<![CDATA[Tetration Forum - All Forums]]> http://math.eretrandre.org/tetrationforum/ Wed, 10 Mar 2010 06:19:22 +0000 MyBB <![CDATA[regular sexp: curve near h=-2 (h=-2 + eps*I)]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=428 Tue, 09 Mar 2010 21:47:45 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=428 The latter means simply to rotate the value of the scröder-function for some real h, say, if we get for x0=1 the schröder-value s(x0)= s(1) = -0.632, then we use the rotation of this value in the complex plane by a parameter k in the following way s(1)(k) = -0.632 * exp(i*2*Pi*k/64) for k=0..64 and plug that value into the inverse schröder-function to get the iterate of desired complex height.
This gives ellipses for -2<real(h)< inf , and if the imaginary part of the h- value is not zero we can even plot for real(h) = -3, real(h) = -4 , ... real(h)-> - inf
This gives the following curves for some h with real heights and k=0..64

[attachment=695]

Then I was interested in the behaviour of the curve where the real part is -2. Does it diverge to imaginary +- infinity? Does it converge to real(-inf)? Or will the two parts of the curve converge to parallels of the x-axis (=constant imaginary part)?
I found the result surprising: not only, that it seems, they converge to an imaginary part of +-c where c = Pi/log(2), but also, that in the plot with exponentially scaled x-axis the stepwidth approximates to a constant value.
The "exponentially scaled" x-axis means simply, that the (negative) x-coordinates go into the exponent in b^x, so for x-> -inf I get b^x->0 instead.
In the following graph I took the very small stepwidth of k/2^32 for the imaginary part of h - and whatever small stepwidth I use, it seems always, that the curve approximates that parallels to the x-axis with distant of c.

[attachment=696]

Hmm. I'm not familiar with things like Riemann-sphere, but I remember something like if we take the plot from the euclidean plane to the surface of a sphere, then the infinities meet at one point. Would such translation be useful here too?

Gottfried]]> The latter means simply to rotate the value of the scröder-function for some real h, say, if we get for x0=1 the schröder-value s(x0)= s(1) = -0.632, then we use the rotation of this value in the complex plane by a parameter k in the following way s(1)(k) = -0.632 * exp(i*2*Pi*k/64) for k=0..64 and plug that value into the inverse schröder-function to get the iterate of desired complex height.
This gives ellipses for -2<real(h)< inf , and if the imaginary part of the h- value is not zero we can even plot for real(h) = -3, real(h) = -4 , ... real(h)-> - inf
This gives the following curves for some h with real heights and k=0..64

[attachment=695]

Then I was interested in the behaviour of the curve where the real part is -2. Does it diverge to imaginary +- infinity? Does it converge to real(-inf)? Or will the two parts of the curve converge to parallels of the x-axis (=constant imaginary part)?
I found the result surprising: not only, that it seems, they converge to an imaginary part of +-c where c = Pi/log(2), but also, that in the plot with exponentially scaled x-axis the stepwidth approximates to a constant value.
The "exponentially scaled" x-axis means simply, that the (negative) x-coordinates go into the exponent in b^x, so for x-> -inf I get b^x->0 instead.
In the following graph I took the very small stepwidth of k/2^32 for the imaginary part of h - and whatever small stepwidth I use, it seems always, that the curve approximates that parallels to the x-axis with distant of c.

[attachment=696]

Hmm. I'm not familiar with things like Riemann-sphere, but I remember something like if we take the plot from the euclidean plane to the surface of a sphere, then the infinities meet at one point. Would such translation be useful here too?

Gottfried]]> <![CDATA[New (sort of) Tex for real Tetration-freaks]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=427 Tue, 02 Mar 2010 16:42:37 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=427
The Logo

:-)

Gottfried]]>
The Logo

:-)

Gottfried]]> <![CDATA[Literature for x^x and Sophomore's Dream]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=426 Tue, 02 Mar 2010 02:33:54 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=426
I'm interested in the second order tetration function, x^x. The integral from 0 to 1 of this function is called the Sophomore's Dream. I was wondering which papers, journals, etc cover either the function or its integral. I've read the respective sections in Dunham as well as Bailey and Borwein, but it's difficult to search for because "x^x" in google scholar just returns everything with "xx," and sophomore's dream just brings up poetry and the like.

