df for Df
    
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/`\``````````````````|
+
|                 / \                 |
−
|`````````````````/```\`````````````````|
+
|                 /   \                 |
−
|````````````````/`````\````````````````|
+
|               /     \               |
−
|```````````````o```````o```````````````|
+
|               o       o               |
−
|``````````````/`\`````/`\``````````````|
+
|             / \     / \             |
−
|`````````````/```\```/```\`````````````|
+
|             /   \   /   \             |
−
|````````````/`````\`/`````\````````````|
+
|           /     \ /     \           |
−
|```````````o```````o```````o```````````|
+
|           o       o       o           |
−
|``````````/`\`````/%\`````/`\``````````|
+
|         / \     /%\     / \         |
−
|`````````/```\```/%%%\```/```\`````````|
+
|         /   \   /%%%\   /   \         |
−
|````````/`````\`/%%%%%\`/`````\````````|
+
|       /     \ /%%%%%\ /     \       |
−
|```````o```````o%%%%%%%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   |
−
|```````|````\`````/`\`````/````|```````|
+
|       |   \     / \     /   |       |
−
|```````|`````\```/```\```/`````|```````|
+
|       |     \   /   \   /     |       |
−
|```````|`du```\`/`````\`/```dv`|```````|
+
|       | du   \ /     \ /   dv |       |
−
|```````o-------o```````o-------o```````|
+
|       o-------o       o-------o       |
−
|````````````````\`````/````````````````|
+
|               \     /               |
−
|`````````````````\```/`````````````````|
+
|                 \   /                 |
−
|``````````````````\`/``````````````````|
+
|                 \ /                 |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|```````````````````````````````````````|
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
 
Figure 1.5.  Remainder rf = Df + df
 
Figure 1.5.  Remainder rf = Df + df
Line 1,489: Line 1,489:     
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/%\``````````````````|
+
|                 /%\                 |
−
|`````````````````/%%%\`````````````````|
+
|                 /%%%\                 |
−
|````````````````/%%%%%\````````````````|
+
|               /%%%%%\               |
−
|```````````````o%%%%%%%o```````````````|
+
|               o%%%%%%%o               |
−
|``````````````/%\%%%%%/%\``````````````|
+
|             /%\%%%%%/%\             |
−
|`````````````/%%%\%%%/%%%\`````````````|
+
|             /%%%\%%%/%%%\             |
−
|````````````/%%%%%\%/%%%%%\````````````|
+
|           /%%%%%\%/%%%%%\           |
−
|```````````o%%%%%%%o%%%%%%%o```````````|
+
|           o%%%%%%%o%%%%%%%o           |
−
|``````````/`\%%%%%/%\%%%%%/`\``````````|
+
|         / \%%%%%/%\%%%%%/ \         |
−
|`````````/```\%%%/%%%\%%%/```\`````````|
+
|         /   \%%%/%%%\%%%/   \         |
−
|````````/`````\%/%%%%%\%/`````\````````|
+
|       /     \%/%%%%%\%/     \       |
−
|```````o```````o%%%%%%%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   |
−
|```````|````\%%%%%/%\%%%%%/````|```````|
+
|       |   \%%%%%/%\%%%%%/   |       |
−
|```````|`````\%%%/%%%\%%%/`````|```````|
+
|       |     \%%%/%%%\%%%/     |       |
−
|```````|`du```\%/%%%%%\%/```dv`|```````|
+
|       | du   \%/%%%%%\%/   dv |       |
−
|```````o-------o%%%%%%%o-------o```````|
+
|       o-------o%%%%%%%o-------o       |
−
|````````````````\%%%%%/````````````````|
+
|               \%%%%%/               |
−
|`````````````````\%%%/`````````````````|
+
|                 \%%%/                 |
−
|``````````````````\%/``````````````````|
+
|                 \%/                 |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|```````````````````````````````````````|
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
 
Figure 2.1.  g = ((u, v))
 
Figure 2.1.  g = ((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%%%%%%%o   |   |
−
|```|```|\%%%%%/`\%%%%%/`\%%%%%/|```|```|
+
|   |   |\%%%%%/ \%%%%%/ \%%%%%/|   |   |
−
|```|```|`\%%%/```\%%%/```\%%%/`|```|```|
+
|   |   | \%%%/   \%%%/   \%%%/ |   |   |
−
|```|`u`|``\%/`````\%/`````\%/``|`v`|```|
+
|   | u | \%/     \%/     \%/ | v |   |
−
|```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
 
