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

Revision as of 18:27, 3 July 2007

6,603 bytes added ,  18:27, 3 July 2007
→‎Taking Aim at Higher Dimensional Targets: wiki table
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|}<br>
   −
<pre>
+
{| align="center" border="1" cellpadding="4" cellspacing="0" style="background:lightcyan; text-align:left; width:96%"
−
Table 59.  Synopsis of Terminology:  Restrictive and Alternative Subtypes
+
|+ '''Table 59.  Synopsis of Terminology:  Restrictive and Alternative Subtypes'''
−
o--------------o----------------------o--------------------o----------------------o
+
|- style="background:paleturquoise"
−
|             | Operator            | Proposition        | Transformation      |
+
| &nbsp;
−
|             |    or                |   or              |    or                |
+
| align="center" | '''Operator<br>or<br>Operand'''
−
|             | Operand              | Component          | Mapping              |
+
| align="center" | '''Proposition<br>or<br>Component'''
−
o--------------o----------------------o--------------------o----------------------o
+
| align="center" | '''Transformation<br>or<br>Mapping'''
−
|             |                     |                   |                      |
+
|-
−
| Operand      | F = <F_1, F_2>       | F_i : <|u,v|> -> B | F : [u, v] -> [x, y] |
+
| valign="top" | Operand
−
|             |                      |                    |                      |
+
| valign="top" |
−
|             | F = <f, g> : U -> X | F_i : B^n -> B    | F : B^n -> B^k      |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
|              |                     |                   |                     |
+
| ''F'' = ‹''F''<sub>1</sub>, ''F''<sub>2</sub>›
−
o--------------o----------------------o--------------------o----------------------o
+
|-
−
|             |                     |                   |                     |
+
| ''F'' = ‹''f'', ''g''› : ''U'' &rarr; ''X''
−
| Tacit        | !e! :               | !e!F_i :           | !e!F :               |
+
|}
−
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> B | [u,v,du,dv]->[x, y] |
+
| valign="top" |
−
|             | (U%->X%)->(EU%->X%)  | B^n x D^n -> B    | [B^n x D^n]->[B^k]   |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
|             |                     |                   |                     |
+
| ''F''<sub>''i''</sub> : 〈''u'', ''v''〉 &rarr; '''B'''
−
o--------------o----------------------o--------------------o----------------------o
+
|-
−
|             |                      |                   |                     |
+
| ''F''<sub>''i''</sub> : '''B'''<sup>''n''</sup> &rarr; '''B'''
−
| Trope        | !h! :                | !h!F_i :           | !h!F :               |
+
|}
−
| Extension    | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [u,v,du,dv]->[dx,dy] |
+
| valign="top" |
−
|             | (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"
−
|             |                     |                   |                     |
+
| ''F'' : [''u'', ''v''] &rarr; [''x'', ''y'']
−
o--------------o----------------------o--------------------o----------------------o
+
|-
−
|             |                      |                   |                     |
+
| ''F'' : '''B'''<sup>''n''</sup> &rarr; '''B'''<sup>''k''</sup>
−
| Enlargement  | E :                  | EF_i :             | EF :                 |
+
|}
−
| 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%"
−
o--------------o----------------------o--------------------o----------------------o
+
| Tacit
−
|             |                      |                    |                      |
+
|-
−
| Difference  | D :                  | DF_i :            | DF :                |
+
| Extension
−
| 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%"
−
o--------------o----------------------o--------------------o----------------------o
+
| <math>\epsilon</math> :
−
|             |                      |                    |                      |
+
|-
−
| Differential | d :                  | dF_i :            | dF :                |
+
| ''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''U''<sup>&nbsp;&bull;</sup>&nbsp;,&nbsp;''X''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;E''X''<sup>&nbsp;&bull;</sup>&nbsp;,
−
| 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]  |
+
| (''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''X''<sup>&nbsp;&bull;</sup>)&nbsp;&rarr;&nbsp;(E''U''<sup>&nbsp;&bull;</sup>&nbsp;&rarr;&nbsp;''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%"
−
| Remainder    | r :                 | rF_i :             | rF :                |
+
| <math>\epsilon</math>''F''<sub>''i''</sub> :
−
| 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]   |
+
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''B'''
−
|             |                     |                   |                      |
+
|-
−
o--------------o----------------------o--------------------o----------------------o
+
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''B'''
−
|             |                      |                    |                      |
+
|}
−
| Radius      | $e$ = <!e!, !h!> :  |                    | $e$F :              |
+
|
−
| Operator    |                      |                    |                      |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->   |
+
| <math>\epsilon</math>''F'' :
−
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
+
|-
−
|              |                      |                    |                      |
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'', ''y'']
−
|             |                      |                    | [B^n x D^n] ->      |
+
|-
−
|              |                      |                    | [B^k x D^k]          |
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>]
−
|              |                      |                    |                      |