Most of the articles on tetration by Galidakis, Kouznetsov, etc are about iterative properties and analytical extension, but not really about the function I'm interested in.

Does anyone have some papers in mind, or know which journal to browse through?

Thanks]]>
I'm interested in the second order tetration function, x^x. The integral from 0 to 1 of this function is called the Sophomore's Dream. I was wondering which papers, journals, etc cover either the function or its integral. I've read the respective sections in Dunham as well as Bailey and Borwein, but it's difficult to search for because "x^x" in google scholar just returns everything with "xx," and sophomore's dream just brings up poetry and the like.

Most of the articles on tetration by Galidakis, Kouznetsov, etc are about iterative properties and analytical extension, but not really about the function I'm interested in.

Does anyone have some papers in mind, or know which journal to browse through?

Thanks]]> <![CDATA[My Tetration Site]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=425 Sun, 28 Feb 2010 00:53:39 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=425 So my website that used to be located here:

http://tetration.itgo.com/

is now located here:

http://tetration.co.cc/

Andrew Robbins]]> So my website that used to be located here:

http://tetration.itgo.com/

is now located here:

http://tetration.co.cc/

Andrew Robbins]]> <![CDATA[using sinh(x) ?]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=424 Sun, 28 Feb 2010 00:22:05 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=424
perhaps not the best one , but sinh(x) comes to my mind.

if we take the half-iterate of sinh(x) by using taylor series , we get a good approximation of the half-iterate of exp(x) for x large.

if this good approximation is analytic in say [e^e,e^e^e^e] we could use that interval and take logs or exp of it to compute the half-iterate for [-oo,+oo] up to a relatively high precision.

in fact , i assume , we can choose our precision if the good approximation is entire , by using bigger intervals and taking logs of it ( as somewhat done above ).

we might even consider the mittag-leffler expansion to avoid problems with non-analytic issues ( thus taking mittag-leffler expansion of the formal taylor series )

thats basicly the idea , but as said maybe sinh(x) isnt the best , on the other hand its probably the easiest.

furthermore some want - and me too actually - that half-iterates have non-negative (2+n)th derivatives and strictly positive zeroth , first and second derivatives.

that last restriction troubles me , especially when i try to get closer to exp(x) than sinh(x) by using function that have non-negative derivatives ... getting conditions on a particular n'th derivate isnt hard but the whole problem is more troublesome ( maybe someone knows a solution to this ? )

i didnt mention that all the above ofcourse is about functions going from R -> R , its obvious , but i just mention it to avoid potential confusion.

also every approximation of exp(x) should equal x at x = 0 only and be larger than id(x).

hope its clear.

maybe you considered this once too ?


greetings my fellow euh ... tetrationalists

regards

tommy1729

ps : showing that all derivatives are positive can sometimes be easy , but in general its hard , see for example " li's criterion " for the RH.
maybe its easy here and i missed a trivial thing ... ( im getting old ? Tongue )]]>
perhaps not the best one , but sinh(x) comes to my mind.

if we take the half-iterate of sinh(x) by using taylor series , we get a good approximation of the half-iterate of exp(x) for x large.

if this good approximation is analytic in say [e^e,e^e^e^e] we could use that interval and take logs or exp of it to compute the half-iterate for [-oo,+oo] up to a relatively high precision.

in fact , i assume , we can choose our precision if the good approximation is entire , by using bigger intervals and taking logs of it ( as somewhat done above ).

we might even consider the mittag-leffler expansion to avoid problems with non-analytic issues ( thus taking mittag-leffler expansion of the formal taylor series )

thats basicly the idea , but as said maybe sinh(x) isnt the best , on the other hand its probably the easiest.

