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