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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)

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→‎Terminological Interlude: wiki table
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Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results.  Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol.  Accordingly, each of the component operator maps W''J'', since their ranges are 1-dimensional (of type '''B'''<sup>1</sup> or '''D'''<sup>1</sup>), can be regarded either as propositions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''' or as logical transformations W''J''&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>.  As a rule, the plan of the Table allows us to name each entry by detaching the adjective at the left of its row and prefixing it to the generic noun at the top of its column.  In one case, however, it is customary to depart from this scheme.  Because the phrase ''differential proposition'', applied to the result d''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''D''', does not distinguish it from the general run of differential propositions ''G''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.
 
Table&nbsp;55 supplies a more detailed outline of terminology for operators and their results.  Here, I list the restrictive subtype (or narrowest defined subtype) that applies to each entity, and I indicate across the span of the Table the whole spectrum of alternative types that color the interpretation of each symbol.  Accordingly, each of the component operator maps W''J'', since their ranges are 1-dimensional (of type '''B'''<sup>1</sup> or '''D'''<sup>1</sup>), can be regarded either as propositions W''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''' or as logical transformations W''J''&nbsp;:&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>.  As a rule, the plan of the Table allows us to name each entry by detaching the adjective at the left of its row and prefixing it to the generic noun at the top of its column.  In one case, however, it is customary to depart from this scheme.  Because the phrase ''differential proposition'', applied to the result d''J''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''D''', does not distinguish it from the general run of differential propositions ''G''&nbsp;:&nbsp;E''U''&nbsp;&rarr;&nbsp;'''B''', it is usual to single out d''J'' as the ''tangent proposition'' of ''J''.
   −
<pre>
+
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
−
Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes
+
|+ '''Table 55.  Synopsis of Terminology:  Restrictive and Alternative Subtypes'''
−
o--------------o----------------------o--------------------o----------------------o
+
|- style="background:paleturquoise"
−
|             | Operator            | Proposition        | Map                  |
+
! &nbsp;
−
o--------------o----------------------o--------------------o----------------------o
+
! Operator
−
|             |                     |                   |                      |
+
! Proposition
−
| Tacit        | !e! :                | !e!J :            | !e!J :              |
+
! Map
−
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x]    |
+
|-
−
|             | (U%->X%)->(EU%->X%)  | B^2 x D^2 -> B    | [B^2 x D^2]->[B^1]  |
+
|
−
|             |                      |                    |                      |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
o--------------o----------------------o--------------------o----------------------o
+
| Tacit
−
|             |                      |                    |                      |
+
|-
−
| Trope        | !h! :                | !h!J :            | !h!J :              |
+
| Extension
−
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
+
|}
−
|             | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D     | [B^2 x D^2]->[D^1]   |
+
|
−
|             |                     |                   |                      |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
o--------------o----------------------o--------------------o----------------------o
+
| <math>\epsilon</math> :
−
|             |                     |                   |                     |
+
|-
−
| Enlargement  | E :                 | EJ :               | EJ :                 |
+
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
−
| Operator    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]   |
+
|-
−
|             | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D    | [B^2 x D^2]->[D^1]   |
+
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>)
−
|             |                     |                   |                     |
+
|}
−
o--------------o----------------------o--------------------o----------------------o
+
|
−
|             |                      |                    |                      |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| Difference  | D :                  | DJ :              | DJ :                |
+
| <math>\epsilon</math>''J'' :
−
| Operator    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]   |
+
|-
−
|             | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D    | [B^2 x D^2]->[D^1]  |
+
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''B'''
−
|             |                     |                   |                     |
+
|-
−
o--------------o----------------------o--------------------o----------------------o
+
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''B'''
−
|              |                      |                    |                      |
+
|}
−
| Differential | d :                 | dJ :               | dJ :                |
+
|
−
| Operator     | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
|             | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D    | [B^2 x D^2]->[D^1]  |
+
| <math>\epsilon</math>''J'' :
−
|             |                      |                   |                     |
+
|-
−
o--------------o----------------------o--------------------o----------------------o
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'']
−
|             |                      |                    |                      |
+
|-
−
