Line 7,671: |
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| 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>
| + | |} |
− | | + | |- |
− | <pre>
| + | | valign="top" | |
− | Table 59. Synopsis of Terminology: Restrictive and Alternative Subtypes
| + | {| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%" |
− | o--------------o----------------------o--------------------o----------------------o
| + | | W |
− | | | Operator | Proposition | Transformation |
| + | |} |
− | | | or | or | or |
| + | | valign="top" | |
− | | | Operand | Component | Mapping |
| + | {| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%" |
− | o--------------o----------------------o--------------------o----------------------o
| + | | W : |
− | | | | | |
| + | |- |
− | | Operand | F = <F_1, F_2> | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
| + | | ''U''<sup> •</sup> → E''U''<sup> •</sup> , |
− | | | | | |
| + | |- |
− | | | F = <f, g> : U -> X | F_i : B^n -> B | F : B^n -> B^k |
| + | | ''X''<sup> •</sup> → E''X''<sup> •</sup> , |
− | | | | | |
| + | |- |
− | o--------------o----------------------o--------------------o----------------------o
| + | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) |
− | | | | | |
| + | |- |
− | | Tacit | !e! : | !e!F_i : | !e!F : |
| + | | → |
− | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
| + | |- |
− | | | (U%->X%)->(EU%->X%) | B^n x D^n -> B | [B^n x D^n]->[B^k] |
| + | | (E''U''<sup> •</sup> → E''X''<sup> •</sup>) , |
− | | | | | |
| + | |- |
− | o--------------o----------------------o--------------------o----------------------o
| + | | for each W in the set: |
− | | | | | |
| + | |- |
− | | Trope | !h! : | !h!F_i : | !h!F : |
| + | | {<math>\epsilon</math>, <math>\eta</math>, E, D, d} |
− | | Extension | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| + | |} |
− | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[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
| + | | Operator |
− | | | | | |
| + | |} |
− | | Enlargement | E : | EF_i : | EF : |
| + | | valign="top" | |
− | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| + | {| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100" |
− | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| + | | |
− | | | | | |
| + | |- |
− | o--------------o----------------------o--------------------o----------------------o
| + | | ['''B'''<sup>''n''</sup>] → ['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] , |
− | | | | | |
| + | |- |
− | | Difference | D : | DF_i : | DF : |
| + | | ['''B'''<sup>''k''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>] , |
− | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| + | |- |
− | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| + | | (['''B'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup>]) |
− | | | | | |
| + | |- |
− | o--------------o----------------------o--------------------o----------------------o
| + | | → |
− | | | | | |
| + | |- |
− | | Differential | d : | dF_i : | dF : |
| + | | (['''B'''<sup>''n''</sup> × '''D'''<sup>''n''</sup>] → ['''B'''<sup>''k''</sup> × '''D'''<sup>''k''</sup>]) |
− | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| + | |- |
− | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| + | | |
− | | | | | |
| + | |- |
− | o--------------o----------------------o--------------------o----------------------o
| + | | |
− | | | | | |
| + | |} |
− | | Remainder | r : | rF_i : | rF : |
| + | |- |
− | | Operator | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
| + | | |
− | | | (U%->X%)->(EU%->dX%) | B^n x D^n -> D | [B^n x D^n]->[D^k] |
| + | {| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%" |
− | | | | | |
| + | | <math>\epsilon</math> |
− | o--------------o----------------------o--------------------o----------------------o
| + | |- |
− | | | | | |
| + | | <math>\eta</math> |
− | | Radius | $e$ = <!e!, !h!> : | | $e$F : |
| + | |- |
− | | Operator | | | |
| + | | E |
− | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| + | |- |
− | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| + | | D |
− | | | | | |
| + | |- |
− | | | | | [B^n x D^n] -> |
| + | | d |
− | | | | | [B^k x D^k] |
| + | |} |
− | | | | | |
| + | | valign="top" | |
− | o--------------o----------------------o--------------------o----------------------o
| + | | colspan="2" | |
− | | | | | |
| + | {| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:60%" |
− | | Secant | $E$ = <!e!, E> : | | $E$F : |
| + | | Tacit Extension Operator || <math>\epsilon</math> |
− | | Operator | | | |
| + | |- |
− | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| + | | Trope Extension Operator || <math>\eta</math> |
− | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| + | |- |
− | | | | | |
| + | | Enlargement Operator || E |
− | | | | | [B^n x D^n] -> |
| + | |- |
− | | | | | [B^k x D^k] |
| + | | Difference Operator || D |
− | | | | | |
| + | |- |
− | o--------------o----------------------o--------------------o----------------------o
| + | | Differential Operator || d |
− | | | | | |
| + | |} |
− | | Chord | $D$ = <!e!, D> : | | $D$F : |
| + | |- |
− | | Operator | | | |
| + | | valign="top" | |
− | | | U%->EU%, X%->EX%, | | [u, v, du, dv] -> |
| + | {| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%" |
− | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| + | | <font face=georgia>'''W'''</font> |
− | | | | | |
| + | |} |
− | | | | | [B^n x D^n] -> |
| + | | valign="top" | |
− | | | | | [B^k x D^k] |
| + | {| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%" |
− | | | | | |
| + | | <font face=georgia>'''W'''</font> : |
− | o--------------o----------------------o--------------------o----------------------o
| + | |- |
− | | | | | |
| + | | ''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''U''<sup> •</sup> = E''U''<sup> •</sup> , |
− | | Tangent | $T$ = <!e!, d> : | dF_i : | $T$F : |
| + | |- |
− | | Functor | | | |
| + | | ''X''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup> = E''X''<sup> •</sup> , |
− | | | U%->EU%, X%->EX%, | <|u,v,du,dv|> -> D | [u, v, du, dv] -> |
| + | |- |
− | | | (U%->X%)->(EU%->EX%) | | [x, y, dx, dy], |
| + | | (''U''<sup> •</sup> → ''X''<sup> •</sup>) |
− | | | | | |
| + | |- |
− | | | | B^n x D^n -> D | [B^n x D^n] -> |
| + | | → |
− | | | | | [B^k x D^k] |
| + | |- |
− | | | | | |
| + | | (<font face=georgia>'''T'''</font>''U''<sup> •</sup> → <font face=georgia>'''T'''</font>''X''<sup> •</sup>) , |
− | o--------------o----------------------o--------------------o----------------------o
| + | |- |
− | </pre> | + | | 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> |
| | | |
| ===Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>=== | | ===Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>=== |