We had a short discussion in the newsroup sci.math recently, which motivated me to put together some thoughts, which came up recently.
I got some nice responses, so I thought it might be interesting for this forum, too.
I had no time to answer to the reply of Ioannis up to now, I hope, Ioannis, you forgive me that delay. (I didn't copy your answer here and hope, you don't take this as an affront. I'll answer to it, once I have something substancial to ask or to say)
Hmm,
I'd like to add two comments. May be, I'm on a wrong track,
because this requires to change view of things a bit and
introduces other weaknesses, of which I'm not aware currently.
The first idea is even a change of a very common view, so
this is a special slippery path. Let me put it here anyway.
a) -----------------------------------------------------
I tend to the opinion, that we should change the view
of tetration, going away from the assumption, that it
means
b, b^b, b^b^b, b^b^b^...
with the dotted line *at the right* and
b^^n = b^b^...^b (n times)
There is already a notation, coined by Andrew Robbins (and I
actually don't know from which sources this notation stems)
which supports my other direction of view:
He defines
{b,x}^^n := b^b^b...^b^x
The idea is, that x is sort of starting value, and tetration
recursively appends bases (not exponents), so a coherent
notation is then for the sequence of partial expressions for
{b,x}^^inf is then
x, b^x, b^b^x, ...^b^b^x
with the dotted line at the left.
This seems to be an super-artificial difference, but has its
own impact. It means, for instance, that the evaluation of
partial expressions is different from (and in opposite direction to)
the common view of partial evaluation, which was discussed
in a thread here already.
The infinite powertower b^b^b^.... cannot take a special
value "at the top" - since there is no top. And in this view
it is true, that (using r=sqrt(2)) the infinite expression
r^r^r^.... approximated by partial expressions converges to 2
and nothing else, so 2 is the only solution (which also reflects
the view of Lambert, Euler and the descendent discussion).
But with this convention we have no tools to include the
occurence (multiple) "fixpoints" in our formula, and must
leave this problem open.
If we redefine tetration as appending bases to a certain
starting value, as in
{b,x}^^n := b^b^b...^b^x
{b,x}^^inf := ...^b^b^x
then also the infinite expression (for n->inf) makes sense
and is better suited to the concept of evaluation of
partial expressions. We may then legally insert the various
fixpoints into x and always have valid expressions.
{b,x0}^^inf := ...^b^b^x0 = x0
{b,x1}^^inf := ...^b^b^x1 = x1
...
where x0,x1,... are the fixpoints.
Note, that this convention is also more coherent with the
lot of research in iteration-theory and theory of dynamical
systems, where always an "initial state" is discussed, to
which then an operator is applied - one time, two times, and
in generalization even fractional or complex "times".
*Only* if this notation is accepted, then the following is
allowed:
{sqrt(2),2}^^inf = 2
{sqrt(2),4}^^inf = 4
...
This is *not* allowed (and we even cannot notate a second solution)
if we write
sqrt(2)^sqrt(2)^... = ??
then we can have only one solution and we have no second
parameter to refer to the different fixpoints.
(See my posting some monthes ago
subject: "sqrt(2)^sqrt(2)^sqrt(2)^... = 2 ? or 4 ?"
where I initiated a discussion of this)
Another spin-off of a redefinition is then the following.
Assume
{b,x}^^h = y
then, if x is already x = b^b^1 = {b,1}^^2, or more
general x = {b,z}^^g then we may do a bit of arithmetic
like
{b,x}^^h = {b,{b,z}^^g}^^h = {b,z}^^(g+h)
*iff* the bases for y and x are the same b.
ANd such an approach complies then easily to the idea
of a dynamical system, where an initial state (here: z)
is modified by an operator g and h times.
Well, *redefining* a common definition is easily a
crankish behave (see the discussion with A.Plutonium, for
instance), but in this case I'll take this risk
and propose a review of our definition.
b) ----------------------------------------------------
If a) is settled, then we may solve for x in
{b,x}^^inf = x
with multiple solutions, assigning branches to the h()-
function this way.