| Remainder    | r :                 | rJ :               | rJ :                 |
+
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>1</sup>]
−
| Operator    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx]    |
+
|}
−
|             | (U%->X%)->(EU%->dX%) | B^2 x D^2 -> D    | [B^2 x D^2]->[D^1]   |
+
|-
−
|             |                      |                    |                      |
+
|
−
o--------------o----------------------o--------------------o----------------------o
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
|             |                      |                    |                      |
+
| Trope
−
| Radius      | $e$ = <!e!, !h!> :  |                    | $e$J :              |
+
|-
−
| Operator    | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
+
| Extension
−
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
+
|}
−
|              |                      |                    |                      |
+
|
−
o--------------o----------------------o--------------------o----------------------o
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
|              |                      |                    |                      |
+
| <math>\eta</math> :
−
| Secant      | $E$ = <!e!, E> :    |                    | $E$J :              |
+
|-
−
| Operator    | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
+
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
−
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
+
|-
−
|              |                      |                    |                      |
+
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
−
o--------------o----------------------o--------------------o----------------------o
+
|}
−
|              |                      |                    |                      |
+
|
−
| Chord        | $D$ = <!e!, D> :    |                    | $D$J :              |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| Operator    | U%->EU%, X%->EX%,    |                    | [u,v,du,dv]->[x, dx] |
+
| <math>\eta</math>''J'' :
−
|              | (U%->X%)->(EU%->EX%) |                    | [B^2 x D^2]->[B x D] |
+
|-
−
|              |                      |                    |                      |
+
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
−
o--------------o----------------------o--------------------o----------------------o
+
|-
−
|              |                      |                    |                      |
+
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
−
| Tangent      | $T$ = <!e!, d> :    | dJ :              | $T$J :              |
+
|}
−
| Functor      | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[x, dx] |
+
|
−
|              | (U%->X%)->(EU%->EX%) | B^2 x D^2 -> D    | [B^2 x D^2]->[B x D] |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
|              |                      |                    |                      |
+
| <math>\eta</math>''J'' :
−
o--------------o----------------------o--------------------o----------------------o
+
|-
−
</pre>
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Enlargement
 +
|-
 +
| Operator
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| E :
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| E''J'' :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| E''J'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Difference
 +
|-
 +
| Operator
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| D :
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| D''J'' :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| D''J'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Differential
 +
|-
 +
| Operator
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| d :
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| d''J'' :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| d''J'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Remainder
 +
|-
 +
| Operator
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| r :
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; d''X''<sup>&nbsp;&bull;</sup>)
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| r''J'' :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| r''J'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>1</sup>]
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Radius
 +
|-
 +
| Operator
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''e'''</font> = ‹<math>\epsilon</math>, <math>\eta</math>› :
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''e'''</font>''J'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Secant
 +
|-
 +
| Operator
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''E'''</font> = ‹<math>\epsilon</math>, E› :
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''E'''</font>''J'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Chord
 +
|-
 +
| Operator
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''D'''</font> = ‹<math>\epsilon</math>, D› :
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|-
 +
| &nbsp;
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''D'''</font>''J'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']
 +
|}
 +
|-
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| Tangent
 +
|-
 +
| Functor
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''T'''</font> = ‹<math>\epsilon</math>, d› :
 +
|-
 +
| ''U''<sup>&nbsp;&bull;</sup> &rarr; E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;&nbsp;''X''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
 +
|-
 +
| (''U''<sup>&nbsp;&bull;</sup> &rarr; ''X''<sup>&nbsp;&bull;</sup>) &rarr; (E''U''<sup>&nbsp;&bull;</sup> &rarr; E''X''<sup>&nbsp;&bull;</sup>)
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| d''J'' :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| <font face=georgia>'''T'''</font>''J'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;d''x'']
 +
|-
 +
| ['''B'''<sup>2</sup>&nbsp;&times;&nbsp;'''D'''<sup>2</sup>]&nbsp;&rarr;&nbsp;['''B'''&nbsp;&times;&nbsp;'''D''']
 +
|}
 +
|}<br>
    
====End of Perfunctory Chatter : Time to Roll the Clip!====
 
====End of Perfunctory Chatter : Time to Roll the Clip!====
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/49169"

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