AENC/resampling_chain Changeset - 0cd16f2a5013 · Centrum Wiskunde & Informatica (CWI)

Changeset - 0cd16f2a5013

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Tom Bannink - 8 years ago 2017-09-10 16:10:16
tombannink@gmail.com

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@@ @@ -173,15 +173,17 @@ @@
 	\newpage
 	\noindent Based on our calculations presented in Table~\ref{tab:coeffs} and Figure~\ref{fig:coeffs_conv_radius} we make the following conjectures:
 	\begin{enumerate}[label=(\roman*)]
 		\item $\forall k\in\mathbb{N}, \forall n\geq 3 : a^{(n)}_k\geq 0$	\label{it:pos}
         (A simpler version: $\forall k>0: a_k^{(3)}=(k+1)(k+2)/6$)
 		\item $\forall k\in\mathbb{N}, \forall n>m\geq 3 : a^{(n)}_k\geq a^{(m)}_k$ \label{it:geq}
-		\item $\forall k\in\mathbb{N}, \forall n,m\geq \max(k,3) : a^{(n)}_k=a^{(m)}_k$ \label{it:const}
+		\item $\forall k\in\mathbb{N}, \forall n,m > \max(k,3) : a^{(n)}_k=a^{(m)}_k$ \label{it:const}
   		\item $\exists p_c=\lim\limits_{k\rightarrow\infty}1\left/\sqrt[k]{a_{k}^{(k+1)}}\right.$ \label{it:lim}
 	\end{enumerate}
 	\colorbox{red}{\ref{it:pos}-\ref{it:geq} is false since $a_{1114}^{(10)}<0$ -- needs to be double checked!}
 	We also conjecture that $p_c\approx0.61$, see Figure~\ref{fig:coeffs_conv_radius}.
 	\begin{figure}[!htb]\centering
 	\includegraphics[width=0.5\textwidth]{coeffs_conv_radius.pdf}
 	%\includegraphics[width=0.5\textwidth]{log_coeffs.pdf}
 	\caption{$1\left/\sqrt[k]{a_{k}^{(k+1)}}\right.$} %$\frac{1}{\sqrt[k]{a_k^{(k+1)}}}$

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