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: 96

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: 1,419

Threads: 345

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

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 non-integer x with

?

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

?

To be more precise we can explicitely give the interpolating polynomials:

,

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

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).

Posts: 1,419

Threads: 345

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
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: 1,419

Threads: 345

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: 1,419

Threads: 345

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

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

Posts: 1,419

Threads: 345

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

iff.

, then:

Discussion:
How and why?

For more discussion

see this thread