[Inquiry] Re: Differential Logic

[Jon Awbrey]

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DLOG.  Note D43

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The Phantom of the Operators:  !h!

| I was wondering what the reason could be,
| when I myself raised my head and everything
| within me seemed drawn towards the Unseen,
| 'which was playing the most perfect music'!
|
| Gaston Leroux, 'The Phantom of the Opera', [Ler, 81]

I now describe an operator whose persistent but elusive action behind the scenes,
whose slightly twisted and ambivalent character, and whose fugitive disposition,
caught somewhere in flight between the arrantly negative and the positive but
errant intent, has cost me some painstaking trouble to detect.  In the end
I shall place it among the other extensions and projections, as a shade
among shadows, of muted tones and motley hue, that adumbrates its own
thematic frame and paradoxically lights the way toward a whole new
spectrum of values.

Given a transformation F : [u_1, ..., u_n] -> [x_1, ..., x_k], we often
need to make a separate treatment of a related family of transformations
of the form F* : [u_1, ..., u_n,  du_1, ..., du_n] -> [dx_1, ..., dx_k].
The operator !h! (Greek eta) is introduced to deal with the simplest one
of these maps:

   !h!F  :  [u_1, ..., u_n,  du_1, ..., du_n]  ->  [dx_1, ..., dx_k]

which is defined by the equations:

o--------------------------------------------------------------------------------o
|                                                                                |
| dx_1   =   !e!F_1 <u_1, ..., u_n,  du_1, ..., du_n>   =   F_1 <u_1, ..., u_n>  |
|                                                                                |
|  ...                                                                           |
|                                                                                |
| dx_k   =   !e!F_k <u_1, ..., u_n,  du_1, ..., du_n>   =   F_k <u_1, ..., u_n>  |
|                                                                                |
o--------------------------------------------------------------------------------o

In effect, the operator !h! is nothing but the stand-alone version of
a procedure that is otherwise invoked subordinate to the work of the
radius operator $e$.  Operating independently, !h! achieves precisely
the same results that the second !e! in <!e!, !e!> accomplishes by
working within the context of its adjuvant thematic frame, "< , >".