To find the various "fixpoints" then can be done by
a process, which reflects the idea of the newton-approximation,
for instance for finding the square-root of a number.
If we want to find x=sqrt(Z) we use an initial guess, say x0,
compute x1 = (Z/x0 + x0)/2 and iterate. This gives diminuishing
intervals for the error and approximates the solution to
arbitray precision.
With complex initial guesses for a fixpoint in the a)-definition
of infinite tetration, we can apply this idea equivalently.
The difference is here, that iterations spiral into the fixpoint
or spiral away - but seemingly always do spirals.
Assume an initial value, say x0, and apply an appropriate
function f(b,x0) and average
x1 = (f(g,x0) + x0)/2
x2 = (f(g,x1) + x1)/2
...
then a partial evaluation of such a spiral up to one approximate
circle gives a center-point near the "mean" of the spiral
(or say: the assumed convergence-point).
y1 = (x1+x2+x3.... xk)
which can then be a better approximation.
Then use x0 = y1 and iterate again up to one complete rotation
and iterate. This is somehow like throwing a lasso, whose
diameter tightens radically.
Depending on the initial guess we may find then the different
fixpoints for b (well, we may implement the Lambert-W-function
allowing different branches as well)
c) This "lasso"-method has another useful consequence: the
initial spiral may even diverge : it will still circle around a
center - and this center may then be nearer to the fixpoint
than the initial guess.
This may then be seen as an equivalent of Euler-summation of
alternating divergent series: the angle of "rotation" here is just
pi, and finding the "center" of a real-valued alternating series
is then just a special case of this "lasso"-process with two
steps:
x1 = something(x0) (negative value)
x2 = something(x1) (positive value again, "circle" closed)
(which is only a sketch here, since Euler-summation employs
a binomial-transform of the values)
So a),b) and c) may be a useful framework for the discussion of
fixpoints for complex tetration. I would like to see a better
formal description here (which I cannot supply due to lack of
knowledge) and then a check, whether such description agrees
with the needs of compatibility of the assumptions/results
with the whole surrounding scene of theory and application of
powerseries.
re a)
I would stay with the established notation, i.e. use
\( \exp_b^{\circ n}(x) \) or \( \exp_b^{[n]}(x) \)
instead of \( \{b,x\}^n \).
re b)
It was not clear what function \( f(b,x_0) \) do you want to use, I would guess \( f(b,x_0)=b^{x_0} \). It is also not clear whether this average method converges. All fixed points of \( b^x \) are repelling except the lower real fixed point (if existing).
re c)
Didnt get the Euler summation for diverging newton method ...
11/03/2007, 10:08 PM (This post was last modified: 11/03/2007, 10:38 PM by Gottfried.)
bo198214 Wrote:re a)
I would stay with the established notation, i.e. use
\( \exp_b^{\circ n}(x) \) or \( \exp_b^{[n]}(x) \)
instead of \( \{b,x\}^n \).
Well, I didn't want to put preferences for this more specialized notation. (as a matter of taste, I even would prefer the "big-T"-notation with parameters, as suggested here in another thread).
My main focus in a) was to point out the conceptual difference between the binary operator notation (two operands) and the notation which involves also the initial-value as a parameter. I think, it has consequences for the definition of partial evaluation, when infinite iteration is involved, as it occurs analoguously with the concept of partial sums for summable infinite series.
bo198214 Wrote:re b)
It was not clear what function \( f(b,x_0) \) do you want to use, I would guess \( f(b,x_0)=b^{x_0} \). It is also not clear whether this average method converges. All fixed points of \( b^x \) are repelling except the lower real fixed point (if existing).
Yeah, this point is an amusing one... I implemented a function out of the tip of my fingers, it worked, wow! but in a review next day, I thought, it shouldn't work at all.... I checked all results, but they were all ok. I even tried some complex parameters, for instance i - still ok :-)
The iteration is simply, with tmp the iterative improved guess
tmp1 = (base^tmp + a*tmp )/(a+1)
tmp = tmp1
where a=2 as a optimizing parameter (for most bases a=1..3 gave convergence, a~2 was average best for a reasonable range of bases)
Initial guess for tmp is the principal complex fixpoint for base=2.