Figure 2.2.  Eg = ((u + du, v + dv))
 
Figure 2.2.  Eg = ((u + du, v + 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   |   |
−
|```|```|\%%%%%/%\`````/%\%%%%%/|```|```|
+
|   |   |\%%%%%/%\     /%\%%%%%/|   |   |
−
|```|```|`\%%%/%%%\```/%%%\%%%/`|```|```|
+
|   |   | \%%%/%%%\   /%%%\%%%/ |   |   |
−
|```|`u`|``\%/%%%%%\`/%%%%%\%/``|`v`|```|
+
|   | u | \%/%%%%%\ /%%%%%\%/ | v |   |
−
|```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
 
Figure 2.3.  Difference Map Dg = g + Eg
 
Figure 2.3.  Difference Map Dg = g + Eg
    
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/`\``````````````````|
+
|                 / \                 |
−
|`````````````````/```\`````````````````|
+
|                 /   \                 |
−
|````````````````/`````\````````````````|
+
|               /     \               |
−
|```````````````o```````o```````````````|
+
|               o       o               |
−
|``````````````/%\`````/%\``````````````|
+
|             /%\     /%\             |
−
|`````````````/%%%\```/%%%\`````````````|
+
|             /%%%\   /%%%\             |
−
|````````````/%%%%%\`/%%%%%\````````````|
+
|           /%%%%%\ /%%%%%\           |
−
|```````````o%%%%%%%o%%%%%%%o```````````|
+
|           o%%%%%%%o%%%%%%%o           |
−
|``````````/%\%%%%%/`\%%%%%/%\``````````|
+
|         /%\%%%%%/ \%%%%%/%\         |
−
|`````````/%%%\%%%/```\%%%/%%%\`````````|
+
|         /%%%\%%%/   \%%%/%%%\         |
−
|````````/%%%%%\%/`````\%/%%%%%\````````|
+
|       /%%%%%\%/     \%/%%%%%\       |
−
|```````o%%%%%%%o```````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   |
−
|```````|````\%%%%%/`\%%%%%/````|```````|
+
|       |   \%%%%%/ \%%%%%/   |       |
−
|```````|`````\%%%/```\%%%/`````|```````|
+
|       |     \%%%/   \%%%/     |       |
−
|```````|`du```\%/`````\%/```dv`|```````|
+
|       | du   \%/     \%/   dv |       |
−
|```````o-------o```````o-------o```````|
+
|       o-------o       o-------o       |
−
|````````````````\`````/````````````````|
+
|               \     /               |
−
|`````````````````\```/`````````````````|
+
|                 \   /                 |
−
|``````````````````\`/``````````````````|
+
|                 \ /                 |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|```````````````````````````````````````|
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
 
Figure 2.4.  Linear Proxy dg for Dg
 
Figure 2.4.  Linear Proxy dg for Dg
    
o---------------------------------------o
 
o---------------------------------------o
−
|```````````````````````````````````````|
+
|                                       |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|``````````````````/`\``````````````````|
+
|                 / \                 |
−
|`````````````````/```\`````````````````|
+
|                 /   \                 |
−
|````````````````/`````\````````````````|
+
|               /     \               |
−
|```````````````o```````o```````````````|
+
|               o       o               |
−
|``````````````/`\`````/`\``````````````|
+
|             / \     / \             |
−
|`````````````/```\```/```\`````````````|
+
|             /   \   /   \             |
−
|````````````/`````\`/`````\````````````|
+
|           /     \ /     \           |
−
|```````````o```````o```````o```````````|
+
|           o       o       o           |
−
|``````````/`\`````/`\`````/`\``````````|
+
|         / \     / \     / \         |
−
|`````````/```\```/```\```/```\`````````|
+
|         /   \   /   \   /   \         |
−
|````````/`````\`/`````\`/`````\````````|
+
|       /     \ /     \ /     \       |
−
|```````o```````o```````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   |
−
|```````|````\`````/`\`````/````|```````|
+
|       |   \     / \     /   |       |
−
|```````|`````\```/```\```/`````|```````|
+
|       |     \   /   \   /     |       |
−
|```````|`du```\`/`````\`/```dv`|```````|
+
|       | du   \ /     \ /   dv |       |
−
|```````o-------o```````o-------o```````|
+
|       o-------o       o-------o       |
−
|````````````````\`````/````````````````|
+
|               \     /               |
−
|`````````````````\```/`````````````````|
+
|                 \   /                 |
−
|``````````````````\`/``````````````````|
+
|                 \ /                 |
−
|```````````````````o```````````````````|
+
|                   o                   |
−
|```````````````````````````````````````|
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
 
