Very nicely explained.
On 12/16/2011 10:18 AM, Marshall Lochbaum wrote: > A right/left inverse under the operation of composition. So if > (+/ @: ri) -: ] > Then ri is a right inverse to +/ , while if > (li @: (+/)) -: ] > li is a left inverse of +/ . > You can see that a right inverse is trivial (try ,: ) but a left inverse > doesn't exist, because +/ discards information. > > Marshall > > On Fri, Dec 16, 2011 at 8:20 AM, Don Guinn<dongu...@gmail.com> wrote: > >> I don't understand what you mean by a right and left inverse to +/ . >> >> On Thu, Dec 15, 2011 at 7:52 AM, Marshall Lochbaum<mwlochb...@gmail.com >>> wrote: >> >>> Yes, except there are an infinite number of choices. In that sense, it's >>> more like trying to find +/^:_1 . It's easy to find a right inverse of >> +/ , >>> but there's no good choice of a left inverse because you'll be wrong in >>> most cases. The situation is a little better for #: , but the question of >>> finding an inverse is still fundamentally unsolvable. >>> We don't really use #: to provide an inverse to #. ; it's more the other >>> way around. We use #: because it allows us to express a number in a >>> different way which is useful for some applications. >>> >>> Marshall >>> >>> On Thu, Dec 15, 2011 at 8:19 AM, Don Guinn<dongu...@gmail.com> wrote: >>> >>>> Isn't the choice of the representation for #: result a lot like picking >>> the >>>> principle root? >>>> >>>> %:*:_2 >>>> 2 >>>> *:%:_2 >>>> _2 >>>> >>>> The solution to the first expression above should really be _2 2 but, >>>> though more correct, is impractical in actual problem solving. A >> similar >>>> thing occurs with circular functions. >>>> >>>> 1 o.0.5+0,o.2 >>>> 0.479426 0.479426 >>>> >>>> And that is why many proofs restrict functions to be single valued. >>>> >>>> On Wed, Dec 14, 2011 at 9:18 PM, Marshall Lochbaum< >> mwlochb...@gmail.com >>>>> wrote: >>>> >>>>> antibase2 has an inverse only for nonnegative numbers, given by #. >>>>> twoscomplement's inverse is [:#. (* _1^0=i.@#)"1 >>>>> signwithbits has inverse #. >>>>> >>>>> Marshall >>>>> >>>>> On Wed, Dec 14, 2011 at 11:10 PM, Kip Murray<k...@math.uh.edu> wrote: >>>>> >>>>>> Thank you, Raul. May we have inverses? >>>>>> >>>>>> On 12/14/2011 9:13 AM, Raul Miller wrote: >>>>>>> The subject line of this thread is arguably wrong -- there are a >>>>>>> variety of "good ways" of decomposing integers to binary. >>>>>>> >>>>>>> That said, it's interesting to think about the various proposals >>>>>>> expressed in terms similar to those which could be used to >>> implement >>>>>>> monadic #: >>>>>>> >>>>>>> antibase2=: #:~ 2 #~ 1 + 2<.@^. 1>.>./@,@:|@:<. >>>>>>> twoscomplement=: #:~ 2 #~ 1 + 2<.@^. 1 +>./@,@:|@:<. >>>>>>> signwithbits=: #:~ 0, 2 #~ 1 + 2<.@^. 1>.>./@,@:|@:<. >>>>>>> >>>>>>> (In all cases the #: here is dyadic, so these definitions are >>>>>>> independent of the definition of monadic #:) >>>>>>> >>>>>>> antibase2 i: 3 >>>>>>> 0 1 >>>>>>> 1 0 >>>>>>> 1 1 >>>>>>> 0 0 >>>>>>> 0 1 >>>>>>> 1 0 >>>>>>> 1 1 >>>>>>> twoscomplement i: 3 >>>>>>> 1 0 1 >>>>>>> 1 1 0 >>>>>>> 1 1 1 >>>>>>> 0 0 0 >>>>>>> 0 0 1 >>>>>>> 0 1 0 >>>>>>> 0 1 1 >>>>>>> signwithbits i: 3 >>>>>>> _1 0 1 >>>>>>> _1 1 0 >>>>>>> _1 1 1 >>>>>>> 0 0 0 >>>>>>> 0 0 1 >>>>>>> 0 1 0 >>>>>>> 0 1 1 >>>>>>> >>>>>>> There's also (* * #:) but that one assumes the antibase2 >>>>>> implementation... >>>>>>> >>>>>> >>> ---------------------------------------------------------------------- >>>>>> 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
