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+--- Thread: open problems survey (/showthread.php?tid=162)
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...}.It's nice!
"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.
I already dreammed: \( 2^-[\infty^-]4^- \)
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 \( (0,0),(1,y_1),\dots,(n,y_n) \) 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 \( x>0 \).
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 \( (0,0), (1,1), (2,2^{1/2}),\dots,(n,n^{1/n}) \) 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).
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
\( \nu_k(x_0)=s^{(k)}(x_0)= \text{ln}(b)^k\sum_{i=0}^\infty\nu_i \cdot \frac{ b^{x_0 i}\cdot i^k}{i!} \) 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 ...
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 \( f(z) \) 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
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 \( \sqrt[3]{w}_s^{(z)} = x \) 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