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 open problems survey bo198214 Administrator Posts: 1,389 Threads: 90 Joined: Aug 2007 06/29/2008, 12:36 PM (This post was last modified: 06/29/2008, 01:32 PM by bo198214.) Conjecture Let $C_N$ be the Carleman matrix of $(x+s)^p-s$ (truncated to N rows and columns), $s>0$, $p>0$ real. Then the set of eigenvalues of $C_N$ converges to the set $\{p^k:k\ge 0\}$ for $N\to\infty$ in the sense that there exist an enumeration $v_{N,k}$ of the Eigenvalues of $C_N$ such that $\lim_{N\to\infty} v_{N,k} = p^k$ for each $k$. Discussion This is about the function $f(x)=x^p$ shifted by $s$. The fixed point 0 is a singularity for $f$ (for non-natural $p$), so $f$ has to be developed at the different point $s$. In the particular case $s=1$ we have the fixed point at 0 and the first derivative is $p$. So the Carleman matrix is triangular and we can solve it exactly, getting $f^{\circ t}(x)=x^{p^t}$. The conjecture is again about the independence of the matrix function method with respect to the development point. $f$ can even be developed at the fixed point 0 in the particular case $p\in\mathbb{N}$. However in this case $f'(0)=0$ except $p=1$ and regular iteration can not be applied, which makes sense as $x^{p^t}$ can for most t not be developed at 0. « Next Oldest | Next Newest »

 Messages In This Thread open problems survey - by bo198214 - 05/17/2008, 10:03 AM eigenvalues of Carleman matrix for b^x, TPID 1 - by bo198214 - 05/17/2008, 10:23 AM Exponential Factorial, TPID 2 - by andydude - 05/26/2008, 03:24 PM eigenvalues of Carleman matrix for (x+s)^p-s, TPID 3 - by bo198214 - 06/29/2008, 12:36 PM Existence of bounded b^z TPID 4 - by bo198214 - 10/08/2008, 04:22 PM sqrt(2) tetrational is completely discontinuous TPID 5 - by bo198214 - 05/01/2009, 09:20 AM Limit of self-super-roots is e^1/e. TPID 6 - by andydude - 10/07/2009, 12:03 AM A conjecture on bounds. TPID 7 - by andydude - 10/23/2009, 05:27 AM Elementary superfunction of polynomial without real fixed points TPID 8 - by bo198214 - 04/25/2010, 10:53 AM Logarithm reciprocal TPID 9 - by bo198214 - 07/20/2010, 05:50 AM Convergence of Eulers and Etas. TPID 10 - by dantheman163 - 10/31/2010, 07:13 PM RE: open problems survey - by nuninho1980 - 10/31/2010, 09:50 PM Tommy's conjecture about Eulers and Etas. TPID 11. - by tommy1729 - 12/01/2010, 03:56 PM convergence of self-tetra-root polynomial interpolation. TPID 12 - by bo198214 - 05/31/2011, 04:54 PM convergence of self-root polynomial interpolation. TPID 13 - by bo198214 - 05/31/2011, 07:02 PM Tommy's conjecture about andrew slog method. TPID 14 - by tommy1729 - 06/01/2011, 06:23 PM Tommy's conjecture TPID 16 - by tommy1729 - 06/07/2014, 10:44 PM Error terms for fake function theory TPID 17 - by tommy1729 - 03/28/2015, 10:48 PM The third super-root TPID 18 - by andydude - 12/25/2015, 06:16 AM

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