ONT Re: Differential Logic
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DLOG. Note D80
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Transformations of Type B^2 -> B^2 (cont.)
Table 67 shows how to compute the analytic series
for F = <f, g> = <((u)(v)), ((u, v))> in terms of
coordinates, and Table 68 recaps these results in
symbolic terms, agreeing with earlier derivations.
Table 67. Computation of an Analytic Series in Terms of Coordinates
o--------o-------o-------o--------o-------o-------o-------o-------o
| u v | du dv | u' v' | f g | Ef Eg | Df Dg | df dg | rf rg |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 0 0 | 0 0 | 0 0 | 0 1 | 0 1 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 0 1 | | 1 0 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 1 0 | | 1 0 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 1 1 | | 1 1 | 1 0 | 0 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 0 1 | 0 0 | 0 1 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 1 0 | | 1 0 | 0 0 | 1 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 1 0 | 0 0 | 1 0 | 1 0 | 1 0 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 1 1 | | 1 1 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 0 0 | | 0 1 | 1 1 | 1 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 0 1 | | 1 0 | 0 0 | 1 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
| | | | | | | | |
| 1 1 | 0 0 | 1 1 | 1 1 | 1 1 | 0 0 | 0 0 | 0 0 |
| | | | | | | | |
| | 0 1 | 1 0 | | 1 0 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 0 | 0 1 | | 1 0 | 0 1 | 0 1 | 0 0 |
| | | | | | | | |
| | 1 1 | 0 0 | | 0 1 | 1 0 | 0 0 | 1 0 |
| | | | | | | | |
o--------o-------o-------o--------o-------o-------o-------o-------o
Table 68. Computation of an Analytic Series in Symbolic Terms
o-----o-----o------------o----------o----------o----------o----------o----------o
| u v | f g | Df | Dg | df | dg | rf | rg |
o-----o-----o------------o----------o----------o----------o----------o----------o
| | | | | | | | |
| 0 0 | 0 1 | ((du)(dv)) | (du, dv) | (du, dv) | (du, dv) | du dv | () |
| | | | | | | | |
| 0 1 | 1 0 | (du) dv | (du, dv) | dv | (du, dv) | du dv | () |
| | | | | | | | |
| 1 0 | 1 0 | du (dv) | (du, dv) | du | (du, dv) | du dv | () |
| | | | | | | | |
| 1 1 | 1 1 | du dv | (du, dv) | () | (du, dv) | du dv | () |
| | | | | | | | |
o-----o-----o------------o----------o----------o----------o----------o----------o
Jon Awbrey
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