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User:Jon Awbrey/SCRATCHPAD (view source)

Revision as of 05:44, 12 February 2009

2,788 bytes added ,  05:44, 12 February 2009
→‎Fact 2.2
Line 3,992: Line 3,992:  
===Fact 2.2===
 
===Fact 2.2===
   −
<pre>
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====Variant 1====
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Fact 2.2
     −
If R c OxSxI,
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<br>
   −
then the following are equivalent:
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{| align="center" cellpadding="0" cellspacing="0" style="border-left:1px solid black; border-top:1px solid black; border-right:1px solid black; border-bottom:1px solid black" width="90%"
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:50px; text-align:center"
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| style="width:80%" | &nbsp;
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| style="width:20%; border-left:1px solid black" | <math>\operatorname{Fact~2.2}</math>
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|}
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|-
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|
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{| align="center" cellpadding="0" cellspacing="0" width="100%"
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|- style="height:50px"
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="10%" style="border-top:1px solid black" | <math>\text{If}\!</math>
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| width="68%" style="border-top:1px solid black" | <math>L ~\subseteq~ O \times S \times I</math>
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| width="20%" style="border-top:1px solid black; border-left:1px solid black" | &nbsp;
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|- style="height:50px"
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| &nbsp;
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| <math>\text{then}\!</math>
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| <math>\text{the following are equivalent:}\!</math>
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| style="border-left:1px solid black" | &nbsp;
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|}
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|-
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|
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{| align="center" cellpadding="0" cellspacing="0" style="text-align:center; width:100%"
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|- style="height:60px"
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| width="2%"  style="border-top:1px solid black" | &nbsp;
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| width="10%" style="border-top:1px solid black" | <math>\operatorname{F2.2a.}</math>
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| width="16%" style="border-top:1px solid black" | <math>\operatorname{Der}^L</math>
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| width="4%"  style="border-top:1px solid black" | <math>=\!</math>
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| width="48%" style="border-top:1px solid black" | <math>\ldots</math>
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| width="20%" style="border-top:1px solid black; border-left:1px solid black; text-align:center" | <math>\operatorname{F2.2a~:~R11a}</math>
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|- style="height:20px"
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| colspan="5" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:40px"
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| &nbsp;
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| <math>\operatorname{F2.2b.}</math>
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| <math>\upharpoonleft \operatorname{Der}^L \upharpoonright</math>
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| <math>=\!</math>
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| <math>\ldots</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F2.2b~:~R11b}</math>
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|- style="height:20px"
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| colspan="5" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F2.2c.}</math>
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| <math>\upharpoonleft \operatorname{Der}^L \upharpoonright</math>
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| <math>=\!</math>
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| <math>\ldots</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F2.2c~:~R11c}</math></p>
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|- style="height:20px"
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| colspan="5" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F2.2d.}</math>
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| <math>\upharpoonleft \operatorname{Der}^L \upharpoonright</math>
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| <math>=\!</math>
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| <math>\ldots</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F2.2d~:~Log}</math>
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|- style="height:20px"
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| colspan="5" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F2.2e.}</math>
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| <math>\upharpoonleft \operatorname{Der}^L \upharpoonright</math>
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| <math>=\!</math>
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| <math>\ldots</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F2.2e~:~Log}</math>
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|- style="height:20px"
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| colspan="5" | &nbsp;
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| style="border-left:1px solid black; text-align:center" | <math>::\!</math>
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|- style="height:60px"
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| &nbsp;
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| <math>\operatorname{F2.2f.}</math>
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| <math>\upharpoonleft \operatorname{Der}^L \upharpoonright</math>
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| <math>=\!</math>
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| <math>\ldots</math>
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| style="border-left:1px solid black; text-align:center" |
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<math>\operatorname{F2.2f~:~$~}</math>
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|}
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|}
   −
F2.2a. DerR = {<x, y> C SxI :
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<br>
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Conj(o C O)
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{Den(R, x)}(o) =
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{Den(R, y)}(o)
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} :R11a
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::
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F2.2b. {DerR} = { {<x, y> C SxI :
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Conj(o C O)
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{Den(R, x)}(o) =
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{Den(R, y)}(o)
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}
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} :R11b
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::
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F2.2c. {DerR} c SxIxB
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:
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{DerR} = {<x, y, v> C SxIxB :
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v =
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[ Conj(o C O)
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{Den(R, x)}(o) =
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{Den(R, y)}(o)
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]
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} :R11c
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::
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F2.2d. {DerR} = {<x, y, v> C SxIxB :
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v =
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Conj(o C O)
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[ {Den(R, x)}(o) =
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{Den(R, y)}(o)
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]
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} :Log
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F2.2e. {DerR} = {<x, y, v> C SxIxB :
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====Variant 2====
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v =
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Conj(o C O)
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(( {Den(R, x)}(o),
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{Den(R, y)}(o)
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))
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} :Log
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F2.2f. {DerR} = {<x, y, v> C SxIxB :
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v =
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Conj(o C O)
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(( {Den(R, x)},
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{Den(R, y)}
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))$(o)
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} :$
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</pre>
      
<br>
 
<br>
Jon Awbrey
12,080

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