open problems survey

[dantheman163]

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.

[nuninho1980]

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!
I already dreammed: \( 2^-[\infty^-]4^- \)

[tommy1729]

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

[bo198214]

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

Does the sequence of interpolating polynomials of the points \( (0,0),(1,y_1),\dots,(n,y_n) \) defined by \( y_n [4] n = n \) pointwise converge to a function \( f \) on (0,oo) (, satisfying \( f(n)=y_n \))?

If it converges:
a) is then the limit function \( f \) analytic, particularly at the point \( x=\eta \)?
b) For \( b\le \eta \) let b[4]x be the regular superexponential at the lower fixpoint. Is then f(x)[4]x = x for non-integer x with \( f(x)\le\eta \)?
c) For \( b> \eta \) 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 \( f(x)>\eta \)?

To be more precise we can explicitely give the interpolating polynomials:
\( f_N(x) = \sum_{n=0}^N \left(x\\n\right) \sum_{m=0}^n \left(n\\m\right) (-1)^{n-m} y_m \),
the question of this post is whether
\( \lim_{n\to\infty} f_n(x) \) exists for each \( x>0 \).

[bo198214]

As simplification of TPID 12, we ask the much simpler question, whether
the sequence of interpolating polynomials for the points \( (0,0), (1,1), (2,2^{1/2}),\dots,(n,n^{1/n}) \) converges towards the function \( x^{1/x} \).
More precise:
Is \( \lim_{n\to\infty} f_n(x)=x^{1/x} \) for each \( x>0 \), where
\( f_N(x)=\sum_{n=0}^N \left(x\\n\right) \sum_{m=0}^n \left(n\\m\right) (-1)^{n-m} m^{1/m} \)?

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).

[tommy1729]

see tid 3 around post 27

http://math.eretrandre.org/tetrationforu...d=3&page=3

\( \nu_k(x_0)=s^{(k)}(x_0)= \text{ln}(b)^k\sum_{i=0}^\infty\nu_i \cdot \frac{ b^{x_0 i}\cdot i^k}{i!} \) for \( k\ge 1 \).

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 ...

[tommy1729]

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

[tommy1729]

TPID 16

Let \( f(z) \) be a nonpolynomial real entire function.
\( f(z) \) has a conjugate primary fixpoint pair : \( L + M i , L - M i. \)
\( f(z) \) has no other primary fixpoints then the conjugate primary fixpoint pair.
For \( t \) between \( 0 \) and \( 1 \) and \( z \) such that \( Re(z) > 1 + L^2 \) we have that
\( f^{[t]}(z) \) is analytic in \( z \).
\( f^{[t]}(x) \) is analytic for all real \( x > 0 \) and all real \( t \ge 0 \) .
If \( f^{[t]}(x) \) is analytic for \( x = 0 \) then :
\( \frac{d^n}{dx^n} f^{[t]}(x) \ge 0 \) for all real \( x \ge 0 \) , all real \( t \ge 0 \) and all integer \( n > 0 \).
Otherwise
\( \frac{d^n}{dx^n} f^{[t]}(x) \ge 0 \) for all real \( x > 0 \) , all real \( t \ge 0 \) and all integer \( n > 0 \).


Are there solutions for \( f(z) \) ?
I conjecture yes.


regards

tommy1729

[tommy1729]

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/tetrationforu...hp?tid=863

How to prove this ?

regards

tommy1729

[andydude]

Conjecture:

Let \( \sqrt[3]{w}_s^{(z)} = x \) iff. \( \exp_x^3(z) = w \), then:

\(
\sqrt[3]{w}_s^{(z)} = \exp \left( \sum_{k=0}^{\infty}
\frac{\log(w)^k}{k!} \sum_{j=0}^{k-1} {k-1 \choose j}(k-j-1)^j(-k)^{k-j-1} z^j \right)
\)

Discussion:

How and why?

For more discussion see this thread