open problems survey

[JmsNxn]

As we're posting our own conjectures here, I thought I'd add mine to this list.

In William Paulsen and Samuel Cowgill's paper; they outline the following uniqueness condition:

\(
F(z)\,\,\text{is holomorphic on}\,\,\mathbb{C}/(-\infty,-2]\\
F(0) = 1\\
F(\overline{z}) = \overline{F(z)}\\
\forall x \in \mathbb{R}\, \,\lim_{y\to\infty} F(x+iy) = L\\
\text{where}\,L\,\text{is the fixed point of}\, \exp\, \text{with minimal imaginary argument}\\
F(z+1) = e^{F(z)}\\
\text{Then necessarily,}\\
F(z)\,\,\text{is Kneser's Tetration}\\
\)

Which they prove, as I believe, completely satisfactorily (Kouznetzov seemed to have doubts).

This question is in two parts:

A). Does there exist a tetration function \( G \) which has all these properties except,

\(
\lim_{y\to\infty} G(x+iy) = \infty\\
\)

B.) Does it satisfy the same uniqueness condition that William Paulsen and Samuel Cowgill proposed? (Albeit, with the different behaviour \( \Im(z) = \infty \)).

---

For A.)--See related threads about the beta method (https://math.eretrandre.org/tetrationfor...p?tid=1314, https://math.eretrandre.org/tetrationfor...p?tid=1334), which seems to point to the beta method not being kneser (I've proved the beta-method converges, but not that it isn't still Kneser's method). But numbers clearly show a discrepancy...

For B.) I don't know at all, but there's a hunch--by looking at Tommy's Gaussian method and the beta method they seem to be one and the same. See: https://math.eretrandre.org/tetrationfor...p?tid=1339 . Furthermore, this conjecture was made by how well different manners of coding the beta method all still gave the same numbers; when I would use different asymptotic shortcuts the same function appears.

I believe I can sketch a proof of A.), but I'd need oversight before it ever became a proof. As for B.), I can't even think of a line of attack.

Regards, James

[tommy1729]

We know that the recurrence for \( b>0 \)
(1) \( f(0)=1 \)
(2) \( f(z+1)=b{f(z)} \)
has \( f(z)=b^z \) as the only entire solution that is bounded on the strip \( S=\{z: 0<\Re(z)\le 1\} \).

The image of \( S \) under \( b^z \) is an annulus for \( b>0 \) and so bounded. We know that for complex \( b\not\ge 0 \) the function \( f(z)=b^z=\exp((\log(z)+2\pi i k)z) \) is not bounded on \( S \) (the image is kinda infinite spiral) for any \( k \). The question remains whether

Conjecture
There is no entire solution \( f \) that satisfies (1) and (2) and is bounded on \( S \) for complex \( b\not\ge 0 \).

the general solution is b^(z + theta(z)) where theta (z) is an entire 1 periodic function.

since theta(z) must reach to  all complex values in its range and period , then so does t(b,z) =   b^theta(z) apart from 0.

since b^z is never zero and does not approach it , this implies t(b,z) b^z is also unbounded.

Afterall bounded and not approaching 0 times a function that is unbounded = unbounded.

conjecture update = b^(z + theta(z)) reaches all nonzero complex numbers.

for more details search for TPID 4.

regards

tommy1729

[Gottfried]

TPID 18: Is "polynomial" tetration (by truncated Carleman-matrices) of some order \(n\) with \( n \to \infty \) asymptotic to the Kneser-solution? 

This is a long outstanding problem of mine. I've some numerical examples since years which seem to suggest that, but had never an idea how to approach at all a proof/disproof of such a conjecture.          

One example where I worked with this is here but I've some more material of comparision in my local excel- and Pari/gp-files which might be worked out further.
My examples with truncations of the Carlemanmatrices from sizes \(4 \times 4,8 \times 8,16 \times 16,32 \times 32 \) up to \(64 \times 64 \) with some random chosen small bases (like \(b=\sqrt 2 , b=4 , b=e \) ) show nice improvement of approximations to 5 or 10 ore more correct digits.

[tommy1729]

TPID 18: Is "polynomial" tetration (by truncated Carleman-matrices) of some order \(n\) with \( n \to \infty \) asymptotic to the Kneser-solution? 

This is a long outstanding problem of mine. I've some numerical examples since years which seem to suggest that, but had never an idea how to approach at all a proof/disproof of such a conjecture.          

One example where I worked with this is here but I've some more material of comparision in my local excel- and Pari/gp-files which might be worked out further.         
My examples with truncations of the Carlemanmatrices from sizes \(4 \times 4,8 \times 8,16 \times 16,32 \times 32 \) up to \(64 \times 64 \) with some random chosen small bases (like \(b=\sqrt 2 , b=4 , b=e \) ) show nice improvement of approximations to 5 or 10 ore more correct digits.

sorry to ask here , but has it been proven that all rising integer sequences (of size) give the same result ?

regards

tommy1729

[Gottfried]

My examples with truncations of the Carlemanmatrices from sizes \(4 \times 4,8 \times 8,16 \times 16,32 \times 32 \) up to \(64 \times 64 \) with some random chosen small bases (like \(b=\sqrt 2 , b=4 , b=e \) ) show nice improvement of approximations to 5 or 10 ore more correct digits.

sorry to ask here , but has it been proven that all rising integer sequences (of size) give the same result ?

