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Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp
Yes.. good exercise to understand that it can not be so simple if these are normal continuos functions, which they of course may be not, or continuity idea itself needs to be changed to include some (un)limited jumps;

But then of course I have no idea how to do it right now. Full stop, and much better idea after good training how to work with hyperop notationSmile

Either f(x) = 0 or f(x) = 1 for all x.... hmm. If that function one could oscillate between those 2 values, it would than be 1 non-continuous function that does the job. Such infinitely oscillating function would also be interesting in terms of providing ultimate discretization for any continuous function since it should oscillate faster than there is difference between 2 real numbers, meaning the period is <smallest real number , frequency > biggest real number, (or, perhaps, straight to imaginary frequency) ?

So that every continuous real function when multiplied (composed?) by this function would be turned in 2 valued function, like bifurcation.

Then, I would like to extend this notion to functions oscillating fast between n constant values and applying such function to continuous function would make it n-furcating, etc. Also, we were looking with Gottfried at behaviour of convergence of certain iterations that exibit simple oscillating, triangular, polygonial apprach to limit point instead of smooth spiralling. There should be a link between those beahviours , n-furcations and oscillating functions.

Anyway, if there is a reason for chaos, it lies in higher speed operations invisible with normal mathematics being hidden in dx and continuity -which is questionable (the continuity ) in my opinion.It is only continuity defined within a set of certain operations. If we move outside those operations, continuity can not be taken for granted, as what seemed continuos when acting on it with slow operations, might prove discontinuos when amplified fast enough.

If tetration transforms some function let us say y=x into discontinuous domains, than the question was y=x truly continuous in all range of x is open. ( just considering variuos behaviours of h(x) at various domains of x)) .

If function h(x) which is just a function applied to x and established certain relationship between sets of values of h(x) and R, and changes its behaviour with jumps just because x changes values "continuosly" , then perhaps x itself (reals) is not continuous, or , at least, should not be considered linearly growing. So, the function or relationship is continuous, but numbers are not.

But I have to read more about Fourier transform and some chaos theory to see where is the clue.


Messages In This Thread
RE: Attempt to formally generalize log, exp functions to 3,4,5..(n,m) log exp - by Ivars - 06/01/2008, 05:02 PM

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