furthermore some want - and me too actually - that half-iterates have non-negative (2+n)th derivatives and strictly positive zeroth , first and second derivatives.

that last restriction troubles me , especially when i try to get closer to exp(x) than sinh(x) by using function that have non-negative derivatives ... getting conditions on a particular n'th derivate isnt hard but the whole problem is more troublesome ( maybe someone knows a solution to this ? )

i didnt mention that all the above ofcourse is about functions going from R -> R , its obvious , but i just mention it to avoid potential confusion.

also every approximation of exp(x) should equal x at x = 0 only and be larger than id(x).

hope its clear.

maybe you considered this once too ?


greetings my fellow euh ... tetrationalists

regards

tommy1729

ps : showing that all derivatives are positive can sometimes be easy , but in general its hard , see for example " li's criterion " for the RH.
maybe its easy here and i missed a trivial thing ... ( im getting old ? Tongue )]]> <![CDATA["Printable view" no more shows graphics]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=423 Tue, 23 Feb 2010 06:33:08 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=423 If you click "printable view" at least graphics which are included as attachments are not shown at my computer.

Ah, well, and also the "attaching graph" behaviour has changed: they are automatically shown in the normal view, without being referenced. The problem is, that we cannot resize them: a line like [img=400x400]...attachment[/img] gives a correct view, but the original attachment-view is also active, so you cannot really adapt the size of graphs. I had to delete three attachments from the myBB-Forum and had to replace the link by an external link to make the pictures sizeable.

Gottfried]]> If you click "printable view" at least graphics which are included as attachments are not shown at my computer.

Ah, well, and also the "attaching graph" behaviour has changed: they are automatically shown in the normal view, without being referenced. The problem is, that we cannot resize them: a line like [img=400x400]...attachment[/img] gives a correct view, but the original attachment-view is also active, so you cannot really adapt the size of graphs. I had to delete three attachments from the myBB-Forum and had to replace the link by an external link to make the pictures sizeable.

Gottfried]]> <![CDATA[[UFO] Attracting Fixpoints or attracting line?]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=422 Mon, 22 Feb 2010 20:31:54 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=422 Feel free to just ignore "UFO"-tagged msgs ...

------------------

I looked for a list of the complex fixpoints of the tetration and whether there are some more interesting regularities. So, for instance the base b=sqrt(2): let's look at the list of the fixpoints.
First: as well known, there is only one attracting fixpoint a=2.
Then we have the repelling fixpoint r0=4.
And then the infinite list of complex fixpoints, all repelling.
Ok, if we apply the iterated log instead of the iterated exponential they become attracting and the fixpoint a0 becomes repelling.

How do we get the different fixpoints r0,r1,r2, ... r_k,... (with each r_k also the conjugate is a fixpoint) ? We use the k'th branch of the log, so
r_k = lim_{n->inf} x = (log(x) + k*2*Pi*I ) / log(b)

When I looked at the plot of that (isolated) fixpoints I asked, whether we could interpolate a line. And indeed, we can simply take any real k to get
r_k = lim_{n->inf} x = (log(x) + k*2*Pi*I ) / log(b)
converging!
So this means, we have in fact an attracting continuous curve, and the usual fixpoints are simply that where k is integer...

I've drawn a plot with a couple of bases:
b = exp(u*exp(-u)) , where if u=log(2) we have our favorite one b=sqrt(2). We get the first two fixpoints
r0 = 4 , r1 ~ 9.09... + i*21.5 etc;
and the fixpoint a = 2, which is repelling if we use logarithmizing.

The plot shows the curves for u = 0.5, u=log(2), u=0.85, u=0.95, where dots are inserted at the integer k's (marking the commonly used fixpoints).

It is interesting to approach the critical base b=e^(1/e), meaning u->1 from below, however I do not yet have a good idea of the geometry of that limiting curves.