In all my heuristics increasing the sizes of the truncations of square(*) Carlemanmatrices the coefficients as well as the results seem to approximate limit values. However, due to extreme need of internal digits-precision I can with reasonable effort only go up to \( 64 \times 64 \) matrices, sometimes a bit more (see fractional iteration of the sine-function, in MO), so this does not really mean much...

(*) for triangular Carlemanmatrix this problem is of much lesser relevance

Example: Carlemanmatrix half-iterate of \(f(x)= 4^x \) for sizes \( d \times d \) for \(d =4,8,12,16,20,24 \) , coefficients c_k for the powerseries seem to converge to limit-values when d is increased:

Code:

.         4x4              8x8                   12x12                16x16              20x20             24x24       |    Kneser(*)    
------------------------------------------------------------------------------------------------------------------------------------------
   0.458786484402      0.457506956745      0.457297994170      0.457244494586      0.457226376487      0.457219121381  |  0.457213238216
    1.01416292699       1.00585184963       1.00570266416       1.00576279604       1.00580782971       1.00583514094  |  1.00588535950
   0.354910001792      0.372463884665      0.374338685963      0.374762250136      0.374890508012      0.374935814278  |  0.374953659250
  0.0423749372335     0.0485034871162     0.0483181418142     0.0480895085885     0.0479650214772     0.0478979652601  |  0.0477946354397
                .  -0.000531678816504   -0.00167964829897   -0.00193989480094   -0.00201097333281   -0.00203167751111  |  -0.00202366092283
                .   0.000528586184651   0.000787045655542   0.000917125824152   0.000980591763273    0.00101283363819  |  0.00105579202994
                .   -3.94783452855E-5    8.17659708660E-5   0.000124080100384   0.000130492347057   0.000128471429488  |  0.000111287079756
                .   -4.44632351059E-6  -0.000137772165702  -0.000205048904088  -0.000233715355744  -0.000246849844561  |  -0.000259506482216
                .                   .    4.27004369289E-5    5.64191495551E-5    6.41783344678E-5    6.96058835299E-5  |  8.20289987826E-5
                .                   .   -2.97008358857E-6    1.57094892640E-5    2.55689613195E-5    2.99812302337E-5  |  3.25772461877E-5
                .                   .   -1.23612708539E-6   -1.52680772749E-5   -2.18517047449E-5   -2.54670434954E-5  |  -3.19939947128E-5
                .                   .    2.28200841656E-7    3.19014711939E-6    1.42388118905E-6    3.52461380591E-7  |  3.94504668958E-7
                .                   .                   .    7.54744807314E-7    4.18053003922E-6    6.04642021637E-6  |  9.15787915121E-6
                .                   .                   .   -5.49383830555E-7   -1.68109328381E-6   -1.70179488684E-6  |  -2.15328520187E-6
                .                   .                   .    1.16400876115E-7   -2.15881955255E-7   -1.03326553912E-6  |  -2.50691103240E-6
                .                   .                   .   -9.14068746483E-9    3.92005894211E-7    7.18657474565E-7  |  1.11459855070E-6
                .                   .                   .                   .   -1.46087335360E-7    2.28851750529E-8  |  7.31756775619E-7
                .                   .                   .                   .    2.60282057394E-8   -1.80477053045E-7  
                .                   .                   .                   .   -1.96351794647E-9    7.64568418432E-8  
                .                   .                   .                   .   9.26756456931E-12   -7.14784297921E-9  
                .                   .                   .                   .                   .   -4.93066217462E-9  
                .                   .                   .                   .                   .    2.09545681167E-9  
                .                   .                   .                   .                   .  -3.50584306552E-10  
                .                   .                   .                   .                   .   2.31284640845E-11

"Kneser(*)" here means: fatou.gp 1708 computed with internal precision 60 digits using "derivnum" by Pari/GP :

Code:

.    for(d=0,16,print(d," ",derivnum(x=0,SLtet(x,0.5),d)/d!))

Note: the "derivnum()"-procedure becomes inaccurate for higher derivatives, so only 16 are documented here!

[Catullus]

Conjecture:

The standard extension of exponentiation is the only extension of exponentiation such that a to the power of x is totally monotonic for all real a between one and zero non inclusive, and for all real x.

The conjecture is proven to be true.

[JmsNxn]

Conjecture:

The standard extension of exponentiation is the only extension of exponentiation, such that a to the power of x is totally monotonic, For all real a between one and zero non inclusive, and for all real a.

OHHHHHH CATULLUS!!!

This is a good one.