+
|}
−
o--------------o----------------------o--------------------o----------------------o
+
|-
−
|              |                      |                    |                      |
+
|
−
| Secant      | $E$ = <!e!, E> :    |                    | $E$F :              |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
| Operator    |                      |                    |                      |
+
| Trope
−
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
+
|-
−
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
+
| Extension
−
|              |                      |                    |                      |
+
|}
−
|              |                      |                    | [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%"
−
|              |                      |                    |                      |
+
| <math>\eta</math> :
−
o--------------o----------------------o--------------------o----------------------o
+
|-
−
|              |                      |                    |                      |
+
| ''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;,
−
| Chord        | $D$ = <!e!, D> :    |                    | $D$F :              |
+
|-
−
| Operator    |                      |                    |                      |
+
| (''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>)
−
|              | U%->EU%, X%->EX%,    |                    | [u, v, du, dv] ->    |
+
|}
−
|              | (U%->X%)->(EU%->EX%) |                    | [x, y, dx, dy],      |
+
|
−
|              |                      |                    |                      |
+
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
−
|              |                      |                    | [B^n x D^n] ->      |
+
| <math>\eta</math>''F''<sub>''i''</sub> :
−
|              |                      |                    | [B^k x D^k]          |
+
|-
−
|              |                      |                    |                      |
+
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
−
o--------------o----------------------o--------------------o----------------------o
+
|-
−
|              |                      |                    |                      |
+
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
−
| Tangent      | $T$ = <!e!, d> :    | dF_i :            | $T$F :              |
+
|}
−
| Functor      |                      |                    |                      |
+
|
−
|              | U%->EU%, X%->EX%,    | <|u,v,du,dv|> -> D | [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],      |
+
| <math>\eta</math>''F'' :
−
|              |                      |                    |                      |
+
|-
−
|              |                      | B^n x D^n -> D    | [B^n x D^n] ->      |
+
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
−
|              |                      |                    | [B^k x D^k]          |
+
|-
−
|              |                      |                    |                      |
+
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</sup>]
−
o--------------o----------------------o--------------------o----------------------o
+
|}
−
</pre>
+
|-
 +
|
 +
{| 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''F''<sub>''i''</sub> :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| E''F'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 +
|-
 +
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</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''F''<sub>''i''</sub> :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| D''F'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 +
|-
 +
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</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''F''<sub>''i''</sub> :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| d''F'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 +
|-
 +
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</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''F''<sub>''i''</sub> :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>&nbsp;&rarr;&nbsp;'''D'''
 +
|}
 +
|
 +
{| align="left" border="0" cellpadding="2" cellspacing="0" style="background:lightcyan; text-align:left; width:100%"
 +
| r''F'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[d''x'', d''y'']
 +
|-
 +
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''D'''<sup>''k''</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>''F'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
 +
|-
 +
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
 +
|}
 +
|-
 +
|
 +
{| 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>''F'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
 +
|-
 +
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
 +
|}
 +
|-
 +
|
 +
{| 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>''F'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
 +
|-
 +
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
 +
|}
 +
|-
 +
|
 +
{| 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''F''<sub>''i''</sub> :
 +
|-
 +
| 〈''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v''〉&nbsp;&rarr;&nbsp;'''D'''
 +
|-
 +
| '''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</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>''F'' :
 +
|-
 +
| [''u'',&nbsp;''v'',&nbsp;d''u'',&nbsp;d''v'']&nbsp;&rarr;&nbsp;[''x'',&nbsp;''y'',&nbsp;d''x'',&nbsp;d''y'']
 +
|-
 +
| ['''B'''<sup>''n''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''n''</sup>]&nbsp;&rarr;&nbsp;['''B'''<sup>''k''</sup>&nbsp;&times;&nbsp;'''D'''<sup>''k''</sup>]
 +
|}
 +
|}<br>
    
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===
 
===Transformations of Type '''B'''<sup>2</sup> &rarr; '''B'''<sup>2</sup>===
Jon Awbrey
12,080

edits

Retrieved from "https://mywikibiz.com/Special:MobileDiff/49259"

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