Generalized arithmetic operator hixidom Junior Fellow  Posts: 20 Threads: 3 Joined: Feb 2014 03/12/2014, 06:43 PM In this thread, there is mention of "Trappmann's Balanced Hyper-operator", and then there is a small section on it at the very end of the FAQ. I guess I finally have to learn about the Lambert W function... tommy1729 Ultimate Fellow     Posts: 1,852 Threads: 399 Joined: Feb 2009 03/21/2014, 10:31 PM https://sites.google.com/site/tommy1729/...e-property We use a uniqueness condition on sexp : for x,y >=0 : sexp(x+yi) is real entire. We could change the base e to base 2 or change tetration to pentation to generalize things. Imho that is the way to do hyperoperation and I believe that answers almost all questions. ( I read your paper ). Imho there are 2 big questions remaining : 1) informally speaking : what lies between tetration and pentation ? Once again I mean the " half-super functions " as has been discussed on this forum before ( mainly by myself and James Nixon ). Let S mean "superfunction of ..." and S^[-1] "is the superfunction of ..." We have S^[-1](f(x)) = f ( f^[-1](x)+1) examples : S(exp(x)) = sexp(x) S^[-1](sexp(x)) = sexp(slog(x)+1) = exp(x) Question : if we say S^[a+b](f(x)) = S^[a](S^[b](f(x)) = S^[b](S^[a](f(x)) Then what is S^[1/2](f(x)) ? Or what is S^[1/2](exp(x)) ? (Question 2 is still under investigation and not formulated yet) regards tommy1729 hixidom Junior Fellow  Posts: 20 Threads: 3 Joined: Feb 2014 03/22/2014, 12:06 AM (This post was last modified: 03/22/2014, 12:09 AM by hixidom.) Quote:what lies between tetration and pentation ? Do we know what lies between addition and multiplication, or multiplication and exponentiation? I would be happy to know those first. I assume they would be simpler to find, but I can also imagine that they would be equally difficult to find. Quote:Question : if we say S^[a+b](f(x)) = S^[a](S^[b](f(x)) = S^[b](S^[a](f(x)) Then what is S^[1/2](f(x)) ? Or what is S^[1/2](exp(x)) ? I found an answer to part of your question. By that I mean I was able to find S^[1/2](exp(x)): By definition: So we are trying to find some function such that If we define such that then , where W is the Lambert W function There is your half-superfunction of exp(x). tommy1729 Ultimate Fellow     Posts: 1,852 Threads: 399 Joined: Feb 2009 03/22/2014, 12:13 AM (03/22/2014, 12:06 AM)hixidom Wrote: Quote:what lies between tetration and pentation ? Do we know what lies between addition and multiplication, or multiplication and exponentiation? I would be happy to know those first. I assume they would be simpler to find, but I can also imagine that they would be equally difficult to find. Quote:Question : if we say S^[a+b](f(x)) = S^[a](S^[b](f(x)) = S^[b](S^[a](f(x)) Then what is S^[1/2](f(x)) ? Or what is S^[1/2](exp(x)) ? I found an answer to part of your question. By that I mean I was able to find S^[1/2](exp(x)): By definition: So we are trying to find some function such that If we define such that then , where W is the Lambert W function There is your half-superfunction of exp(x). Sorry for not using tex before but By definition: that is sufficient to see your answer is wrong ... Sorry. regards tommy1729 hixidom Junior Fellow  Posts: 20 Threads: 3 Joined: Feb 2014 03/22/2014, 12:42 AM (This post was last modified: 03/22/2014, 12:47 AM by hixidom.) Ah. I see. By half-superfunction, I thought you meant the superfunction of exp(x), S(x,n), evaluated at n=1/2. But I guess you're talking about the superfunction of S(x,n), , evaluated at m=1/2. Since S is a function of 2 variables, I guess I have to ask... Is , , or ? MphLee Long Time Fellow    Posts: 373 Threads: 29 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: means that takes a function and gives a function calles superfunction of such that satisfies 1) since there are infinite solution for (infinite superfunctions) that means that 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 the -based superfunction of the function that satifies two requirements 1) 2) and we have 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 (iteration of ) that would mean having 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 is a function and is the half superfunction. example : let define and we have (multiplication is the 0-based superfunction of addition) so we search for a such that and that if we should have (maybe...) and I apologize if I did some mistakes. MSE MphLee Mother Law $$(\sigma+1)0=\sigma (\sigma+1)$$ S Law $$\bigcirc_f^{\lambda}\square_f^{\lambda^+}(g)=\square_g^{\lambda}\bigcirc_g^{\lambda^+}(f)$$ hixidom Junior Fellow  Posts: 20 Threads: 3 Joined: Feb 2014 06/11/2014, 05:10 PM (This post was last modified: 06/11/2014, 05:11 PM by hixidom.) So here is a link to the updated document. I've added a little bit on non-integer iteration of the [x] operator as well as [x] for non-integer x. I used the results to write matlab code that plots over ranges of a, n, and x values. The plots are also in the document. There are still some limitations, but the expansion method (See http://arxiv.org/pdf/hep-th/9707206v2.pdf, pg.31) seems to work very well for x<3 and a,n<2. blows up for larger values of a and/or n, as expected, and the expansion produces poor results for x>3, since I currently only know inverse operations for , , and , and so my expansions for non-integer x are limited to 4 terms. Plot over a: Plot over n: Plot over x: « Next Oldest | Next Newest »

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