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 Generalized arithmetic operator MphLee Fellow Posts: 95 Threads: 7 Joined: May 2013 03/22/2014, 09:38 AM (This post was last modified: 03/22/2014, 03:01 PM by MphLee.) Superfunction is a multivalued function defined over a set of functions not over a set of numbers: $S(f)=F$ means that $S$ takes a function $f$ and gives a function $S(f)=F$ calles superfunction of $f$ such that $F$ satisfies 1) $F(x+1)=f(F(x))$ since there are infinite solution for $F$ (infinite superfunctions) that means that $S(f)=F$ is multivalued and then is not a function at all and we have to put some restrictions: using Trapmann-Kouznetsov terminology used in their paper "5+ methods..." we call $S_u(f)=F_u$ the $u$-based superfunction of $f$ the function $F_u(x)$ that satifies two requirements 1) $F_u(x+1)=f(F_u(x))$ 2) $F_u(0)=u$ and we have $F_u(n)=f^{\circ n}(u)$ In this way we obtain uniqueness over the naturals: in fact superfunction is equivalent to the "definition by recursion" that is unique . But is not over the reals... there we need more requirments. Obviously this is still not enough to achieve the uniqueness of $F_u$ (iteration of $f$) that would mean having $S$ to be a function over a set of functions (not multivalued). By the way I guess that Trapmann and Kouznetsov tried to find such additionals requirments but my math level is not enough to understand it. Anyways we have that $S^{\circ -1}$ is a function and $S^{\circ 1/2}$ is the half superfunction. example : let define $add_b(x)=b+x$ and $mul_b(x)=bx$ we have $S_0(add_b)=mul_b$ (multiplication is the 0-based superfunction of addition) so we search for a $S^{\circ 1/2}$ such that $S^{\circ 1/2}(S^{\circ 1/2}(add_b))=mul_b$ and that if $b[1,5]x=hyper-(1,5)_b(x)$ we should have (maybe...) $S^{\circ 1/2}(add_b)=hyper-(1,5)_b$ and $S^{\circ 1/2}(hyper-(1,5)_b)=mul_b$ I apologize if I did some mistakes. MathStackExchange account:MphLee « Next Oldest | Next Newest »

 Messages In This Thread Generalized arithmetic operator - by hixidom - 03/11/2014, 03:52 AM RE: Generalized arithmetic operator - by JmsNxn - 03/11/2014, 03:15 PM RE: Generalized arithmetic operator - by hixidom - 03/11/2014, 06:24 PM RE: Generalized arithmetic operator - by MphLee - 03/11/2014, 10:49 PM RE: Generalized arithmetic operator - by hixidom - 03/11/2014, 11:20 PM RE: Generalized arithmetic operator - by MphLee - 03/12/2014, 11:18 AM RE: Generalized arithmetic operator - by JmsNxn - 03/12/2014, 02:59 AM RE: Generalized arithmetic operator - by hixidom - 03/12/2014, 04:37 AM RE: Generalized arithmetic operator - by MphLee - 03/12/2014, 06:19 PM RE: Generalized arithmetic operator - by hixidom - 03/12/2014, 06:43 PM RE: Generalized arithmetic operator - by tommy1729 - 03/21/2014, 10:31 PM RE: Generalized arithmetic operator - by hixidom - 03/22/2014, 12:06 AM RE: Generalized arithmetic operator - by tommy1729 - 03/22/2014, 12:13 AM RE: Generalized arithmetic operator - by hixidom - 03/22/2014, 12:42 AM RE: Generalized arithmetic operator - by hixidom - 06/11/2014, 05:10 PM

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