Here are the original plot, the value of the imaginary part scaled by asinh() (which means nearly a log-scaling, but allows the zero- and negativ values to be rescaled) and two details near the real fixpoints to see the smoothness of the curves.

Gottfried

[Image: fixpointline_1.png]

[Image: fixpointline_2.png]

[Image: fixpointline_3.png]]]> Feel free to just ignore "UFO"-tagged msgs ...

------------------

I looked for a list of the complex fixpoints of the tetration and whether there are some more interesting regularities. So, for instance the base b=sqrt(2): let's look at the list of the fixpoints.
First: as well known, there is only one attracting fixpoint a=2.
Then we have the repelling fixpoint r0=4.
And then the infinite list of complex fixpoints, all repelling.
Ok, if we apply the iterated log instead of the iterated exponential they become attracting and the fixpoint a0 becomes repelling.

How do we get the different fixpoints r0,r1,r2, ... r_k,... (with each r_k also the conjugate is a fixpoint) ? We use the k'th branch of the log, so
r_k = lim_{n->inf} x = (log(x) + k*2*Pi*I ) / log(b)

When I looked at the plot of that (isolated) fixpoints I asked, whether we could interpolate a line. And indeed, we can simply take any real k to get
r_k = lim_{n->inf} x = (log(x) + k*2*Pi*I ) / log(b)
converging!
So this means, we have in fact an attracting continuous curve, and the usual fixpoints are simply that where k is integer...

I've drawn a plot with a couple of bases:
b = exp(u*exp(-u)) , where if u=log(2) we have our favorite one b=sqrt(2). We get the first two fixpoints
r0 = 4 , r1 ~ 9.09... + i*21.5 etc;
and the fixpoint a = 2, which is repelling if we use logarithmizing.

The plot shows the curves for u = 0.5, u=log(2), u=0.85, u=0.95, where dots are inserted at the integer k's (marking the commonly used fixpoints).

It is interesting to approach the critical base b=e^(1/e), meaning u->1 from below, however I do not yet have a good idea of the geometry of that limiting curves.

Here are the original plot, the value of the imaginary part scaled by asinh() (which means nearly a log-scaling, but allows the zero- and negativ values to be rescaled) and two details near the real fixpoints to see the smoothness of the curves.

Gottfried

[Image: fixpointline_1.png]

[Image: fixpointline_2.png]

[Image: fixpointline_3.png]]]> <![CDATA[Arithmetic in the height-parameter (sums, series)]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=420 Thu, 04 Feb 2010 17:08:00 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=420
using the notation and we do arithmetic in the height (or "iteration") parameter like

What about infinite series instead of a sum?
If we have a sufficient method for continuous tetration, then, for instance we should get


For "nice" bases 1<= b <=ß (= exp(exp(-1)) ) we should get even more; I don't see any reason, why the series in the height-parameter should not be allowed to diverge.

So, with base b=sqrt(2) the following expression



seems to make sense to me because the partial evaluations converge to y=2 , for instance if we choose x_0 = 1.

On the other hand, the analytical continuation for the geometric series with constant quotient q at q=2 gives



But -substitued this into the height-parameter- then we should also have



where we see a contradiction.
So it seems to me that tetration can restrict the validity/range-of-usability of analytic continuation (if we do not have another explanation/notion/way-of-handling of such effects).

---

In a second view, this extends to other functions similarly. We have another inequality if a divergent series is used in the exponent of the power-function, for instance, to base e:




Just another plot of meditations... <phew>

Gottfried]]>
using the notation and we do arithmetic in the height (or "iteration") parameter like

What about infinite series instead of a sum?
If we have a sufficient method for continuous tetration, then, for instance we should get


For "nice" bases 1<= b <=ß (= exp(exp(-1)) ) we should get even more; I don't see any reason, why the series in the height-parameter should not be allowed to diverge.

So, with base b=sqrt(2) the following expression



seems to make sense to me because the partial evaluations converge to y=2 , for instance if we choose x_0 = 1.

