Negate, Augment, Denominate
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NAD. Note 1
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This old puzzle continues to come up in various contexts of discussion.
It's been too long and I can no longer remember what it had to do with
modular forms, but I wnated to keep thinking about it, so until I work
up the quantum to go look it up, I'll break it out for working on here.
Let C be the complex plane.
Let R be the real line.
Let * be the Kleene star.
Define A : C -> C such that A : x ~> x+1.
Define D : C -> C such that D : x ~> 1/x.
Define N : C -> C such that N : x ~> - x.
By way of mnemonics:
"A" is for Augment,
"D" is for Denominate,
"N" is for Negate.
For some reason that I can't remember either,
I used to write these operators on the right.
For example, pick a number on the real line,
let us say, x = 3. Then we have this data:
xA = 4
xD = 1/3 = 0.3*
xN = -3
Among the important compound operators
are several trigrams like NAD, DNA, etc.
For example:
xNAD = 1/(1-x)
Formally speaking, one's formal knee jerks to give this:
xNAD = 1 + x + x^2 + x^3 + ...
But! The second equation holds
only when the series converges.
For example:
(1/3)NAD = 1/(1-(1/3)) = 1/(2/3) = 3/2,
= 1 + (1/3) + (1/3)^2 + (1/3)^3 + ...
But:
3NAD = 1/(1-3) = -1/2
=/= 1 + 3 + 3^2 + 3^3 + ...
Anyway, I hope this is right -- it's been a while.
Exercise for the reader:
What can you say in general about x(NAD)^k
for k = 1, 2, 3, ...?
Jon Awbrey
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inquiry e-lab: http://stderr.org/pipermail/inquiry/
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