I would like to read the pdf. But I do not feel like looking up my password.
-- Raul On Thu, Jun 6, 2013 at 4:13 PM, Bo Jacoby <[email protected]> wrote: > @ Raul: Yes, one identity may have many proofs. > Why don't you want to download the PFD? What is the facebook account problem? > - Bo > > > > > >>________________________________ >> Fra: Raul Miller <[email protected]> >>Til: Programming forum <[email protected]> >>Sendt: 22:01 torsdag den 6. juni 2013 >>Emne: Re: [Jprogramming] Finding repeated substrings >> >> >>Note that you can have many proofs for the same identity. >> >>Also, I did not download your pdf, because I did not feel like signing >>into my facebook account. So, for example, I do not know how your >>identities treat the relationship between the pascal and sierpinski >>triangles. >> >>-- >>Raul >> >>On Thu, Jun 6, 2013 at 3:37 PM, Bo Jacoby <[email protected]> wrote: >>> Yes, Roger, but if you exclude the special cases, the remaining number of >>> identities to learn is, practically speaking, finite. I have a collection >>> of identities in >>> http://www.academia.edu/3247833/Statistical_induction_and_prediction >>> to supplement those in Concrete Mathematics. >>> >>> - Bo >>> >>> >>> >>> >>>>________________________________ >>>> Fra: Roger Hui <[email protected]> >>>>Til: Programming forum <[email protected]> >>>>Sendt: 18:41 torsdag den 6. juni 2013 >>>>Emne: Re: [Jprogramming] Finding repeated substrings >>>> >>>> >>>>To quote Graham, Knuth, & Patashnik, *Concrete Mathematics*, page 155: The >>>>numbers in Pascal's triangle satisfy, practically speaking, infinitely many >>>>identities, so it's not too surprising that we can find some surprising >>>>relationships by looking closely. >>>> >>>>The relationship you quoted, (>:x)!y ←→ +/x!i.y, can be generalized into a >>>>theorem that I called Pascal's >>>>Ladder<http://www.jsoftware.com/jwiki/Essays/Pascal%27s%20Ladder> (I >>>>like that better than "Hockey Stick Theorem"). >>>> >>>> >>>>On Thu, Jun 6, 2013 at 9:18 AM, Bo Jacoby <[email protected]> wrote: >>>> >>>>> The theorem that 2!y is equal to +/i.y is a special case of the more >>>>> general theorem that (>:x)!y is equal to +/x!i.y >>>>> >>>>> - Bo >>>>> >>>>> >>>>> >________________________________ >>>>> > Fra: Roger Hui <[email protected]> >>>>> >Til: Programming forum <[email protected]> >>>>> >Sendt: 16:41 torsdag den 6. juni 2013 >>>>> >Emne: Re: [Jprogramming] Finding repeated substrings >>>>> > >>>>> > >>>>> >There is a proof of a very similar theorem in section 1.4 of *Notation as >>>>> a >>>>> >Tool of Thought <http://www.jsoftware.com/papers/tot.htm>*. (The >>>>> >difference is that index origin is 1 in the paper.) >>>>> > >>>>> > >>>>> > >>>>> >On Thu, Jun 6, 2013 at 6:44 AM, Raul Miller <[email protected]> >>>>> wrote: >>>>> > >>>>> >> Caution: this code can give an incomplete result. For example, I do >>>>> >> not believe it will find 'aabaab'. Rather than fix this, I'll defer to >>>>> >> other solutions in this thread (which I imagine properly address this >>>>> >> issue). >>>>> >> >>>>> >> If anyone wants to take this code and fix it, the first instance of 2 >>>>> >> -~/\ ] should be replaced with a mechanism that treats all >>>>> >> combinations of 2 (and not just adjacent pairs). >>>>> >> >>>>> >> (And on that note, I Tracy Harms recently directed my attention to a >>>>> >> page with a beautiful proof that 2&! is +/@i. - that concept would be >>>>> >> useful, here, I think. I wish I had recorded the url of that page. But >>>>> >> the gist of my thought is that it should be possible to go from y and >>>>> >> a member of i.2!y to a unique pair of two numbers in the range i.y, >>>>> >> and that might be a nice way of implementing this "combinations of 2" >>>>> >> function.) >>>>> >> >>>>> >> FYI, >>>>> >> >>>>> >> -- >>>>> >> Raul >>>>> >> >>>>> >---------------------------------------------------------------------- >>>>> >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 >>---------------------------------------------------------------------- >>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