On the other hand, the analytical continuation for the geometric series with constant quotient q at q=2 gives



But -substitued this into the height-parameter- then we should also have



where we see a contradiction.
So it seems to me that tetration can restrict the validity/range-of-usability of analytic continuation (if we do not have another explanation/notion/way-of-handling of such effects).

---

In a second view, this extends to other functions similarly. We have another inequality if a divergent series is used in the exponent of the power-function, for instance, to base e:




Just another plot of meditations... <phew>

Gottfried]]> <![CDATA[image recognition, decaptcha & bots]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=419 Thu, 04 Feb 2010 10:03:50 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=419 And then one day later the next bot was registered on the forum.

How can that happen?
Now I realized (see the thread on stopforumspam) that the captchas are solved by humans, e.g. certain services (e.g. keepvid.c0m) require "as payment" to solve a captcha. These solved captures are sold to bot advertisers (see e.g. decaptcher.c0m: 1000 solved captchas for 2$)

Fortunately there is now a stopforumspam-plugin for myBB (our forumsoftware), which I installed and which dont let recognized spam bots register. Lets see how well this works.]]> And then one day later the next bot was registered on the forum.

How can that happen?
Now I realized (see the thread on stopforumspam) that the captchas are solved by humans, e.g. certain services (e.g. keepvid.c0m) require "as payment" to solve a captcha. These solved captures are sold to bot advertisers (see e.g. decaptcher.c0m: 1000 solved captchas for 2$)

Fortunately there is now a stopforumspam-plugin for myBB (our forumsoftware), which I installed and which dont let recognized spam bots register. Lets see how well this works.]]> <![CDATA[Branch points of superlog]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=418 Wed, 03 Feb 2010 23:00:07 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=418
What are the different branch points of the superlog? There look to be others that are not fixed points of . What are these other branch points, and what sort of special behavior, if any, of the integer iteration for , do they undergo?]]>
What are the different branch points of the superlog? There look to be others that are not fixed points of . What are these other branch points, and what sort of special behavior, if any, of the integer iteration for , do they undergo?]]> <![CDATA[Partial Differential Equation for power-towers]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=417 Mon, 01 Feb 2010 21:05:41 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=417
The equation is as following:

Define




And



Where



Then every satisfies:



I don't understand anything about pdes so I don't know if this says any thing, but I wanted to share it with you guys.

I guess we can somehow use this equation to extend natural iteration monomial for larger bases, but I don't really know.

Kobi]]>
The equation is as following:

Define




And



Where



Then every satisfies:



I don't understand anything about pdes so I don't know if this says any thing, but I wanted to share it with you guys.

I guess we can somehow use this equation to extend natural iteration monomial for larger bases, but I don't really know.

Kobi]]> <![CDATA[new results from complex dynamics]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=416 Mon, 25 Jan 2010 19:30:48 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=416
Complex dynamics is a currently flourishing field and uses a different terminology than we use. For example what we call "Abel function", they call it "Fatou coordinates".

The keyword is "parabolic implosion", you even find online articles when googeling. I will base my explanations on the article of Mitsushiro Shishikura in the book "The mandelbrot set, theme and variations". His article in the book is named "Bifurcation of parabolic fixed points", and is rather advanced.

The basic idea is the following:
We start with a parabolic fixed point of a holomorphic function , i.e. and .
If we slightly perturb this function by a complex , , then the one fixed point splits up into several fixed points (at least 2). But the perturbed function still behaves quite similar to the original function.

From the classic theory about Abel functions at a parabolic fixed point (basically developed by Ecalle), we know that there is a Leau-Fatou-flower (see the online book of Milnor: "dynamics in one complex variable" for detailed explanation), i.e. alternating attracting and repulsing petals, of a flower with center in . The number of petals is where is the number of vanishing derivatives in after the first derivative (which is 1).

For example for the function with . Its second derivative is non-zero, so and hence there are two petals. And indeed one petal covers the real axis , where is repelling and the other petal covers the real axis where is attracting.

