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 open problems survey bo198214 Administrator Posts: 1,386 Threads: 90 Joined: Aug 2007 05/17/2008, 10:23 AM (This post was last modified: 06/29/2008, 10:57 AM by bo198214.) Conjecture Let $B_n$ be the Bell/Carleman matrix of $f(x)=b^x$, $1, truncated to $n$ rows and columns. There is an enumeration $(v_{n,k})_{1\le k\le n}$ of the eigenvalues of $B_n$ such that $\lim_{n\to\infty} v_{n,k} = \ln(a)^k$ where $a$ is the lower real fixed point of $b^x$. Explanation This is a key question for deciding whether the diagonalization/matrix power method is independent of its development point. A more general question would be under which circumstances the eigenvalues of the Carleman matrix converge to the powers of an attracting fixed point. It seems they dont do for $f(x)=x^2+x-1/16$ which has an attracting fixed point at $-1/4$ and a repelling fixed point at $1/4$. Also the behaviour of the eigenvalues for $e^{-e} seems unclear. However there is only one real repelling fixed point in this case. Notes $b=a^{1/a}$ $\ln(a)$ is the derivative of $b^x$ at $a$: $\frac{\partial b^x}{\partial x}|_{x=a}=\ln(b)b^x|_{x=a}=\ln(b)a=\ln(b^a)=\ln(a)$ « 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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