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Revision as of 02:20, 24 June 2007

347 bytes removed ,  02:20, 24 June 2007
→‎Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators
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| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>]
 
| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>]
 
|-
 
|-
−
| valign="top" | <p>W</p>
+
|-
−
| valign="top" | <p>W&nbsp;:</p>
+
| valign="top" |
−
<p>''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,</p>
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
<p>''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,</p>
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| W
−
<p>(''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)&nbsp;&rarr;<br>
+
|}
−
E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>)&nbsp;,</p>
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| valign="top" |
−
<p>for each W in the set:<br>
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
{<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d}</p>
+
| W&nbsp;:
−
| valign="top" | <p>Operator</p>
+
|-
−
| valign="top" | <p>&nbsp;<p>
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| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,
−
<p>['''B'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;,</p>
+
|-
−
<p>['''B'''<sup>1</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>1</sup>&nbsp;&times;&nbsp;'''D'''<sup>1</sup>]&nbsp;,</p>
+
| ''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
−
<p>(['''B'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>1</sup>])&nbsp;&rarr;<br>
+
|-
−
(['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>1</sup>&nbsp;&times;&nbsp;'''D'''<sup>1</sup>])</p>
+
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)
−
<p>&nbsp;<br>&nbsp;</p>
+
|-
 +
| &rarr;
 +
|-
 +
| (E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>)&nbsp;,
 +
|-
 +
| for each W in the set:
 +
|-
 +
| {<math>\epsilon</math>,&nbsp;<math>\eta</math>,&nbsp;E,&nbsp;D,&nbsp;d}
 +
|}
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Operator
 +
|}
 +
| valign="top" |
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
 +
| &nbsp;
 +
|-
 +
| ['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>],
 +
|-
 +
| ['''B'''<sup>1</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>],
 +
|-
 +
| (['''B'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup>])
 +
|-
 +
| &rarr;
 +
|-
 +
| (['''B'''<sup>2</sup> &times; '''D'''<sup>2</sup>] &rarr; ['''B'''<sup>1</sup> &times; '''D'''<sup>1</sup>])
 +
|-
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|}
 
|-
 
|-
 
|
 
|
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|}
 
|}
 
|-
 
|-
−
| &nbsp;
+
|
−
| &nbsp;
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| &nbsp;
+
| <font face=georgia>'''e'''</font>
−
| &nbsp;
   
|-
 
|-
−
| <font face="lucida calligraphy">A<font>
+
| <font face=georgia>'''E'''</font>
−
| {''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}
  −
| Alphabet
  −
| [''n''] = '''n'''
   
|-
 
|-
−
| ''A''<sub>''i''</sub>
+
| <font face=georgia>'''D'''</font>
−
| {(''a''<sub>''i''</sub>), ''a''<sub>''i''</sub>}
  −
| Dimension ''i''
  −
| '''B'''
   
|-
 
|-
−
| ''A''
+
| <font face=georgia>'''T'''</font>
−
|
+
|}
−
〈<font face="lucida calligraphy">A</font>〉<br>
+
| valign="top" | &nbsp;
−
〈''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>〉<br>
+
| colspan="2"  |
−
{‹''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>›}<br>
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
−
''A''<sub>1</sub> &times; &hellip; &times; ''A''<sub>''n''</sub><br>
+
| Radius Operator || <font face=georgia>'''e'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;<math>\eta</math>›
−
&prod;<sub>''i''</sub> ''A''<sub>''i''</sub>
  −
|
  −
Set of cells,<br>
  −
coordinate tuples,<br>
  −
points, or vectors<br>
  −
in the universe<br>
  −
of discourse
  −
| '''B'''<sup>''n''</sup>
   
|-
 
|-
−
| ''A''*
+
| Secant Operator || <font face=georgia>'''E'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;E›
−
| (hom : ''A'' &rarr; '''B''')
  −
| Linear functions
  −
| ('''B'''<sup>''n''</sup>)* = '''B'''<sup>''n''</sup>
   
|-
 
|-
−
| ''A''^
+
| Chord Operator  || <font face=georgia>'''D'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;D›
−
| (''A'' &rarr; '''B''')
  −
| Boolean functions
  −
| '''B'''<sup>''n''</sup> &rarr; '''B'''
   
|-
 
|-
−
| ''A''<sup>&bull;</sup>
+
| Tangent Functor || <font face=georgia>'''T'''</font>&nbsp;=&nbsp;‹<math>\epsilon</math>,&nbsp;d›
−
|
+
|}
−
[<font face="lucida calligraphy">A</font>]<br>
  −
(''A'', ''A''^)<br>
  −
(''A'' +&rarr; '''B''')<br>
  −
(''A'', (''A'' &rarr; '''B'''))<br>
  −
[''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>]
  −
|
  −
Universe of discourse<br>
  −
based on the features<br>
  −
{''a''<sub>1</sub>, &hellip;, ''a''<sub>''n''</sub>}
  −
|
  −
('''B'''<sup>''n''</sup>, ('''B'''<sup>''n''</sup> &rarr; '''B'''))<br>
  −
('''B'''<sup>''n''</sup> +&rarr; '''B''')<br>
  −
['''B'''<sup>''n''</sup>]
   
|}<br>
 
|}<br>
−
  −
<pre>
  −
| $e$  || Radius Operator    $e$ = <!e!, !h!>
  −
| $E$  || Secant Operator    $E$ = <!e!,  E >
  −
| $D$  || Chord Operator    $D$ = <!e!,  D >
  −
| $T$  || Tangent Functor    $T$ = <!e!,  d >
  −
</pre>
      
===Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes===
 
===Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes===
Jon Awbrey
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