If we slightly perturb this function by adding or by just increasing the base a little then the one parabolic fixed point splits up into two conjugate repelling fixed points. If we slightly perturb with then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling).

The classic unperturbed theory says now that we can develop an injective Abel function/Fatou coordinate on each petal. And this Abel function is uniquely determined by the demand that the resulting fractional/complex iterates have an asymptotic power series development in . I always call it here the regular Abel function and the regular iterates.

The current complex dynamics theory says that for small enough epsilon, we find a sickle between the perturbed fixed point pair, and an injective Fatou coordinate on that sickle, which has also a not so explicit formulated uniqueness. Moreover this Fatou coordinate depends holomorphically on !

In some other post on this forum I mentioned that the regular Abel function/regular iterate of at the lower fixed point for is probably not holomorphically continuable in to . However if one uses the perturbed Fatou coordinates for and not the regular ones at the lower fixed point, then it (and the resulting fractional iterates) depends holomorphically on !
I have to revalidate this statement, but it appears very promising.

As to the uniqueness argument, it indeed appears to be similar to the one I developed in my article.]]>
Complex dynamics is a currently flourishing field and uses a different terminology than we use. For example what we call "Abel function", they call it "Fatou coordinates".

The keyword is "parabolic implosion", you even find online articles when googeling. I will base my explanations on the article of Mitsushiro Shishikura in the book "The mandelbrot set, theme and variations". His article in the book is named "Bifurcation of parabolic fixed points", and is rather advanced.

The basic idea is the following:
We start with a parabolic fixed point of a holomorphic function , i.e. and .
If we slightly perturb this function by a complex , , then the one fixed point splits up into several fixed points (at least 2). But the perturbed function still behaves quite similar to the original function.

From the classic theory about Abel functions at a parabolic fixed point (basically developed by Ecalle), we know that there is a Leau-Fatou-flower (see the online book of Milnor: "dynamics in one complex variable" for detailed explanation), i.e. alternating attracting and repulsing petals, of a flower with center in . The number of petals is where is the number of vanishing derivatives in after the first derivative (which is 1).

For example for the function with . Its second derivative is non-zero, so and hence there are two petals. And indeed one petal covers the real axis , where is repelling and the other petal covers the real axis where is attracting.

If we slightly perturb this function by adding or by just increasing the base a little then the one parabolic fixed point splits up into two conjugate repelling fixed points. If we slightly perturb with then the parabolic fixed point splits up into two real fixed points on the real axis (one attracting, one repelling).

The classic unperturbed theory says now that we can develop an injective Abel function/Fatou coordinate on each petal. And this Abel function is uniquely determined by the demand that the resulting fractional/complex iterates have an asymptotic power series development in . I always call it here the regular Abel function and the regular iterates.

The current complex dynamics theory says that for small enough epsilon, we find a sickle between the perturbed fixed point pair, and an injective Fatou coordinate on that sickle, which has also a not so explicit formulated uniqueness. Moreover this Fatou coordinate depends holomorphically on !

In some other post on this forum I mentioned that the regular Abel function/regular iterate of at the lower fixed point for is probably not holomorphically continuable in to . However if one uses the perturbed Fatou coordinates for and not the regular ones at the lower fixed point, then it (and the resulting fractional iterates) depends holomorphically on !
I have to revalidate this statement, but it appears very promising.

As to the uniqueness argument, it indeed appears to be similar to the one I developed in my article.]]> <![CDATA[Matrix-method is healthy? (Post # 4000!)]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=415 Sat, 23 Jan 2010 09:54:55 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=415 When I goggled for "matrix stack" I found the following advertisement:

matrix stack, See:

Code:
Matrix Quantenheilung
Seminare ab 280.- Euro
Der schnellste Weg zum Erfolg
w w.naturheilpraxis-frenze
meaning:
"matrix-quantum-healing, the fastest path to success." (can't translate the "naturheilpraxis")

--
Also : this is the 4000'th post in this forum, a lot of thoughts here (beside the jokes (and the spam recently), indeed!
Thanks especially to Henryk!