This gives after some iterations values for a spiral, and if one round is reached (I check the next local minimum of distances to the first value), the next initial guess is the center of this approximate round/circle.
Why I was sceptical: a positive error in tmp seems to expand when averaging base^tmp and tmp - but, well, we're in complex numbers and so a rotation is involved, put it aside for later consideration...
bo198214 Wrote:re c)
Didnt get the Euler summation for diverging newton method ...
Hmm, c) is a bit speculative. Ok, also the Euler-sum analogy is not best chosen.
I see two problems with my b) and c):
First, from my arguments Cesaro-sum were a better analogy, because I use averages of untransformed values.
Second - it's not exactly the analogy to evaluate partial sums, which always sum from the first term (possibly including a regular transformation). The lasso-method is rather an approximation utility, which helps to improve guesses iteratively. May be, that is also the reason, that I was unable to transfer the method into my other matrix-context. ("Gut, daß wir mal darüber geredet haben" :-) )
In my contour-plot http://math.eretrandre.org/tetrationforu...432#pid432 one sees the lines as borders of the leaves, where each of this lines contain the whole set of positive reals (may be restricted to be greater than a certain bound) - so the lines are the (curves of) loci of the fixpoints for that real numbers. The line of the biggest leaf markes the set of fixpoints, that you may refer to as the principal fixpoints or something like that - I'll look at Gianfranco's wording in his reply tomorrow.
(I'm a bit sick today, so I'll stop the msg here. If you like and want to try/improve the Pari/Gp-function I could just upload the script)
11/04/2007, 03:23 PM (This post was last modified: 11/04/2007, 03:25 PM by GFR.)
Gottfried Wrote:........
I'll look at Gianfranco's wording in his reply tomorrow.
........
Gottfried
Hey Gottfried!!
How did you know that I was going to post a reply, ... to-day ?! As a matter of fact, ... I am doing it. Please forgive my wording if I am wrong. I should like to comment your nice push-down tower description. It is a thought I also have some time ago with KAR (He will come soon in the picture, as soon as I shall post a thread about ... zeration!!). Please look in this annex (sorry, no zeration, only ... tetrational number notation). I think that you are right, concerning the "last floor" of the inhomogeneous (incomplete) towers. Here, attached, you will find a proposal for their use. I hope you will find it interesting.
GFR Wrote:How did you know that I was going to post a reply, ... to-day ?!
Hi Gianfranco -
as a double sagittarius sometimes you hit the point without knowing it; and as a practionizer of Zen ... one learns, that the goal attracts the arrow if you simply let it happen... :-)
Well - good that happened. I've read your article: nice coincidence of ideas. Thanks!
I think, it has even the characteristic of a paradigm-change to move from the view as "appending exponents" to the view of "appending bases" - this seems to be much more important than the difficulties of introducing another notation. The question is: how do we embed our view and handling of tetration in the case of infinite height in the very basic and important concept of approximation via partial evaluation (partial sums). In the context of divergent summation, for instance, this seems the main and indispensable prerequisite of all statements about "the value" of an infinite divergent series.
In the understanding of "appending exponents" there is no differing top-exponent in the infinite case, and we have only one fixpoint.
In the understanding of "appending bases" we can deal with all fixpoint, constituting a more complete framework.
I think this is in fact a (however sophisiticated but not superfluous) question of definition of the operation.
Gottfried Wrote:I think, it has even the characteristic of a paradigm-change to move from the view as "appending exponents" to the view of "appending bases" - this seems to be much more important than the difficulties of introducing another notation. The question is: how do we embed our view and handling of tetration in the case of infinite height in the very basic and important concept of approximation via partial evaluation (partial sums).
"paradigm-change" is perhaps a bit exaggarated.
