Between the ops - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Between the ops (/showthread.php?tid=1160) Between the ops - Xorter - 03/29/2017 I suppose that between evertyhing is there something. E. g. xy = x*y (multiplication). I suppose that between every operators must be there a multiplication, let us call it operational multiplication: □ For example: xy = x*y 123 = 1*100 + 2*10 + 3 ¬x = ¬ □ x ¬¬x = ¬ □ ¬ □ x (x = y) = (x □ = □ y) etc ... And as the "normal" multiplication has power and functional multiplication (f o c = f( c )) has functional power (f^oN = f o f o ... o f) as the operational multiplication has operational power and roots: O □ O □ ... □ O = O^□N For instance: (¬¬x)^□0.5 = x or ¬x, because id id x = x and ¬¬x = x, right? What do you think, is it exist or not? Can we substitute operational multiplication with other multiplication, like the functional or not? RE: Between the ops - Gottfried - 03/30/2017 Hmm, Xorter, I'm not sure I do actually understand what you're after. But well, in case this is in line...           I've one time considered the relation between $a + b$ and $a * b$ as interpolatable. The key was that $a * b = \log( \exp(a) + \exp(b) )$   and the addition can be written as $\log^0( \exp^0(a) + \exp^0(b) )$ compared with the multiplication $a * b = \log^1( \exp^1(a) + \exp^1(b) )$ . So the idea was to define a continuum of fractional-order operators between "+" and "*" based on fractional iterates of $\log()$ and $\exp()$ .                      Well, this has surely a lot of issues, for instance do we want to have that fractional-order operations associative, commutative and so on, remembering, that such properties are reduced when we extrapolate to higher/lower orders by higher/lower iterates of the $\log()$ and $\exp()$ -functions. Also I didn't find this really promising/interesting so I did no more engage much in this ansatz. Gottfried RE: Between the ops - Xorter - 04/01/2017 (03/30/2017, 07:39 PM)Gottfried Wrote: Hmm, Xorter, I'm not sure I do actually understand what you're after. But well, in case this is in line...           I've one time considered the relation between $a + b$ and $a * b$ as interpolatable. The key was that $a * b = \log( \exp(a) + \exp(b) )$   and the addition can be written as $\log^0( \exp^0(a) + \exp^0(b) )$ compared with the multiplication $a * b = \log^1( \exp^1(a) + \exp^1(b) )$ . So the idea was to define a continuum of fractional-order operators between "+" and "*" based on fractional iterates of $\log()$ and $\exp()$ .                      Well, this has surely a lot of issues, for instance do we want to have that fractional-order operations associative, commutative and so on, remembering, that such properties are reduced when we extrapolate to higher/lower orders by higher/lower iterates of the $\log()$ and $\exp()$ -functions. Also I didn't find this really promising/interesting so I did no more engage much in this ansatz. Gottfried Gottfried! I am afraid your topic is not so relevant or close to mine, to this one. But never mind (but please, open a new topic for it). Maybe I was not so clear. Of course, I am answering for you according to my knowledge. If I understand your point, then x[.5]y should be equal to log^o.5(exp^o.5(x)+exp^o.5(y)) (between the addition and multiplication), right? 3+3 = 6 3*3 = 9 And look at this function: https://www.dropbox.com/s/xr0swdwug44abtm/log%5Eo.5.jpg?dl=0 The fractional (half-)iterate of exp and log functions according to whose graph I created 3[.5]3 is about 4.62197 < 6 which is a paradox if we suppose that x[z]y < x[w]y if z