Gottfried]]> When I goggled for "matrix stack" I found the following advertisement:

matrix stack, See:

Code:
Matrix Quantenheilung
Seminare ab 280.- Euro
Der schnellste Weg zum Erfolg
w w.naturheilpraxis-frenze
meaning:
"matrix-quantum-healing, the fastest path to success." (can't translate the "naturheilpraxis")

--
Also : this is the 4000'th post in this forum, a lot of thoughts here (beside the jokes (and the spam recently), indeed!
Thanks especially to Henryk!

Gottfried]]> <![CDATA[new method in progress ...]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=414 Sun, 17 Jan 2010 22:49:17 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=414
it is probably Coo ...

its still in progress ... just letting you guys know first , you deserve that Smile

im working on number theory right now ...

if my number theory ideas and papers get accepted i will write a paper about tetration.

plz forgive me for giving priority to number theory.

tommy1729]]>
it is probably Coo ...

its still in progress ... just letting you guys know first , you deserve that Smile

im working on number theory right now ...

if my number theory ideas and papers get accepted i will write a paper about tetration.

plz forgive me for giving priority to number theory.

tommy1729]]> <![CDATA[coo half-iterate div @ fix]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=412 Fri, 08 Jan 2010 00:36:07 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=412 <![CDATA[our beloved automated participiants....]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=411 Wed, 06 Jan 2010 09:35:38 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=411 I couldn't imagine, what is the benefit for that authors - they spam us with msgs, which are not yet obviously advertizing. Today I found some interesting discussion in a forum, which may enlighten us in this regard. I converted it to a short form and pdf-format.
While the arguments in that discussion seem nearly sufficient for an explanation of that disgusting phenomen, I'm still suspicious, whether there may be other interests, possibly activated at a later time (B.ot-n.ets etc), but this may be subject of further discussion, not necessarily in our individual forum.

If someone else is able to detect such fake-memberships in our forum, I'd like to propose to open some list. What I did so far was to google for pseudos - if you find 200 forums having the same pseudo, all joined the same day, then very likely that one should be a fake. Also we had some msg's recently with typically vague formulations, valid for any forum ("whea, I like good discussions") If such phrases extend with some more typical wording, one can find such phrases via google. If this identifies another some dozen identically worded msgs, we've likely got another fake.

Here is the pdf of the forum-discussion :[attachment=688]

Gottfried]]> I couldn't imagine, what is the benefit for that authors - they spam us with msgs, which are not yet obviously advertizing. Today I found some interesting discussion in a forum, which may enlighten us in this regard. I converted it to a short form and pdf-format.
While the arguments in that discussion seem nearly sufficient for an explanation of that disgusting phenomen, I'm still suspicious, whether there may be other interests, possibly activated at a later time (B.ot-n.ets etc), but this may be subject of further discussion, not necessarily in our individual forum.

If someone else is able to detect such fake-memberships in our forum, I'd like to propose to open some list. What I did so far was to google for pseudos - if you find 200 forums having the same pseudo, all joined the same day, then very likely that one should be a fake. Also we had some msg's recently with typically vague formulations, valid for any forum ("whea, I like good discussions") If such phrases extend with some more typical wording, one can find such phrases via google. If this identifies another some dozen identically worded msgs, we've likely got another fake.

Here is the pdf of the forum-discussion :[attachment=688]

Gottfried]]> <![CDATA[A MAPLE QUESTION]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=410 Mon, 04 Jan 2010 20:16:16 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=410 <![CDATA[Borel summation and other continuation/summability methods for continuum sums]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=409 Tue, 29 Dec 2009 10:55:06 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=409
Here's a new possibility for tetration. It's based on the use of Borel summation and other types of methods to attempt to extend continuum sums to a wider domain.

