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<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+| | 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+
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map ''F'' : ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F'' : ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>].
But that's it, and no further. Neglect of these distinctions in range and target universes of higher dimensions is bound to cause a hopeless confusion. To guard against these adverse prospects, Tables 58 and 59 lay the groundwork for discussing a typical map ''F'' : ['''B'''<sup>2</sup>] → ['''B'''<sup>2</sup>], and begin to pave the way, to some extent, for discussing any transformation of the form ''F'' : ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>].
−{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
−Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators
+|+ '''Table 58. Cast of Characters: Expansive Subtypes of Objects and Operators'''
−|- style="background:paleturquoise"
−| Item | Notation | Description | Type |
+! Item
−! Notation
−| | | | |
+! Description
−| U% | = [u, v] | Source Universe | [B^n] |
+! Type
−| | | | |
+|-
−| valign="top" | ''U''<sup> •</sup>
−| | | | |
+| valign="top" | <font face="courier new">= </font>[''u'', ''v'']
−| X% | = [x, y] | Target Universe | [B^k] |
+| valign="top" | Source Universe
−| | = [f, g] | | |
+| valign="top" | ['''B'''<sup>''n''</sup>]
−| | | | |
+|-
−| valign="top" | ''X''<sup> •</sup>
−| | | | |
+| valign="top" |
−| EU% | = [u, v, du, dv] | Extended | [B^n x D^n] |
+{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| | | Source Universe | |
+| <font face="courier new">= </font>[''x'', ''y'']
−| | | | |
+|-
−| <font face="courier new">= </font>[''f'', ''g'']
−| | | | |
+|}
−| EX% | = [x, y, dx, dy] | Extended | [B^k x D^k] |
+| valign="top" | Target Universe
−| | = [f, g, df, dg] | Target Universe | |
+| valign="top" | ['''B'''<sup>''k''</sup>]
−| | | | |
+|-
−| valign="top" | E''U''<sup> •</sup>
−| | | | |
+| valign="top" | <font face="courier new">= </font>[''u'', ''v'', d''u'', d''v'']
−| F | F = <f, g> : U% -> X% | Transformation, | [B^n] -> [B^k] |
+| valign="top" | Extended Source Universe
−| | | or Mapping | |
+| valign="top" | ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>]
−| | | | |
+|-
−| valign="top" | E''X''<sup> •</sup>
−| | | | |
+| valign="top" |
−{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| <font face="courier new">= </font>[''x'', ''y'', d''x'', d''y'']
−|-
−| <font face="courier new">= </font>[''f'', ''g'', d''f'', d''g'']
−|}
−| valign="top" | Extended Target Universe
−| valign="top" | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
−|-
−| ''F''
−| ''F'' = ‹''f'', ''g''› : ''U''<sup> •</sup> → ''X''<sup> •</sup>
−| Transformation, or Mapping
−| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
−|-
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−|
−|-
−| ''f''
−|-
−| ''g''
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| ''f'', ''g'' : ''U'' → '''B'''
−|-
−| ''f'' : ''U'' → [''x''] ⊆ ''X''<sup> •</sup>
−|-
−| ''g'' : ''U'' → [''y''] ⊆ ''X''<sup> •</sup>
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−| Proposition
−|}
−| valign="top" |
−{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
−| '''B'''<sup>''n''</sup> → '''B'''
−|-
−| ∈ ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> → '''B''')
−|-
−| = ('''B'''<sup>''n''</sup> +→ '''B''') = ['''B'''<sup>''n''</sup>]
−</pre>
+|}
+|-
+| valign="top" |
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| W
+|}
+| valign="top" |
+{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
+| W :
+|-
+| ''U''<sup> •</sup> → E''U''<sup> •</sup> ,
+|-
+| ''X''<sup> •</sup> → E''X''<sup> •</sup> ,
+|-
+| (''U''<sup> •</sup> → ''X''<sup> •</sup>)
+|-
+| →
+|-
+| (E''U''<sup> •</sup> → E''X''<sup> •</sup>) ,
+|-
+| for each W in the set:
+|-
+| {<math>\epsilon</math>, <math>\eta</math>, E, D, 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"
+|
+|-
+| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
+|-
+| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
+|-
+| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
+|-
+| →
+|-
+| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>])
+|-
+|
+|-
+|
+|}
+|-
+|
+{| 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>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] ,
+|-
+| ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] ,
+|-
+| (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>])
+|-
+| →
+|-
+| (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</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>
<pre>
