The proof refers back to identity A1, and there it says +/5⍴6 ←→ 6×5 and +/⍳5 ←→ (6×5)÷2, and A1 is +/⍳n ←→ ((n+1)×n)÷2. A close reading of the text (including the figures) surrounding A1 indicates that the RHS could easily (and more "naturally") be 5×6 and (5×6)÷2, respectively, and A1 is more naturally +/⍳n ←→ (n×(n+1))÷2.
On Mon, Jul 27, 2015 at 7:40 AM, Roger Hui <[email protected]> wrote: > I had in fact marked the Lemma line and the immediately following line in > the Errata section of the webpage. > > On Mon, Jul 27, 2015 at 7:36 AM, 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
