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open problems survey - Printable Version

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Convergence of Eulers and Etas. TPID 10 - dantheman163 - 10/31/2010

We have the sequence of "Eulers" {2.718281828,3.0885322718...} and the sequence of "Etas" {1.44466786,1.6353244967...}.

"Eulers" and "Etas" can be defined as the x-coordinate and y-coordinate of the maximum of the nth order self root.

Conjecture:

The limit of the sequence of "Eulers" is 4.
The limit of the sequence of "Etas" is 2.

Some discussion can be found here

If you can find a better name for these sequences feel free to use it.


RE: open problems survey - nuninho1980 - 10/31/2010

(10/31/2010, 07:13 PM)dantheman163 Wrote: We have the sequence of "Eulers" {2.718281828,3.0885322718...} and the sequence of "Etas" {1.44466786,1.6353244967...}.

"Eulers" and "Etas" can be defined as the x-coordinate and y-coordinate of the maximum of the nth order self root.

Conjecture:

The limit of the sequence of "Eulers" is 4.
The limit of the sequence of "Etas" is 2.

Some discussion can be found here

If you can find a better name for these sequences feel free to use it.
It's nice! Smile
I already dreammed: Big Grin



Tommy's conjecture about Eulers and Etas. TPID 11. - tommy1729 - 12/01/2010

(10/31/2010, 07:13 PM)dantheman163 Wrote: We have the sequence of "Eulers" {2.718281828,3.0885322718...} and the sequence of "Etas" {1.44466786,1.6353244967...}.

"Eulers" and "Etas" can be defined as the x-coordinate and y-coordinate of the maximum of the nth order self root.

Conjecture:

The limit of the sequence of "Eulers" is 4.
The limit of the sequence of "Etas" is 2.

Some discussion can be found here

If you can find a better name for these sequences feel free to use it.

let the "Eulers" be eul(n) and the "Etas" be et(n).

now i conjecture :

1) et(n)^2 < eul(n-1)

2) lim n-> oo (et(n)^2 - eul(n-1)) / (et(n-1)^2 - eul(n-2)) = 1

regards

tommy1729


convergence of self-tetra-root polynomial interpolation. TPID 12 - bo198214 - 05/31/2011

In generalization of (the already solved) TPID 6 and following this thread of Andrew:

Does the sequence of interpolating polynomials of the points defined by pointwise converge to a function on (0,oo) (, satisfying )?

If it converges:
a) is then the limit function analytic, particularly at the point ?
b) For let b[4]x be the regular superexponential at the lower fixpoint. Is then f(x)[4]x = x for non-integer x with ?
c) For let b[4]x be the super-exponential via Kneser/perturbed Fatou coordinates. Is then f(x)[4]x = x for non-integer x with ?

To be more precise we can explicitely give the interpolating polynomials:
,
the question of this post is whether
exists for each .



convergence of self-root polynomial interpolation. TPID 13 - bo198214 - 05/31/2011

As simplification of TPID 12, we ask the much simpler question, whether
the sequence of interpolating polynomials for the points converges towards the function .
More precise:
Is for each , where
?

a) Is that still true if we omit a certain number of points from the beginning of the sequence. For example omitting (0,0) and (1,1).


Tommy's conjecture about andrew slog method. TPID 14 - tommy1729 - 06/01/2011

see tid 3 around post 27

http://math.eretrandre.org/tetrationforum/showthread.php?tid=3&page=3

for .

the conjecture is that if we replace x_0 by 0 we have described the same superlog as long as 0 is still in the radius of x_0 and x_0 is still in the radius of 0.

this might relate to tpid 1 and tpid 3 though ...


RE: x↑↑x = -1, TPID 15 - tommy1729 - 05/27/2014

(05/27/2014, 08:54 PM)KingDevyn Wrote: What are some possible answers to the equation x↑↑x = -1? Must a new type of number be conceptualized similar to the answer to the equation x*x = -1? Or can it be proved that this answer lies within the real and complex planes?

Seems it cannot be a negative real.
There are reasons for it...

I think you better start a thread instead of ask here.

regards

tommy1729


Tommy's conjecture TPID 16 - tommy1729 - 06/07/2014

TPID 16

Let be a nonpolynomial real entire function.
has a conjugate primary fixpoint pair :
has no other primary fixpoints then the conjugate primary fixpoint pair.
For between and and such that we have that
is analytic in .
is analytic for all real and all real .
If is analytic for then :
for all real , all real and all integer .
Otherwise
for all real , all real and all integer .


Are there solutions for ?
I conjecture yes.


regards

tommy1729


Error terms for fake function theory TPID 17 - tommy1729 - 03/28/2015

TPID 17

Let f(x) be a real-entire function such that for x > 0 we have

f(x) > 0 , f ' (x) > 0 and f " (x) > 0 and also

0 < D^m f(x) < D^(m-1) f(x).

Then when we use the S9 method from fake function theory to approximate the Taylor series

fake f(x) = a_0 + a_1 x + a_2 x^2 + ...

by setting a_n x^n = f(x) ( as S9 does )

we get an approximation to the true Taylor series

f(x) = t_0 + t_1 x + ...

such that

(a_n / t_n) ^2 + (t_n / a_n)^2 = O(n ln(n+1)).

Where O is big-O notation.

reference : http://math.eretrandre.org/tetrationforum/showthread.php?tid=863

How to prove this ?

regards

tommy1729


The third super-root TPID 18 - andydude - 12/25/2015

Conjecture:

Let iff. , then:



Discussion:

How and why?

For more discussion see this thread