Not sure. I suppose instead of
-@^.@(+/&.:*:)
we could write:
^.@%@(+/&.:*:)
or even:
^.@(+/&.:(*: :. (^&_0.5) ) )
But I'm not sure what this buys us.
-----Original Message-----
From: [email protected]
[mailto:[email protected]] On Behalf Of km
Sent: Monday, December 16, 2013 7:49 PM
To: [email protected]
Subject: Re: [Jprogramming] A complex question?
Dan, I haven't been following this thread, but know that minus the logarithm
of a positive number is the logarithm of the reciprocal. Is that relevant?
^. 1r4 1r2 1 2 4
_1.38629 _0.693147 0 0.693147 1.38629
--Kip
Sent from my iPad
> On Dec 16, 2013, at 3:42 PM, Dan Bron <[email protected]> wrote:
>
> Raul wrote:
>> Is there a better way of doing this?
>> {: +. r.inv j./1 1
>
> Marshall responded:
>> You can also use (+/&.:*:) in place of |@j./ , leaving you with
>> -@^.@(+/&.:*:)"1
>
> Raul wrote:
>> Experimenting: the - is necessary and the ^. is not necessary.
>> (I do not get a hexagon without the minus, I do get a hexagon without
>> the ^.).
>
>> Immediately after writing this I realized the - is also unnecessary -
>> changing >./ to <./
>
> What I love is that through some simple trig and a few experiments, we
> got from {:@+.@(r.^:_1)@(j./) to +/&.:*: .
>
> I suppose I find this particularly gratifying because I spent some
> time trying to restate Raul's phrase in terms of simple arithmetic
> operations, staying entirely in the real domain, and I eventually
> reproduced Marshall's verb. Having spent so much time "simplifying",
> when I got the final, irreducible result, I wondered at the need for
> -@^. , and what its physical interpretation was.
>
> Raul's original verb could be rendered in English as "the length
> component of a polar coordinate (initially specified in Cartesian
> terms)". Why should that length be expressed as the negative log of a
> distance? Why not, as Don put it, "the raw distance"?
>
> I know there are subtle and beautiful connections between the
> trigonometric and exponential functions, and the e hidden in r. is one
> expression of that. But I'm still not seeing the fundamental physical
interpretation.
> In other words, I wasn't surprised with the -@^. disappeared in Raul's
> use case; I might've been more surprised if it'd persisted.
>
> Anyone want to help me see it? Maybe the best illustration would be a
> concrete use case where the -@^. isn't superfluous - one where where
> it is not only necessary, but inevitable?
>
> That is, a use case where -@^. has obvious physical interpretation,
> when applied to the distance. Ideally one like Raul's, which
> ultimately didn't involve complex numbers (i.e. a real-valued binary
> [dyadic] operation on real numbers).
>
> -Dan
>
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