ONT Re: Differential Analytic Turing Automata

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DATA.  Note 16

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| for equalities are so weighed
| that curiosity in neither can
| make choice of either's moiety.
|
| King Lear, Sc.1.5-7, (Quarto)

| for qualities are so weighed
| that curiosity in neither can
| make choice of either's moiety.
|
| King Lear, 1.1.5-6, (Folio)

Justifying a notion of approximation is a little more
involved in general, and especially in these discrete
logical spaces, than it would be expedient for people
in a hurry to tangle with right now.  I will just say
that there are "naive" or "obvious" notions and there
are "sophisticated" or "subtle" notions that we might
choose among.  The later would engage us in trying to
construct proper logical analogues of Lie derivatives,
and so let's save that for when we have become subtle
or sophisticated or both.  Against or toward that day,
as you wish, let's begin with an option in plain view.

Figure 4.1 illustrates one way of ranging over the cells of the
underlying universe U% = [u, v] and selecting at each cell the
linear proposition in dU% = [du, dv] that best approximates
the patch of the difference map Df that is located there,
yielding the following formula for the differential df.

df  =  uv.0 + u(v).du + (u)v.dv + (u)(v).(du, dv)

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Figure 4.1.  df = linear approx to Df

Figure 4.2 illustrates one way of ranging over the cells of the
underlying universe U% = [u, v] and selecting at each cell the
linear proposition in dU% = [du, dv] that best approximates
the patch of the difference map Dg that is located there,
yielding the following formula for the differential dg.

dg  =  uv.(du, dv) + u(v).(du, dv) + (u)v.(du, dv) + (u)(v).(du, dv)

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Figure 4.2.  dg = linear approx to Dg

Well, g, that was easy, seeing as how Dg
is already linear at each locus, dg = Dg.

Jon Awbrey

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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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