Grzegorczyk hierarchy vs Iterated differential equations? - Printable Version +- Tetration Forum (https://math.eretrandre.org/tetrationforum) +-- Forum: Tetration and Related Topics (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=1) +--- Forum: Mathematical and General Discussion (https://math.eretrandre.org/tetrationforum/forumdisplay.php?fid=3) +--- Thread: Grzegorczyk hierarchy vs Iterated differential equations? (/showthread.php?tid=945) Grzegorczyk hierarchy vs Iterated differential equations? - MphLee - 01/03/2015 Inspired by JmsNxn's thread (http://math.eretrandre.org/tetrationforum/showthread.php?tid=818&pid=7139#pid7139) about the continuum sum I repost this obsevation about the link between the fractional calculus and the Hyperoperations. I guess that there can be interesting links... and probably is not the wrong way to approach the problem. I just found some results about something similar. M. Campagnolo, C. Moore -Upper and Lower Bounds on Continuous-Time Computation In this text I found a relation betwen a hierarchy of real valued function and the Grzegorczyk hierarchy. The interesting relations are betwen a hierarchy called $\mathcal{G}_n+\theta_k$ and the hierarchy $\mathcal{E}_n$: Quote:1-Any function in $\mathcal{G}_n+\theta_k$ is computable in $\mathcal{E}_n$ 2-If $f\in \mathcal{G}_n+\theta_k$ then $f$ is the extension to the reals of some $f^{*}:\mathbb{N}\rightarrow\mathbb{N}$ then $f^{*}\in \mathcal{E}_n$ 3-the converse holds: if $f$ is a function on the naturals of rank $n$ it has an extension in $\mathcal{G}_n+\theta_k$ ------------------- The interesting thing is that the various levels of $\mathcal{G}_n+\theta_k$ are defined via iterated solution of a special kind of functional equation...and that maybe can be linked with your knowledge in this field... Definition-$\mathcal{G}_3+\theta_k$ is defined as follow Quote:I-the constants $0$,$1$,$-1$ and $\pi$, the projection functions, $\theta_k$ are in $\mathcal{G}_3+\theta_k$ II-$\mathcal{G}_3+\theta_k$ is closed composition and linear integration in a recursive way we define $\mathcal{G}_{n+1}+\theta_k$ Quote:III- $\mathcal{G}_{n+1}+\theta_k$ contains the functions in $\mathcal{G}_{n}+\theta_k$ IV- $\mathcal{G}_{n+1}+\theta_k$ in we can find all the solutions to the equation (2) in this text ( http://languagelog.ldc.upenn.edu/myl/DK/CampagnoloMoore.pdf ) applied to the functions in $\mathcal{G}_{n}+\theta_k$ V-$\mathcal{G}_{n+1}+\theta_k$ is closed under composition and linear integration $\theta_k(x):=x^k\theta(x)$ and $\theta(x):=0$ if $x \le 0$ $\theta(x):=1$ if $x \gt 1$