I didnt know of anybody who thinks of tetration as appending exponents. The right-bracketing rather forces one to use appending of bases, which is equivalent to \( {^nb}=f^{\circ n}(1) \) for \( f(x)=b^x \).
I also dont know what you mean by different partial evaluation, we already know that \( {^nb}=\{b,1\}^n \) in your notation. So (finite) tetration is just a particular case of appending bases. So the partial evaluations of \( {^\infty x} \) is exactly about appending bases. Or what do you mean by partial evaluation?
11/06/2007, 07:16 PM (This post was last modified: 01/06/2010, 07:43 AM by Gottfried.)
bo198214 Wrote:
Gottfried Wrote:I think, it has even the characteristic of a paradigm-change to move from the view as "appending exponents" to the view of "appending bases" - this seems to be much more important than the difficulties of introducing another notation. The question is: how do we embed our view and handling of tetration in the case of infinite height in the very basic and important concept of approximation via partial evaluation (partial sums).
"paradigm-change" is perhaps a bit exaggarated.
I didnt know of anybody who thinks of tetration as appending exponents. The right-bracketing rather forces one to use appending of bases, which is equivalent to \( {^nb}=f^{\circ n}(1) \) for \( f(x)=b^x \).
I also dont know what you mean by different partial evaluation, we already know that \( {^nb}=\{b,1\}^n \) in your notation. So (finite) tetration is just a particular case of appending bases. So the partial evaluations of \( {^\infty x} \) is exactly about appending bases. Or what do you mean by partial evaluation?
In a sci.math thread I posed this question, when I started metabolizing the fixpoint idea: what is sqrt(2)^sqrt(2)^... =2? = 4?
(I can't find the google-id, here is another one)
And then showed, that the initial assumption of any of these possibility could be valid, and that this made me think....
The argument was, that partial evaluation had to proceed *always* from sqrt(2), (sqrt(2)^sqrt(2)), ... and this way converges to 2 and thus no other possibility occured. (Either by argument or by my own mind only the analogy to partial sums of inifnite series appeared) That was made as a very important point, otherwise we would introduce inconsistencies.
I finally agreed to the arguments, but the discussion put a seed in my thoughts which evolved then to the view, that it is really a different *definition* or -to make it more explicite- a different paradigm in regard to the view of tetration. (well - may be this expression is a bit harsh)
Hi everyone , this is Andrewan. I am professionally an engineer. At present, I am staying in Moscow. I find forum discussion, a very interesting stuff to do. I used to hang out with these forums during my spare time. Many topics have got a healthy discussion over here. I think it would be most enjoyable while doing discussions in this forums.And pleased to meet all the other members of this site.
Well the above analysis is partially true as it won't fulfil the conditions in all cases in sqrt cases. I find the argument is also agreeing with this analysis. But some more analogy is required here.
Wish you all a Prosperous New year in this site. Smile
(01/06/2010, 05:31 AM)andrewan Wrote: Hi everyone , this is Andrewan. I am professionally an engineer. At present, I am staying in Moscow. I find forum discussion, a very interesting stuff to do. I used to hang out with these forums during my spare time. Many topics have got a healthy discussion over here. I think it would be most enjoyable while doing discussions in this forums.And pleased to meet all the other members of this site.
Well the above analysis is partially true as it won't fulfil the conditions in all cases in sqrt cases. I find the argument is also agreeing with this analysis. But some more analogy is required here.
Wish you all a Prosperous New year in this site. Smile
search "I find forum discussion, a very interesting stuff to do. I used" using google. About 20 times exactly the same msg at this day in 20 forums.
But I think, even this method of identical copying (or sending one unchanged msg) shall be changed soon. I came across a nice program called "Billy" with which you could do fairly lively discussion (like ELIZA, but much much nicer) so there is no problem to generate such messages having their keywords and exchange some tokens like subject, professional description etc. The program "SciGen" even can generate full fake-scientific articles, of so good formal quality that some were already published...
