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
