[Jprogramming] A complex question?

[km]

   NB. Identities for ^.@* and ^.@% and ^.@^ with complex arguments
   
   prodid =: 1 = [: ^ ^.@* - +&^.
   
   quotid =: 1 = [: ^ ^.@% - -&^.
   
   pwrid =: 1 = [: ^ ^.@^ - (* ^.)~
   
   NB. Remark ^ is periodic with period 0j2p1
   
   periodtest =: 2 : '(v"0 y) = v"0 y+m'
   
   0j2p1 periodtest ^ 2 0j3 4j_5
1 1 1
   
   ]xx =: 2 3j_4 5j6
2 3j_4 5j6
   
   ]yy =: _3 _4j_5 6j7
_3 _4j_5 6j7
   
   xx prodid"0 yy
1 1 1
   
   xx quotid"0 yy
1 1 1
   
   xx pwrid"0 yy
1 1 1

   NB. Reciprocals are covered by 1 quotid"0 yy and 0 = ^. 1

--Kip Murray

Sent from my iPad

> On Dec 17, 2013, at 2:09 AM, "Linda Alvord" <lindaalvord at verizon.net> wrote:
> 
> Is this a true statement?
> 
>   NB. Taking the logarithm of the reciprocal of a 
>   NB. number changes the sign of all parts of the logarithm 
>   NB. the original number.
> 
>   N=: 4 2 1 0 0.5 0.25
> 
>   ^. N
> 1.38629 0.693147 0 __ _0.693147 _1.38629
> 
>   ]R=: % N
> 0.25 0.5 1 _ 2 4
> 
>   ^. R
> _1.38629 _0.693147 0 _ 0.693147 1.38629
> 
>   C=:1j1 _1j1 _1j_1 1j_1 
> 
>   ^.C
> 0.346574j0.785398 0.346574j2.35619 0.346574j_2.35619 0.346574j_0.785398
> 
>   ]RC=: % C
> 0.5j_0.5 _0.5j_0.5 _0.5j0.5 0.5j0.5
> 
>   ^. RC
> _0.346574j_0.785398 _0.346574j_2.35619 _0.346574j2.35619 _0.346574j0.785398
> 
> Linda
> 
> From: programming-bounces at forums.jsoftware.com
> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of km
> Sent: Monday, December 16, 2013 10:54 PM
> To: programming at jsoftware.com
> Subject: Re: [Jprogramming] A complex question?
> 
> Off topic: I now know that taking the logarithm of the reciprocal of a
> non-zero real number changes the sign of the real part of the logarithm of
> the original number:
> 
>   csrp NB. change sign of real part
> (1r2 * -@(+ +) + (- +))"0
>   csrp 1j2 _3j_4
> _1j2 3j_4
>   ]rr =: 1 % 4 2 1 0.5 0.25  NB. non-zero reals
> 0.25 0.5 1 2 4
>   (^.@% -: csrp@^.) rr
> 1
>   (^.@% -: csrp@^.) -rr
> 1
> 
> --Kip
> 
> Sent from my iPad
> 
>> On Dec 16, 2013, at 7:02 PM, "Dan Bron" <j at bron.us> wrote:
>> 
>> 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: programming-bounces at forums.jsoftware.com
>> [mailto:programming-bounces at forums.jsoftware.com] On Behalf Of km
>> Sent: Monday, December 16, 2013 7:49 PM
>> To: programming at jsoftware.com
>> 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 <j at bron.us> 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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