Physical Applications of Tetration
Ioannis
have any of you come across any references which link tetration to any
physical processes?
I have checked out Knoebel's article "Exponentials Reiterated", but the
applications he cites in biology are sort of lukewarm and are only indirectly
related to tetration.
I tried to look into nuclear fission but it looks as though such processes are
mainly growing as simple exponentials. Tetration seems to be too "fast" (or
too powerful if you wish) to describe anything physical in this universe.
I was thinking that perhaps this was a good reason why the hierarchy of
operators {+,*,^,^^,...} has a discrepancy at "^^". Maybe tetration is too
powerful a process to describe anything physical.
If any of you have come across any actual physical applications, (such as
something "growing" tetrationally, for example), I would appreciate the
reference.
Many thanks,
--
I.N. Galidakis --- http://ioannis.virtualcomposer2000.com/
Gottfried Helms
I found a text called "WexZal" , which deals with the x^^2 term.
Don't know about the relevance regarding your question. It was some
years ago, so I don't know, whether this document was continued,
or whether it is still online at all.
cite from Preface of WexZal:
>
> This book is about the solution to and properties of the Coupled
> Exponent equation (y=x^x). The solution to this equation is called the
> "Coupled Root function". This work details our research efforts since
> 1975. Included are computers/calculators used, evolution of ideas,
> history of our efforts and still outstanding problems. We have organized
> the work into different topics such as "Applications", "Solving logarithmic
> Equations", "Integration", etc. to make it easier for the reader to find
> a topic. This is a work where the appendices and tables are (in some ways)
> more important then the text itself. The text is to explain the theory;
> the tables have the actual items of interest.
> Our goal in writing this book is to show the (in our opinion) interesting
> things we found and to encourage research into this topic as we feel this
> is one area that has been mostly overlooked. We feel that the Coupled
> Root function has many hidden properties that have the potential to be
> useful. Two such applications have been found so far: Ballistics (internal
> & external) and automobile acceleration. There is no doubt other areas where
> the Coupled Root could be used.
HTH -
Gottfried Helms
JoeSh...@aol.com
deeper than just "nothing physical grows fast enough" -- it is that
there is no known satisfactory way of making tetration continuous
(that is, no canonical way that doesn't depend on arbitrary choices).
If there were some physical process which involved a generalized
"functional square root" then that would already be enough to
"explain" tetration (that is, if you could find f(x) so that f(f(x))
was an exponential, and g(x) so that g(g(x))=f(x), etc., then you
already can interpolate to get a set of functions indexed by the
reals, which can be regarded as a 2-variable function, which when the
other variable is fixed specializes to tetration).
There is not even a good example in physics that I know of for a "half-
exponential" -- a function which arises in physics whose rate of
growth is the functional square root of an exponential -- this is much
weaker than having a generalized functional square root, so it would
not solve the problem of interpreting tetration physically, but it is
a prerequisite that may be easier.
-- Joe Shipman
tc...@lsa.umich.edu
Ioannis <morp...@olympus.mons> wrote:
>I was thinking that perhaps this was a good reason why the hierarchy of
>operators {+,*,^,^^,...} has a discrepancy at "^^". Maybe tetration is too
>powerful a process to describe anything physical.
All of physics reduces to optimizing the action. O.K., maybe that's an
overstatement, but it seems hard to construct an action-minimization
problem whose solution is tetration. Is that a good enough reason?
--
Tim Chow tchow-at-alum-dot-mit-dot-edu
The range of our projectiles---even ... the artillery---however great, will
never exceed four of those miles of which as many thousand separate us from
the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences
bo198214
> I don't have an answer, but would like to point out that the issue is
> deeper than just "nothing physical grows fast enough" -- it is that
> there is no known satisfactory way of making tetration continuous
> (that is, no canonical way that doesn't depend on arbitrary choices).
Hm, at least for a similar topic exists a unique continuation.
We know that for those higher operations the bracketing is somewhat
arbitrary.
Tetration is *right-bracketed* x^(x^(x^....))), while the *left-
bracketing*
(..(x^x)^..)^x has already a standard continuation, i.e. x^(x^(r-1)).
But I want to point on something I would call *middle-bracketing*. We
want an operation # such that
x # 2 = x^x
x # 4 = (x^x)^(x^x) = (x#2)^(x#2)
x # 8 = (x#4)^(x#4)
....
x # 2^(n+1) = (x # 2^n )^(x # 2^n)
For doing this let $ be the operation defined inductively by
x $ 0 = x
x $ (n+1) = (x$n)^(x$n)
Then we can easily see that x $ n = x # 2^n. The interesting thing
however is that we can regard x$n as the n-times application of the
function f(x)=x^x, i.e. x $ n = f^n(x) := fo...of(x).
Now there is a unique analytic iteration group (though to put out all
details would take too much room here) for f, especially f^(1/n) has
exactly one analytic strictly increasing solution on x>1 such that it
has asymptotically at (the fixed point) 1 the unique formal
powerseries f^(1/n).
Hence we can define x$r in a unique way as f^r(x).
And then we go back, defining x#r = x $ (log_2 r).
x#r is then also a quite rapidly increasing operation (though probably
slower than x^^r but faster than any x^(x^(...x^r)) for a fixed number
of x's). Of course nobody can tell physics applications for *this*
function, if it was unknown yet.
There is a bunch of similar operations #, by letting the above f for
example be f(x)=x^(x^x) or f(x)=x^(x^2) and then taking the log_3
instead of log_2, or doing so with other bracketings of n x's and
taking log_n instead.
Ioannis
news:euj55l$a5u$1...@news.ks.uiuc.edu...
>
> I don't have an answer, but would like to point out that the issue is
> deeper than just "nothing physical grows fast enough" -- it is that
> there is no known satisfactory way of making tetration continuous
> (that is, no canonical way that doesn't depend on arbitrary choices).
Why is this "the issue"? Many exponential models in physics and biology simply
manifest only with discrete values. Continuation is not necessary for the
understanding of the model. Consider cancer for example: The cell division
pattern follows the sequence 1,2,4,8,16,32,...,2^n,...
The previous process does not have to exhibit a continuous behavior to be
understood. Cells are distinct entities, and their division is a discrete
process. In particular, the function f(x) = 2^x = exp(ln(2)*x) is far from
necessary to understand cancer.
Something similar holds for the number of neutrons produced in nuclear fission
and fusion. The total numbers of nucleons produced, are in all cases simple
exponential sequences of time.
The above processes are completely understood (barring the cause in the case
of cancer of course) without resorting to a continuous function.
In a similar spirit, I cannot see why some phenomenon in nature cannot grow as
1,2,4,16,65536,....,2^^n,... or even as [f(n)], where f(f(x))=exp(x), and
where [] is the integer part function, particularly since in this case f(x) <
exp(x).
[rest snipped for brevity and many thanks for all the responses]
> -- Joe Shipman
Daniel Geisler
> In article <euh43o$gp...@news.ks.uiuc.edu>,
I agree that tetration is an unlikely action, but what about tetration
as the machinery upon which the action runs? Wouldn't removing the
integrals from the Feynman Path Integral result in tetration? In This
Week's Finds in Mathematical Physics (Week 251) John Baez discusses
foils for quantum mechanics; see
http://tetration.org/tetration_net/tetration_net_Physics.htm for my
own thoughts on tetration as such a foil. If removing the integrals
from the FPI cause it the degenerate into tetration, maybe the FPI is
a simple variant of tetration. My question is this; is there any way
to exclude tetration as a alternate basis of the FPI?
Thanks,
Daniel