The intention of the author was to prove ((n+1)×n)÷2, which led to the more complicated (+/(n+1)⍴n)÷2 instead of the simpler (+/n⍴n+1)÷2. I believe the author would have been amenable to changing his intention to prove (n×(n+1))÷2, all the way back to the section around A1.
The one instance of BC is a typo in the original paper and is noted in the errata section. For the transcription I decided that ordinary majuscules in APL code map to minuscules and underscored majuscules map to majuscules. This mapping was a common manoeuver when APL interfaces joined the wider world in the 1980's. In the original paper the BC that you mentioned are underscored majuscules. On Mon, Jul 27, 2015 at 8:30 AM, June Kim (김창준) <[email protected]> wrote: > Hi > > Thank you, Roger. I didn't know there was the errata at the end. That would > definitely help. > > I am aware that (+/(n⍴n+1))÷2 and (+/((n+1)⍴n))÷2 are equivalent but my > guess was the author's intention here could've been the former. > > One more typo that I found is bc. According to the facsimile version: > > http://www.eecg.toronto.edu/~jzhu/csc326/readings/iverson.pdf > > all the names that refer to the binomical coefficient function in the paper > are in upper cases, not lower cases. > > However, in the web version, it's somtimes BC and other times bc. > > Best, > > June > > On Mon, Jul 27, 2015 at 11:36 PM, Roger Hui <[email protected]> > wrote: > > > +/⍳n > > +/⌽⍳n + is associative and commutative > > ((+/⍳n)+(+/⌽⍳n))÷2 (x+x)÷2←→x > > (+/(⍳n)+(⌽⍳n))÷2 + is associative and commutative > > (+/((n+1)⍴n))÷2 Lemma > > ((n+1)×n)÷2 Definition of × > > > > If a typo is any deviation in transcription from the printed paper to the > > web page, I can tell you that there was one: > > > > (+/(⍳n)+(⌽⍳n))÷2 + is associative and commutative (current web > > page) > > (+/((⍳n)+(⌽⍳n)))÷2 + is associative and commutative (original > > text) > > > > I will correct this in the web page shortly. > > > > (The extra parens are unnecessary and make the expression harder to read, > > but they were there in the original.) > > > > Regarding the "Lemma" line: I agree that (⍳n)+(⌽⍳n) is more naturally > > n⍴n+1, since both are vectors of length n and each vector element is 1+n. > > That is, they are identical vectors. (n+1)⍴n has the same sum but is a > > different vector, and it is unnecessary in the proof to change the > vector. > > Nevertheless, ((n+1)⍴n) is what was in the original paper, > > > > (If the Lemma line is (+/n⍴n+1)÷2, that would lead to (n×(n+1))÷2 as the > > last line.) > > > > p.s. This is one of the areas in the paper where index-origin 0 would > have > > improved things. (There are no areas I know of there 0-origin would make > > things more complicated.) In 0-origin the sum is equivalent to 2!n. As > it > > is, in 1-origin, the sum is equivalent to 2!n+1. > > > > Another thing that would have improved things, is if # (tally) were > > available and used. > > > > > > > > > > On Mon, Jul 27, 2015 at 6:20 AM, June Kim (김창준) <[email protected]> > > wrote: > > > > > Hello > > > > > > I am still translating Iverson's paper. It really takes time. One more > > > question: > > > > > > <quote> > > > +/⍳n+/⌽⍳n+ is associative and commutative((+/⍳n)+(+/⌽⍳n))÷2(x+x)÷2←→x > > > (+/(⍳n)+(⌽⍳n))÷2+ is associative and commutative(+/((n+1)⍴n))÷2Lemma > > > ((n+1)×n)÷2Definition of × > > > > > > The fourth annotation above concerns an identity which, after > observation > > > of the pattern in the special case (⍳5)+(⌽⍳5) , might be considered > > obvious > > > or might be considered worthy of formal proof in a separate lemma. > > > > > > http://www.jsoftware.com/papers/tot.htm > > > > > > </quote> > > > > > > In the above, I think more natural way of thinking is, the fifth line > > > should've been instead (+/(n⍴n+1))÷2,since (⍳n)+(⌽⍳n) is identical to > > > n⍴n+1. > > > > > > Do you also think this was a typo? Or is there any other thing that I > am > > > missing? > > > > > > June > > > ---------------------------------------------------------------------- > > > For information about J forums see http://www.jsoftware.com/forums.htm > > ---------------------------------------------------------------------- > > For information about J forums see http://www.jsoftware.com/forums.htm > > > ---------------------------------------------------------------------- > For information about J forums see http://www.jsoftware.com/forums.htm > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