Figure 2.5.  Remainder rg = Dg + dg
 
Figure 2.5.  Remainder rg = Dg + dg
Line 1,713: Line 1,713:  
Orbit 1.  u v
 
Orbit 1.  u v
 
o-----o-----o
 
o-----o-----o
−
| ` ` | d d |
+
|     | d d |
 
| u v | u v |
 
| u v | u v |
 
o=====o=====o
 
o=====o=====o
Line 1,730: Line 1,730:  
Orbit 2.  (u v)
 
Orbit 2.  (u v)
 
o-----o-----o-----o
 
o-----o-----o-----o
−
| ` ` | ` ` | d d |
+
|     |     | d d |
−
| ` ` | d d | 2 2 |
+
|     | d d | 2 2 |
 
| u v | u v | u v |
 
| u v | u v | u v |
 
o=====o=====o=====o
 
o=====o=====o=====o
Line 2,066: Line 2,066:     
o---------------------------------------o
 
o---------------------------------------o
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                       |
−
| ` ` ` ` ` ` ` ` p ` q ` ` ` ` ` ` ` ` |
+
|                 p   q                 |
−
| ` ` ` ` ` ` ` ` o---o ` ` ` ` ` ` ` ` |
+
|                 o---o                 |
−
| ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` ` |
+
|                 |                     |
−
| ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` ` |
+
|                 @                     |
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
−
| ` ` ` ` ` ` ` ( p ( q ))` ` ` ` ` ` ` |
+
|               ( p ( q ))             |
 
o---------------------------------------o
 
o---------------------------------------o
   Line 2,098: Line 2,098:     
o---------------------------------------o
 
o---------------------------------------o
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                       |
−
| ` ` ` ` ` ` ` ` p ` q ` ` ` ` ` ` ` ` |
+
|                 p   q                 |
−
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` |
+
|                 o   o                 |
−
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` |
+
|                 \ /                 |
−
| ` ` ` ` ` ` ` ` ` o ` ` ` ` ` ` ` ` ` |
+
|                   o                   |
−
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
+
|                   |                   |
−
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
+
|                   @                   |
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
−
| ` ` ` ` ` ` ` ((p) (q)) ` ` ` ` ` ` ` |
+
|               ((p) (q))               |
 
o---------------------------------------o
 
o---------------------------------------o
   Line 2,133: Line 2,133:     
o---------------------------------------o
 
o---------------------------------------o
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                       |
−
| ` ` ` ` e_1 ` e_2 ` ... ` e_k ` ` ` ` |
+
|         e_1   e_2   ...   e_k         |
−
| ` ` ` ` `o` ` `o` ` ` ` ` `o` ` ` ` ` |
+
|         o     o           o         |
−
| ` ` ` ` `|` ` `|` ` ` ` ` `|` ` ` ` ` |
+
|         |     |           |         |
−
| ` ` ` ` `o-----o--- ... ---o` ` ` ` ` |
+
|         o-----o--- ... ---o         |
−
| ` ` ` ` ` \ ` ` ` ` ` ` ` / ` ` ` ` ` |
+
|           \               /           |
−
| ` ` ` ` ` `\` ` ` ` ` ` `/` ` ` ` ` ` |
+
|           \             /           |
−
| ` ` ` ` ` ` \ ` ` ` ` ` / ` ` ` ` ` ` |
+
|             \           /             |
−
| ` ` ` ` ` ` `\` ` ` ` `/` ` ` ` ` ` ` |
+
|             \         /             |
−
| ` ` ` ` ` ` ` \ ` ` ` / ` ` ` ` ` ` ` |
+
|               \       /               |
−
| ` ` ` ` ` ` ` `\` ` `/` ` ` ` ` ` ` ` |
+
|               \     /               |
−
| ` ` ` ` ` ` ` ` \ ` / ` ` ` ` ` ` ` ` |
+
|                 \   /                 |
−
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` |
+
|                 \ /                 |
−
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
+
|                   @                   |
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
−
| ` ` ` ((e_1),(e_2),(...),(e_k)) ` ` ` |
+
|       ((e_1),(e_2),(...),(e_k))       |
 
