open problems survey  Printable Version + Tetration Forum (https://math.eretrandre.org/tetrationforum) + Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) + Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) + Thread: open problems survey (/showthread.php?tid=162) Pages:
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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 xcoordinate and ycoordinate 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! I already dreammed: 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...}. let the "Eulers" be eul(n) and the "Etas" be et(n). now i conjecture : 1) et(n)^2 < eul(n1) 2) lim n> oo (et(n)^2  eul(n1)) / (et(n1)^2  eul(n2)) = 1 regards tommy1729 convergence of selftetraroot 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 noninteger x with ? c) For let b[4]x be the superexponential via Kneser/perturbed Fatou coordinates. Is then f(x)[4]x = x for noninteger 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 selfroot 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 realentire 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^(m1) 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 bigO notation. reference : http://math.eretrandre.org/tetrationforum/showthread.php?tid=863 How to prove this ? regards tommy1729 The third superroot TPID 18  andydude  12/25/2015 Conjecture: Let iff. , then: Discussion: How and why? For more discussion see this thread 