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MyWikiBiz, Author Your Legacy — Wednesday April 02, 2025
Jump to navigationJump to search→Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
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|-
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <math>\epsilon</math>
−|-
−| <math>\eta</math>
−|-
−| E
−|-
−| D
−|-
−| d
−|}
−| valign="top" |
−| colspan="2" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
−| Tacit Extension Operator || <math>\epsilon</math>
−|-
−| Trope Extension Operator || <math>\eta</math>
−|-
−| Enlargement Operator || E
−|-
−| Difference Operator || D
−|-
−| Differential Operator || d
−|}
−|-
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <font face=georgia>'''W'''</font>
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <font face=georgia>'''W'''</font> :
−|-
−| ''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> ,
−|-
−| ''X''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup> = E''X''<sup> •</sup> ,
−|-
−| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
−|-
−| →
−|-
−| (<font face=georgia>'''T'''</font>''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup>) ,
−|-
−| for each <font face=georgia>'''W'''</font> in the set:
−|-
−| {<font face=georgia>'''e'''</font>, <font face=georgia>'''E'''</font>, <font face=georgia>'''D'''</font>, <font face=georgia>'''T'''</font>}
−|}
−| 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"
−|
−|-
−| ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] ,
−|-
−| ['''B'''<sup>1</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>] ,
−|-
−| (['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>])
−|-
−| →
−|-
−| (['''B'''<sup>2</sup> × '''D'''<sup>2</sup>] → ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>])
−|-
−|
−|-
−|
−|}
−|-
−|
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <font face=georgia>'''e'''</font>
−|-
−| <font face=georgia>'''E'''</font>
−|-
−| <font face=georgia>'''D'''</font>
−|-
−| <font face=georgia>'''T'''</font>
−|}
−| valign="top" |
−| colspan="2" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%"
−| Radius Operator || <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>›
−|-
−| Secant Operator || <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E›
−|-
−| Chord Operator || <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D›
−|-
−| Tangent Functor || <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d›
−|}
−|}<br>
−<pre>
−-------------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
−| !e! | | Tacit Extension Operator !e!
−| !h! | | Trope Extension Operator !h!
−| E | | Enlargement Operator E
−| D | | Difference Operator D
−| d | | Differential Operator d
−-------------o
−| $W$ | $W$ : | Operator | |
−| | U% -> $T$U% = EU%, | | [B^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
−| $e$ | | Radius Operator $e$ = <!e!, !h!> |
−| $E$ | | Secant Operator $E$ = <!e!, E > |
−| $D$ | | Chord Operator $D$ = <!e!, D > |
−| $T$ | | Tangent Functor $T$ = <!e!, d > |
−-------------o
−</pre>
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
−|+ '''Table 54. Cast of Characters: Expansive Subtypes of Objects and Operators'''
−|- style="background:paleturquoise"
−! Item
−! Notation
−! Description
−! Type
−|-
−| ''U''<sup> •</sup>
−| = [''u'', ''v'']
−| Source Universe
−| ['''B'''<sup>2</sup>]
−|-
−| ''X''<sup> •</sup>
−| = [''x'']
−| Target Universe
−| ['''B'''<sup>1</sup>]
−|-
−| E''U''<sup> •</sup>
−| = [''u'', ''v'', d''u'', d''v'']
−| Extended Source Universe
−| ['''B'''<sup>2</sup> × '''D'''<sup>2</sup>]
−|-
−| E''X''<sup> •</sup>
−| = [''x'', d''x'']
−| Extended Target Universe
−| ['''B'''<sup>1</sup> × '''D'''<sup>1</sup>]
−|-
−| ''J''
−| ''J'' : ''U'' → '''B'''
−| Proposition
−| ('''B'''<sup>2</sup> → '''B''') ∈ ['''B'''<sup>2</sup>]
−|-
−| ''J''
−| ''J'' : ''U''<sup> •</sup> → ''X''<sup> •</sup>
−| Transformation, or Mapping
−| ['''B'''<sup>2</sup>] → ['''B'''<sup>1</sup>]
| = ('''B'''<sup>''n''</sup> +→ '''B''') = ['''B'''<sup>''n''</sup>]
| = ('''B'''<sup>''n''</sup> +→ '''B''') = ['''B'''<sup>''n''</sup>]
|}
|}
−|-
|-
| valign="top" |
| valign="top" |
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|}
|}
|}<br>
|}<br>
+
+<pre>
+-------------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
+| !e! | | Tacit Extension Operator !e!
+| !h! | | Trope Extension Operator !h!
+| E | | Enlargement Operator E
+| D | | Difference Operator D
+| d | | Differential Operator d
+-------------o
+| $W$ | $W$ : | Operator | |
+| | U% -> $T$U% = EU%, | | [B^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
+| $e$ | | Radius Operator $e$ = <!e!, !h!> |
+| $E$ | | Secant Operator $E$ = <!e!, E > |
+| $D$ | | Chord Operator $D$ = <!e!, D > |
+| $T$ | | Tangent Functor $T$ = <!e!, d > |
+-------------o
+</pre>
===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
===Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes===