o---------------------------------------o
 
o---------------------------------------o
   Line 2,256: Line 2,256:     
o---------------------------------------o
 
o---------------------------------------o
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                       |
−
| ` ` ` ` p<i>_r<j> ` p<i+1>_r<j>_s<k>` |
+
|         p<i>_r<j>   p<i+1>_r<j>_s<k> |
−
| ` ` ` ` ` ` ` ` o ` o ` ` ` ` ` ` ` ` |
+
|                 o   o                 |
−
| ` ` ` ` ` ` ` ` `\`/` ` ` ` ` ` ` ` ` |
+
|                 \ /                 |
−
| ` `p<i>_r<j>_s<k> o ` ` ` ` ` ` ` ` ` |
+
|   p<i>_r<j>_s<k> o                   |
−
| ` ` ` ` ` ` ` ` ` | ` ` ` ` ` ` ` ` ` |
+
|                   |                   |
−
| ` ` ` ` ` ` ` ` ` @ ` ` ` ` ` ` ` ` ` |
+
|                   @                   |
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                       |
 
o---------------------------------------o
 
o---------------------------------------o
   Line 2,412: Line 2,412:     
o-------------------------------------------------o
 
o-------------------------------------------------o
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                                 |
−
| p0_q0 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
| p0_q0                                           |
−
| `p0_r1` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
| p0_r1                                         |
−
| ` p0_r0_s#` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|   p0_r0_s#                                     |
−
| ` `p0_r1_s0 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|   p0_r1_s0                                     |
−
| ` ` p0_r2_s#` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|     p0_r2_s#                                   |
−
| ` ` `p1_q0` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|     p1_q0                                     |
−
| ` ` ` p1_r2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|       p1_r2                                     |
−
| ` ` ` `p1_r2_s# ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|       p1_r2_s#                                 |
−
| ` ` ` ` p1_r0_s#` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|         p1_r0_s#                               |
−
| ` ` ` ` `p1_r1_s0 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|         p1_r1_s0                               |
−
| ` ` ` ` ` p2_q# ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|           p2_q#                                 |
−
| ` ` ` ` ` `p2_r1` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|           p2_r1                               |
−
| ` ` ` ` ` ` p2_r0_s#` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|             p2_r0_s#                           |
−
| ` ` ` ` ` ` `p2_r1_s0 ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|             p2_r1_s0                           |
−
| ` ` ` ` ` ` ` p2_r2_s#` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|               p2_r2_s#                         |
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                                 |
 
o-------------------------------------------------o
 
o-------------------------------------------------o
   Line 2,466: Line 2,466:     
o-------------------------------------------------o
 
o-------------------------------------------------o
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                                 |
−
| p0_q0 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
| p0_q0                                           |
−
| `p0_r1` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
| p0_r1                                         |
−
| ` p0_r0_s#` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|   p0_r0_s#                                     |
−
| ` `p0_r1_s1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|   p0_r1_s1                                     |
−
| ` ` p0_r2_s#` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|     p0_r2_s#                                   |
−
| ` ` `p1_q1` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|     p1_q1                                     |
−
| ` ` ` p1_r2 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|       p1_r2                                     |
−
| ` ` ` `p1_r2_s# ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|       p1_r2_s#                                 |
−
| ` ` ` ` p1_r0_s#` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|         p1_r0_s#                               |
−
| ` ` ` ` `p1_r1_s1 ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|         p1_r1_s1                               |
−
| ` ` ` ` ` p2_q* ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|           p2_q*                                 |
−
| ` ` ` ` ` `p2_r1` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|           p2_r1                               |
−
| ` ` ` ` ` ` p2_r0_s#` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|             p2_r0_s#                           |
−
| ` ` ` ` ` ` `p2_r1_s1 ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|             p2_r1_s1                           |
−
| ` ` ` ` ` ` ` p2_r2_s#` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|               p2_r2_s#                         |
−
| ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` ` |
+
|                                                 |
 
o-------------------------------------------------o
 
o-------------------------------------------------o
  
Jon Awbrey
12,122

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/80910"