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 open problems survey dantheman163 Junior Fellow Posts: 13 Threads: 3 Joined: Oct 2009 10/31/2010, 07:13 PM (This post was last modified: 05/31/2011, 04:51 PM by bo198214.) 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 Fellow Posts: 95 Threads: 6 Joined: Apr 2009 10/31/2010, 09:50 PM (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! I already dreammed: $2^-[\infty^-]4^-$ tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 12/01/2010, 03:56 PM (This post was last modified: 05/31/2011, 04:52 PM by bo198214.) (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 bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/31/2011, 04:54 PM (This post was last modified: 05/31/2011, 07:04 PM by 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 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 05/31/2011, 07:02 PM (This post was last modified: 05/31/2011, 07:05 PM by 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 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 06/01/2011, 06:23 PM (This post was last modified: 06/29/2011, 12:08 PM by 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 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 05/27/2014, 11:33 PM (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 tommy1729 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 06/07/2014, 10:44 PM (This post was last modified: 06/18/2014, 11:48 PM by 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 Ultimate Fellow Posts: 1,370 Threads: 335 Joined: Feb 2009 03/28/2015, 10:48 PM (This post was last modified: 03/28/2015, 10:59 PM by 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 Long Time Fellow Posts: 509 Threads: 44 Joined: Aug 2007 12/25/2015, 06:16 AM 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 « Next Oldest | Next Newest »

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