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