I'd suggest an old thread by me on MathOverflow, but it's lost in all the overflow. To summarize the result:

Every totally monotonic function \(f\) can be described, using a unique measure \(\mu\), such:

\[
f(x) = \int_0^\infty e^{-xt}\,d\mu\\
\]

So... since \(f(x) = e^{-x}\) the exponential satisfies this formula for \(x>0\) when \(\mu(x) = \delta\) for the Dirac \(\delta\)-function, and we have a mole at this point. We've uniquely determined the exponential.

Essentially this means once you assume totally monotonic, you get a Laplace transform expression, then just use the Laplace transform to solve it. Since this process is reversible, it's unique.

Your conjecture is true.

[Leo.W]

Conjecture:
For any locally-analytic complex function \(f:\mathbb{C}\to\mathbb{C}\), analytically continued whether multivalued or singlevalued, there always lies a fixed point \(L\), which satisfies either:
\(f'(z)\) 's value at \(L\) exists and not in the form \(q\in\mathbb{Q}\cap(0,1),f'(L)=e^{2\pi qi}\)
or: there's a conjugacy g of f by h: \(g=h^{-1}fh\) which maps \(L\to{L_0}\), where \(f'(L)\) doesn't exists but some directional derivative \(\lambda=f'(L_0)\) exists, or the limit \(\lambda=\lim_{a\to{L}}{f'(a)}\) exists, and \(\lambda\) not in the form \(q\in\mathbb{Q}\cap(0,1),\lambda=e^{2\pi qi}\)
This may not be true.
Sub-conjecture: Only take polynomial functions instead of any such f into consideration, the result is true.
This is proved correct for linear functions and quadratic functions.

Conjecture:
Denote P as the successor operator. For any operator T, which can act on a number or another operator, despite multivalued-ness, there always exist an operator Q or M which fits QT=PQ or TM=MP where multiplication refers to compositions between operator.
A specific example is hyperoperation.
An equivalent expression is for any operator T, there always lies exp(T) or log(T).

[tommy1729]

Conjecture:
For any locally-analytic complex function \(f:\mathbb{C}\to\mathbb{C}\), analytically continued whether multivalued or singlevalued, there always lies a fixed point \(L\), which satisfies either:
\(f'(z)\) 's value at \(L\) exists and not in the form \(q\in\mathbb{Q}\cap(0,1),f'(L)=e^{2\pi qi}\)
or: there's a conjugacy g of f by h: \(g=h^{-1}fh\) which maps \(L\to{L_0}\), where \(f'(L)\) doesn't exists but some directional derivative \(\lambda=f'(L_0)\) exists, or the limit \(\lambda=\lim_{a\to{L}}{f'(a)}\) exists, and \(\lambda\) not in the form \(q\in\mathbb{Q}\cap(0,1),\lambda=e^{2\pi qi}\)
This may not be true.
Sub-conjecture: Only take polynomial functions instead of any such f into consideration, the result is true.
This is proved correct for linear functions and quadratic functions.

Conjecture:
Denote P as the successor operator. For any operator T, which can act on a number or another operator, despite multivalued-ness, there always exist an operator Q or M which fits QT=PQ or TM=MP where multiplication refers to compositions between operator.
A specific example is hyperoperation.
An equivalent expression is for any operator T, there always lies exp(T) or log(T).

is there a related thread ?

why is this important ?

ps : yay the open problems section is on the top of the page again

regards

tommy1729

[Leo.W]

is there a related thread ?
...
tommy1729

The first one follows 
https://math.eretrandre.org/tetrationfor...p?tid=1318
on page 4 & 5 me and james discussed about the generalized iterations that has such fixed point and multiplier which is "non-constructable", for example \(f(z)=-z+z^2\), it's almost impossible to generate \(f^{\frac{1}{2}}(z)\) from the fixed point z=0 because its multiplier is -1 in the form \(\lambda=e^{2\pi i\frac{1}{2}}\), but can still generate the half iterate at z=2. It matters because if it's true then we can always generate any iterations of any such functions as we want, otherwise it'll leave us with tons of heavy arduous computative problems. You can try to build such function \(f^{\frac{1}{2}}(z)=\pm iz+O(z^2)\) and may won't succeed.

The second is a generalized idea, for example hyperoperations.
We define S[f](z) or S as an operator whose results are the superfunction families of f, then hyperoperations can be expressed as a branch cut of \(S^n[f](z)\), precisely \(a[n]b\in{S^n[a[0]b]}\) wrt b.
And also, if we define a S operator for S operator, or let's call it T: \(T[S[f](z)](z+1)=S[T[S[f](z)](z)](z)\), then we have hyperhyperoperators (idk the exact name lol but such concept exists widely especially in googleplex-ology(?))
And more, for example any functional equation can be seen as \(H[f(z)](z)=0\), H is an operator, thus if we find exp(H) and log(H), we get immediately the solution by \(f(z)=H^{-1}[0](z)=e^{-\log(H)}[0](z)\)