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+--- 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 \( \mathcal{G}_n+\theta_k \)
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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 \( \mathcal{G}_3+\theta_k \)in a recursive way we define \( \mathcal{G}_{n+1}+\theta_k \)
II-\( \mathcal{G}_3+\theta_k \) is closed composition and linear integration
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 \( x \gt 1 \)