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Directory:Jon Awbrey/Papers/Differential Logic and Dynamic Systems (view source)
Revision as of 14:43, 15 June 2007
, 14:43, 15 June 2007→Transformations of Type '''B'''<sup>2</sup> → '''B'''<sup>2</sup>
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components ''f'' and ''g'' that we earlier used on ''J''. This work is recorded in Appendix 1 and a summary of the results is presented in Tables 66-i and 66-ii.
But these initial formulas are purely definitional, and help us little in understanding either the purpose of the operators or the meaning of their results. Working symbolically, let us apply the same method to the separate components ''f'' and ''g'' that we earlier used on ''J''. This work is recorded in Appendix 1 and a summary of the results is presented in Tables 66-i and 66-ii.
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<font face="courier new">
Table 66-i. Computation Summary for f<u, v> = ((u)(v))
{| align="center" border="1" cellpadding="8" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:96%"
|+ Table 66-i. Computation Summary for ''f''‹''u'', ''v''› = ((''u'')(''v''))
| |
|
| !e!f = uv. 1 + u(v). 1 + (u)v. 1 + (u)(v). 0 |
{| align="left" border="0" cellpadding="1" cellspacing="0" style="background:lightcyan; font-weight:bold; text-align:center; width:100%"
| |
| <math>\epsilon</math>''f''
| Ef = uv. (du dv) + u(v). (du (dv)) + (u)v.((du) dv) + (u)(v).((du)(dv)) |
| = || ''uv'' || <math>\cdot</math> || 1
| |
| + || ''u''(''v'') || <math>\cdot</math> || 1
| Df = uv. du dv + u(v). du (dv) + (u)v. (du) dv + (u)(v).((du)(dv)) |
| + || (''u'')''v'' || <math>\cdot</math> || 1
| |
| + || (''u'')(''v'') || <math>\cdot</math> || 0
| df = uv. 0 + u(v). du + (u)v. dv + (u)(v). (du, dv) |
|-
| |
| E''f''
| rf = uv. du dv + u(v). du dv + (u)v. du dv + (u)(v). du dv |
| = || ''uv'' || <math>\cdot</math> || (d''u'' d''v'')
| |
| + || ''u''(''v'') || <math>\cdot</math> || (d''u (d''v''))
| + || (''u'')''v'' || <math>\cdot</math> || ((d''u'') d''v'')
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| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|-
| D''f''
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' (d''v'')
| + || (''u'')''v'' || <math>\cdot</math> || (d''u'') d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || ((d''u'')(d''v''))
|-
| d''f''
| = || ''uv'' || <math>\cdot</math> || 0
| + || ''u''(''v'') || <math>\cdot</math> || d''u''
| + || (''u'')''v'' || <math>\cdot</math> || d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || (d''u'', d''v'')
|-
| r''f''
| = || ''uv'' || <math>\cdot</math> || d''u'' d''v''
| + || ''u''(''v'') || <math>\cdot</math> || d''u'' d''v''
| + || (''u'')''v'' || <math>\cdot</math> || d''u'' d''v''
| + || (''u'')(''v'') || <math>\cdot</math> || d''u'' d''v''
|}
|}
</font><br>
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