@ Raul. I didn't know that downloading the PDF requested your password. Too bad. What can be done? - Bo
>________________________________ > Fra: Raul Miller <[email protected]> >Til: Programming forum <[email protected]> >Sendt: 22:19 torsdag den 6. juni 2013 >Emne: Re: [Jprogramming] Finding repeated substrings > > >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 > > ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
