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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 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 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... 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<w, right? (2exp^o.5(3) = ~11.9136) I am afraid the formula we would like to use is wrong. To be honest I thought of the operators between operators, like between two negations... RE: Between the ops - Gottfried - 04/01/2017 Upp, so I seem to have missed the point completely. Sorry! Gottfried RE: Between the ops - Xorter - 04/01/2017 (04/01/2017, 12:10 PM)Gottfried Wrote: Upp, so I seem to have missed the point completely. Sorry! No problem, my friend. This is why I am trying to explain it clearlier. Well, I suppose that an operator must exist between the others (other ops and numbers). For instance: between two tensors: O O x = O^2 x = O □ O x, but what is this? Is it the same as which is between a number and the '='?: So 2 □ = This is not so clear like the fractional iterates of hyper ops. But, I think this way is more closeable/easier. |