01/26/2008, 03:52 PM
Dear Ivars!
Concerning:
In my comments, I chose to start defining y = b # x as the tetrational function to be analyzed (b-tetra-x), for 0 < b < e^(-e) [i.e.: in 0 ..... Beta, where Beta is 0.06598036..]. Generally, it is a ternary relation among three variables (y, b, x) or a binary operation (y, x) with parameter b.
Moreover, if parameter b is a positive constant number (for ... any constant b > 0), y = b # x is (or .. we hope that it is) a function y = f(x), which we expect to be ... continuous and smooth. :>
Having said that, let me try to fix our ideas in examining the y = b#x "function":
- x: is the independent variable, the "prototype" of which could be ... the independently running "time";
- b: is the tetration "base", provisionally considered as a constant, similarly to what we admit when we study y = b ^ x;
- y: is the result of the operation (or the "function");
- h: is used in the literature to indicate what I am calling "the infinite tower height" i.e.: h = (x->+oo)lim(b # x) = b # oo; we must observe that b # +oo can be real or complex, finite or infinite (variable with b), while b ^ +oo is always +oo, as far as I know, for b>0. ;> In some cases, as we may see, it can be non-determinated, between two extreme values. Well, the latter case describes exactly the situation when 0 < b < Beta [b = e^(-e)].
In this framework, we discovered that h = b#oo is itself a "function" of b, inverse of b = y-root(y), with complex solutions for b > Eta, double real solutions (a two-valued real function) for 1 < b < Eta, one real solution for Beta < b < 1 and undetermined values in 0 < b < Beta.
Actually, in 0 < b < Beta, we know three values of h, which we might identify as:
- h/mid = y/mid = - plog[-ln(x)/ln(x);
- h/sup = (x->+oo)lim[y/sup];
- h/inf = (x->+oo)lim[y/inf].
Then, attention, please! What I mean is that: I believe that function y = b # x, for b in the 0 .... Beta domain, oscillates, with oscillations between y/inf and y/sup, asymptotically decreasing towards h/inf and h/sup. These decreasing values are always and only verified for integer values of x (odd and even, respectively). The "jumps" among them are justified by the reationship:
y(x+1) = b^y(x) and are detectable at a minimum distance of one unit in the x axis. Also: the h/inf and h/sup are the asymptotic values of y/inf and y/sup, respectively, for x -> +oo. Think of a similar situation of a sinus function having, for x -> +oo, a residual oscillation around 0, between h/inf and h/sup. Its asymptotic behaviour, for x -< +oo, could be described as being undeterminad between h/inf and h/sup, with an average at y = 0. Non-orthodox, but true
!
As you see, no infinitesimal distance on x axis between the results should, in any case, appear. Great Euler should have said:
"Functio non facit saltus". But, probably he never did it.
This I what I was trying to say. As somebody once said: "Please pay attention to it, because it might be interesting, but dont take it for granted, because ... I might be wrong!"
GFR
Concerning:
Ivars Wrote:Ref.:Ops, ... sorry again! I shall try now to be as clear as possible, but I believe that the "plots" are more explicit than any linguistic explanation, particularly when the available terminology is not final and agreed upon, as in our present situation. Nevertheless, let's try it, hoping not to create furher confusion. :<
GFR Wrote:@ IVARSI was studying it so long but its still above my head. I wanted to see oscillating behaviour as n=infinity (both odd and even) in region x<e^-e- so that the jumps between 2 infinite tetration of x where one is odd, one is even, next odd again etc. happen via rotation in perpendicular plane. As both h odd and h even as n-> infinity are continuous functions, if we infinitely backtetrate them , there has to be a distance on x axis between the results, but its infinitesimal.
Please see the attachment, in which I try to imagine the situation represented in a 3-dimensional environment (y,b,x). The plots of y=b#x are the (appromimated and simulated) real projections on the yx plane of the y=d#x complex helicoidal "functions". The transition (yellow, off-limit) zone on the yb diagram is (in my ... simple opinion) the graph of the max-min span of the undetermined y values, when x -> oo. Persintent oscillations. Difficult to describe, ... see the plots.
.....
In my comments, I chose to start defining y = b # x as the tetrational function to be analyzed (b-tetra-x), for 0 < b < e^(-e) [i.e.: in 0 ..... Beta, where Beta is 0.06598036..]. Generally, it is a ternary relation among three variables (y, b, x) or a binary operation (y, x) with parameter b.
Moreover, if parameter b is a positive constant number (for ... any constant b > 0), y = b # x is (or .. we hope that it is) a function y = f(x), which we expect to be ... continuous and smooth. :>
Having said that, let me try to fix our ideas in examining the y = b#x "function":
- x: is the independent variable, the "prototype" of which could be ... the independently running "time";
- b: is the tetration "base", provisionally considered as a constant, similarly to what we admit when we study y = b ^ x;
- y: is the result of the operation (or the "function");
- h: is used in the literature to indicate what I am calling "the infinite tower height" i.e.: h = (x->+oo)lim(b # x) = b # oo; we must observe that b # +oo can be real or complex, finite or infinite (variable with b), while b ^ +oo is always +oo, as far as I know, for b>0. ;> In some cases, as we may see, it can be non-determinated, between two extreme values. Well, the latter case describes exactly the situation when 0 < b < Beta [b = e^(-e)].
In this framework, we discovered that h = b#oo is itself a "function" of b, inverse of b = y-root(y), with complex solutions for b > Eta, double real solutions (a two-valued real function) for 1 < b < Eta, one real solution for Beta < b < 1 and undetermined values in 0 < b < Beta.
Actually, in 0 < b < Beta, we know three values of h, which we might identify as:
- h/mid = y/mid = - plog[-ln(x)/ln(x);
- h/sup = (x->+oo)lim[y/sup];
- h/inf = (x->+oo)lim[y/inf].
Then, attention, please! What I mean is that: I believe that function y = b # x, for b in the 0 .... Beta domain, oscillates, with oscillations between y/inf and y/sup, asymptotically decreasing towards h/inf and h/sup. These decreasing values are always and only verified for integer values of x (odd and even, respectively). The "jumps" among them are justified by the reationship:
y(x+1) = b^y(x) and are detectable at a minimum distance of one unit in the x axis. Also: the h/inf and h/sup are the asymptotic values of y/inf and y/sup, respectively, for x -> +oo. Think of a similar situation of a sinus function having, for x -> +oo, a residual oscillation around 0, between h/inf and h/sup. Its asymptotic behaviour, for x -< +oo, could be described as being undeterminad between h/inf and h/sup, with an average at y = 0. Non-orthodox, but true
!As you see, no infinitesimal distance on x axis between the results should, in any case, appear. Great Euler should have said:
"Functio non facit saltus". But, probably he never did it.
This I what I was trying to say. As somebody once said: "Please pay attention to it, because it might be interesting, but dont take it for granted, because ... I might be wrong!"
GFR