04/01/2015, 03:20 PM

(03/31/2015, 09:16 PM)MphLee Wrote:(03/31/2015, 08:50 PM)JmsNxn Wrote: The goal is to do it in a much more general setting. I am working very hard on cleaning it up and making sure all the proofs pop out like clock work from the more general schema.I can't really follow some of the step but I think I got what your going for. The complex analysis that youre using is really crazy for me atm.

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I'm just trying to rigorize everything now. The skeleton is there, I just need to prop it up with some muscular rigor.

Anyways I always believed that one of the ways to reach the solution was inside operator theory (aka application of the analysis to higher order functions).

Quote: is a contraction operator, this proves invaluable to the generalization.

About the contraction operator, the concept is new for me (Operator theory is still new for me) but maybe I'm missing something...Did you present a normed space of functions in your paper? Because the contraction op. need the operator norm in order to be defined, and operator norm is defined using the norm(s) of the space(s) (if I recall correctly it is the operator norm is the smallest number that bounds the operator on its domain). Wich norm is used usually in these kind of frameworks? (sorry if the question is stupid).

To answer your question:

Quote:Wich norm is used usually in these kind of frameworks?

It is usually defined formally on any metric space. This can be generalized to many different types of mathematical objects. Sometimes they're functions, sometimes they're points on a plane, sometimes they're operators themselves acting on more operators.

The normed space part can be cleverly avoided. We do not need a complicated metric for the functions we are investigating. It's a certain type of contraction operator--namely one that works on points on a plane, not on the functions themselves.

For example in the paper we note that as for all in the immediate basin of attraction. This is equivalent to the statement that in the basin for some . We are contracting the points, based off the operator . It's a little tricky to wrap your head around but I'll explain it much more clearly in the next paper.

We do not have to create a normed space that the super function operator acts on, but we do have to talk about values converging to a fixed point under the super function operator. This will be equivalent to the values "contracting" under the point wise norm. Namely the family of functions we use are those such that as , with some additional conditions attached to . The notation is the notation for the super function operator that I am using. This also implies under the pointwise norm it is a type of contraction mapping. We don't need to induce a metric anymore than the traditional one, but again, the trick is hidden. I'm close to getting a well rounded rough copy out. I don't want to give away all the details at the moment but it's simpler than it seems.

I just have to make sure I don't have any holes in my ideas. I'm going to talk to some of my professors to make sure it all makes sense and the advanced theorems I need to reference are not used incorrectly. This is going to be more complicated than my last two papers, and at a few parts I feel like I'm out of my depth, but I'll manage. It will probably take a month or two to get this out well enough that I'm willing to post it.