New terminological standard for superfunctions.

[JmsNxn]

yeah but it does not matter if we solve 

f(2x) = exp(f(x))

or 

g(x+1) = exp(g(x))

essentially those equations are equivalent.
And even though they might not hold everywhere , where they do also transfers when you change one into the other.

Instead of using new names, just write out the equations explicitly.
In particular if we want to publish a readable paper we should define things as clear as possible rather then making up new terminology or symbols.

Unless you find different methods for different solutions of those different morphisms I see no usefulness ?

I might be wrong about those branches ... Not sure what equations were valid there ... Maybe my memory plays tricks on me.

But If I may advertise one of my questions ; special cases of similar functional equations are still unsolved :

https://math.stackexchange.com/questions...ght2-1-fz2

That seems to be within the spirit of these functional equations.

Sorry for being hard on a new valuable member or posting this at the wrong place , but im surprised at such a poll.

regards

tommy1729

Again, Tommy. I think you misunderstand the point.

What do we call a map, such that,



If we are in a space where, inversion (and hence, conjugation) doesn't make sense.

This is just to agree upon terminology. Mphlee is working on the categorical aspect, and he's wondering if there's a good word; or trying to find an agreed upon word.

This is incredibly relevant to this forum, because it's the principle of the base change function; but we're not necessarily assuming invertability.

[tommy1729]

yeah but it does not matter if we solve 

f(2x) = exp(f(x))

or 

g(x+1) = exp(g(x))

essentially those equations are equivalent.
And even though they might not hold everywhere , where they do also transfers when you change one into the other.

Instead of using new names, just write out the equations explicitly.

...

Again, Tommy. I think you misunderstand the point.

What do we call a map, such that,



If we are in a space where, inversion (and hence, conjugation) doesn't make sense.

This is just to agree upon terminology. Mphlee is working on the categorical aspect, and he's wondering if there's a good word; or trying to find an agreed upon word.

This is incredibly relevant to this forum, because it's the principle of the base change function; but we're not necessarily assuming invertability.

Yeah I misunderstand the point or I miss the point assuming there is one.

You talk about space and inversion.

What space ??

What inversion ??

I can define the exponential function for square matrices , real numbers , complex numbers and a few other numbers.
In general use taylor series.

But for nonassociative numbers or things that are not numbers I have no idea how to define them.

I can not define exp(x) when x is a space , operator , set , ... 
I only know exp(x) when x is a ( traditional type ) number ( including a finite set like mod p or a cardinal number )

And what about inversion ?
Set inversion ? group inversion ? functional inversion ? multiplicative inversion ?  
And since we are apparantly not talking about complex numbers or riemann surfaces , what do you mean by conjugation ?

So before I replied I started reading about " space " and exp but I was not able to find a definition for exp of a " general " or " nontrivial " space.

And do we even still have exp(-x) = 1/exp(x) since we apparantly "lost" inversion ??
So we also lost substraction ?

And how is all this space thing important for tetration and base change anyway ?

Btw I thought we were here to find (complex-)analytic tetration , so why go into non-analytic subjects and then call them crucial ??

Sorry to ask.

regards

tommy1729

[MphLee]

Yeah I misunderstand the point or I miss the point assuming there is one.

You talk about space and inversion.

What space ??

What inversion ??

[...]

Btw I thought we were here to find (complex-)analytic tetration , so why go into non-analytic subjects and then call them crucial ??

Sorry to ask.

regards

tommy1729

There is indeed a point but this is not "the point" (excuse me for this joke xD). The crucial thing: is this "point" relevant for your approach to the subject?
I admit that probably I didn't underline too much the link with classical complex/analytic iteration theory. Sometimes mathematicans should ad a grain of marketing in their expositions to get clearly the point. I suggest you to read my thread on the generalized superfuntion tricks.

Start from the assumption that my effort is foundational in its goal, algebraic in its essence and philosophical in its inspiration. You can regard it as an effort to unify and find a shared lingua franca for describing many constructions that we can observe are common in Kneser's, Walker's, limit solutions to functional equations, Abel/Schroeder equations, iteration theory, Matrix methods and James' theory of compositional calculus. I'm not going to explain it to you right now since I'd like first to make you sure that there is or there is not "a point" at all.

Is it crucial? Maybe? Maybe not. Is it useful? Idk, can the design of a simplified conceptual framework that unifies a bunch of apparently different constructions and definitions be valuable or useful? Yes!? It is indeed what a mathematician does..

Do I have a new method to compute complex iterations of exp? Nope.
Am I here to find complex-analytic tetration? Nope. I'm here to gain insight from the users' approaches to iteration and hyperoperations. I'm here because I want to understand deeply the nature of iteration and composition with the ultimate goal of expressing a true theory of non-integer ranks (hyperoperations with non-integer ranks).

Am I off-topic here at the tetration forum? Idk, I hope not.

When James talks of "spaces" he just means "sets of functions" with some nice structure on it (like composition and/or function inversion).
For example the real numbers constitute a space in some sense: points are real numbers and upon them there are metric, topological and field structures also a vector space structure and R-algebra and so on. Functional analysis is full of function spaces. We also have monoids of functions under composition, groups of invertible functions, rings of polynomials, ecc.

What I'm trying to do is to make those term "space" and "structure" formal and precise. I'm also trying to discover if other fields of math play with our same structures but under different names.

I hope to be helpful.
Regards