@ 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
