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Continuum sum = Continuum product
#1
One of my earliest teenage conjectures was this :

In the context of nonnegative real numbers consider

A = A

A + B = AB

A + B + C = ABC

...

It was clear that A could not be 0.
A = 2 works fine.

2 = 2

2 + 2 = 2*2

So to go to the next variable D we need to solve

2 + 2 + D = 2*2*D

or simpler

4 + D = 4D

4 = 3D

D = 4/3

And basicly to go further we need to solve recursively

D0 + D1 = D0*D1

D0 = (D0 - 1)D1
D1 = D0/(D0 - 1)

Now this clearly is an iterated Möbius transform which we understand well.

Notice the fixpoint

X = X/(X - 1)

X - 1 = 1 => X = 2.

We ignore the solution 0 here because as said in the beginning 0 "fails".

This is basicly the solution for

"discrete sum = discrete product"

Now the conjecture is

" Continuum sum = Continuum product "

Is given by the continuum iteration of X/(X-1).

Basicly just handwaving and saying what is true for positive integers should be true for positive reals.

***

I called equations involving continuum sums and/or continuum products " chaotic equations " or "chaos equations ".

I was also aware of the idea that f ' (x) = continuum product [ f(x) ] had tetration as a solution.

I tried to find some books about " chaos equations " but failed to find any.

I assumed I would learn about it later , which never happened.

Since I found the sinh method I forgot about it as it appeared the " best solution ".
At that time I did not know what complex analytic meant hence it seemed perfect. Unfortunately this somewhat made me forget about these " chaos equations ".
I also somewhat forgot about those " chaos equations " by the failure of finding closed forms solutions and the fear that all solutions would be as nonstandard as tetration. And ofcourse the lack of serious theory or tools to work with this kind of stuff.

I spoke to a few people about this and I think I mentioned it at sci.math but that did not help much.

Here and Now seems right to mention this stuff again.

Problem is these " chaos equations " require more than just a method to compute a continuum sum ; probably comparable to being able to do differentiation and solving PDE.

regards

tommy1729
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Messages In This Thread
Continuum sum = Continuum product - by tommy1729 - 08/21/2013, 11:27 PM
RE: Continuum sum = Continuum product - by JmsNxn - 08/22/2013, 04:01 PM

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