Now as you may know, the continuum sum from Faulhaber's formula:

(here we use )

where is a holomorphic function, only works for a very limited range of functions. Namely, it won't work for non-entire functions, or (it seems) entire functions not of exponential type or of exponential type greater than . Tetration fails both requirements: it is not entire ( for nonnegative integer is a singularity, a branch point in fact, and there may be others too), and except for , it is not of exponential type due to insanely rapid growth. These are well-known.

Because of this, the inner sums will not converge. So the question arises: could one assign a meaning to the formula even when that is the case? Note that we already know some continuum sums for functions that fail this criterion, e.g.



(even for !)



etc. and for functions that can be represented via exponential series, i.e. ,

.

So we would expect that any such divergent/continuation/summability method that's up to the task should preserve these, while allowing us to continuum-sum more things, and it should be "natural" in some way (what that means is, of course, the biggest question).

One option I've thought of is Borel summation. It works like:



where



if this can be analytically continued to all and grows at most exponentially. Thus we get the "regularized Faulhaber coefficients"



with the inner sum analytically continued, so the continuum sum is . Another method that might be useful is the one I mentioned here:

http://math.eretrandre.org/tetrationforu...93#pid4293

Perhaps it could give a still wider range of functions. There are some functions for which these do not appear to work -- consider . The first coefficient of the continuum sum by the Faulhaber's formula gives the divergent sum which does not look to be Borel summable. However it seems it could work for other functions, and the big question is, of course, could it work for tetration, and if so, do the obtained extensions of tetration agree with the ones already made, yet enable expansion to a much wider variety of bases?]]>
Here's a new possibility for tetration. It's based on the use of Borel summation and other types of methods to attempt to extend continuum sums to a wider domain.

Now as you may know, the continuum sum from Faulhaber's formula:

(here we use )

where is a holomorphic function, only works for a very limited range of functions. Namely, it won't work for non-entire functions, or (it seems) entire functions not of exponential type or of exponential type greater than . Tetration fails both requirements: it is not entire ( for nonnegative integer is a singularity, a branch point in fact, and there may be others too), and except for , it is not of exponential type due to insanely rapid growth. These are well-known.

Because of this, the inner sums will not converge. So the question arises: could one assign a meaning to the formula even when that is the case? Note that we already know some continuum sums for functions that fail this criterion, e.g.



(even for !)



etc. and for functions that can be represented via exponential series, i.e. ,

.

So we would expect that any such divergent/continuation/summability method that's up to the task should preserve these, while allowing us to continuum-sum more things, and it should be "natural" in some way (what that means is, of course, the biggest question).

One option I've thought of is Borel summation. It works like:



where



if this can be analytically continued to all and grows at most exponentially. Thus we get the "regularized Faulhaber coefficients"



with the inner sum analytically continued, so the continuum sum is . Another method that might be useful is the one I mentioned here:

http://math.eretrandre.org/tetrationforu...93#pid4293

Perhaps it could give a still wider range of functions. There are some functions for which these do not appear to work -- consider . The first coefficient of the continuum sum by the Faulhaber's formula gives the divergent sum which does not look to be Borel summable. However it seems it could work for other functions, and the big question is, of course, could it work for tetration, and if so, do the obtained extensions of tetration agree with the ones already made, yet enable expansion to a much wider variety of bases?]]> <![CDATA[This formula should work for base<1]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=408 Fri, 25 Dec 2009 10:04:06 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=408

I have tried it in Mathematica. This is for example, a plot for b=0.5

For b=Sqrt(2) it also gives good results.]]>

I have tried it in Mathematica. This is for example, a plot for b=0.5

For b=Sqrt(2) it also gives good results.]]> <![CDATA[A simple formula for superlogarithm of arbitrary integer]]> http://math.eretrandre.org/tetrationforum/showthread.php?tid=407 Sun, 20 Dec 2009 08:55:08 +0100 http://math.eretrandre.org/tetrationforum/showthread.php?tid=407

where
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where
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