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 > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
