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
