Line 7,671:
Line 7,671:
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>].
−
<pre>
+
{| 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'''
−
o------o-------------------------o------------------o----------------------------o
+
|- style="background:paleturquoise"
−
| Item | Notation | Description | Type |
+
! Item
−
o------o-------------------------o------------------o----------------------------o
+
! Notation
−
| | | | |
+
! Description
−
| U% | = [u, v] | Source Universe | [B^n] |
+
! Type
−
| | | | |
+
|-
−
o------o-------------------------o------------------o----------------------------o
+
| 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>]
−
| | | | |
+
|-
−
o------o-------------------------o------------------o----------------------------o
+
| 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'']
−
| | | | |
+
|-
−
o------o-------------------------o------------------o----------------------------o
+
| <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>]
−
| | | | |
+
|-
−
o------o-------------------------o------------------o----------------------------o
+
| 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>]
−
| | | | |
+
|-
−
o------o-------------------------o------------------o----------------------------o
+
| valign="top" | E''X''<sup> •</sup>
−
| | | | |
+
| valign="top" |
−
| | f, g : U -> B | Proposition, | B^n -> B |
+
{| align="left" border="0" cellpadding="0" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| | | special case | |
+
| <font face="courier new">= </font>[''x'', ''y'', d''x'', d''y'']
−
| f | f : U -> [x] c X% | of a mapping, | c (B^n, B^n -> B) |
+
|-
−
| | | or component | |
+
| <font face="courier new">= </font>[''f'', ''g'', d''f'', d''g'']
−
| g | g : U -> [y] c X% | of a mapping. | = (B^n +-> B) = [B^n] |
+
|}
−
| | | | |
+
| valign="top" | Extended Target Universe
−
o------o-------------------------o------------------o----------------------------o
+
| valign="top" | ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]
−
| | | | |
+
|-
−
| W | W : | Operator | |
+
| ''F''
−
| | U% -> EU%, | | [B^n] -> [B^n x D^n], |
+
| ''F'' = ‹''f'', ''g''› : ''U''<sup> •</sup> → ''X''<sup> •</sup>
−
| | X% -> EX%, | | [B^k] -> [B^k x D^k], |
+
| Transformation, or Mapping
−
| | (U%->X%)->(EU%->EX%), | | ([B^n] -> [B^k]) |
+
| ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]
−
| | for each W among: | | -> |
+
|-
−
| | !e!, !h!, E, D, d | | ([B^n x D^n]->[B^k x D^k]) |
+
| valign="top" |
−
| | | | |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
o------o-------------------------o------------------o----------------------------o
+
|
−
| | | |
+
|-
−
| !e! | | Tacit Extension Operator !e! |
+
| ''f''
−
| !h! | | Trope Extension Operator !h! |
+
|-
−
| E | | Enlargement Operator E |
+
| ''g''
−
| D | | Difference Operator D |
+
|}
−
| d | | Differential Operator d |
+
| valign="top" |
−
| | | |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
o------o-------------------------o------------------o----------------------------o
+
| ''f'', ''g'' : ''U'' → '''B'''
−
| | | | |
+
|-
−
| $W$ | $W$ : | Operator | |
+
| ''f'' : ''U'' → [''x''] ⊆ ''X''<sup> •</sup>
−
| | U% -> $T$U% = EU%, | | [B^n] -> [B^n x D^n], |
+
|-
−
| | X% -> $T$X% = EX%, | | [B^k] -> [B^k x D^k], |
+
| ''g'' : ''U'' → [''y''] ⊆ ''X''<sup> •</sup>
−
| | (U%->X%)->($T$U%->$T$X%)| | ([B^n] -> [B^k]) |
+
|}
−
| | for each $W$ among: | | -> |
+
| valign="top" |
−
| | $e$, $E$, $D$, $T$ | | ([B^n x D^n]->[B^k x D^k]) |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| | | | |
+
| Proposition
−
o------o-------------------------o------------------o----------------------------o
+
|}
−
| | | |
+
| valign="top" |
−
| $e$ | | Radius Operator $e$ = <!e!, !h!> |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100"
−
| $E$ | | Secant Operator $E$ = <!e!, E > |
+
| '''B'''<sup>''n''</sup> → '''B'''
−
| $D$ | | Chord Operator $D$ = <!e!, D > |
+
|-
−
| $T$ | | Tangent Functor $T$ = <!e!, d > |
+
| ∈ ('''B'''<sup>''n''</sup>, '''B'''<sup>''n''</sup> → '''B''')
−
| | | |
+
|-
−
o------o-------------------------o-----------------------------------------------o
+
| = ('''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>