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.
Posts: 95
Threads: 6
Joined: Apr 2009
(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:
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
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
,(1,y_1),\dots,(n,y_n))
defined by
pointwise converge to a function
on (0,oo) (, satisfying
=y_n)
)?
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
\le\eta)
?
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
>\eta)
?
To be more precise we can explicitely give the interpolating polynomials:
 = \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
)
exists for each
.
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
, (1,1), (2,2^{1/2}),\dots,(n,n^{1/n}))
converges towards the function
.
More precise:
Is
=x^{1/x})
for each
, where
=\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).
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
=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
.
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 ...
Posts: 1,370
Threads: 335
Joined: Feb 2009
(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
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
)
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
 > 1 + L^2)
we have that
)
is analytic in
.
)
is analytic for all real
and all real
.
If
is analytic for
then :
 \ge 0)
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
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
Posts: 509
Threads: 44
Joined: Aug 2007
Conjecture:
Let
} = x)
iff.
 = w)
, then:
} = \exp \left( \sum_{k=0}^{\infty} <br />
\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)<br />
)
Discussion:
How and why?
For more discussion
see this thread
